FROM SINGULAR CHAINS TO ALEXANDER DUALITY

JESPER M. MØLLER

Abstract. Some comments about singular homology and cohomology based on Hatcher’s book [6]

Contents

1. Singular chains 2 2. Singular chains 3 3. The long exact sequence of a pair 3 4. Homotopy invariance 4 5. Excision 6 5.2. Linear chains 6 5.4. Cones on linear chains 6 5.6. Subdivision of linear chains 7 5.7. Iterated subdivision of linear chains 8 5.14. Subdivision for general spaces 9 5.16. A general form of excision 9 5.20. The long exact sequence of a quotient 11 6. Δ-complexes [3, V.§7] 11 6.5. The simplicial chain complex of a Δ-set 12 7. The degree of a self-map of the sphere 13 8. Cellular homology of CW-complexes 14 8.1. Homology of an n-cellular extension 14 8.3. The cellular chain complex 15 8.11. Compact surfaces 16 8.14. Real projective space 18 8.15. Complex projective space 20 8.16. Lens spaces 20 9. The Eilenberg–MacLane Complexes K(G, 1) [2, Chp II] [6, Chp 1.B] 21 10. Cohomology 22 10.1. Evaluation 22 10.1. Cup and cap products 22 10.5. Relative cup and cap products 23 10.6. UCT 24 10.10. The simplicial cochain complex of a Δ-set 24 10.18. Cellular cohomology of CW-complexes 29 11. Orientation of manifolds 29 11.1. Local homology groups 29 11.4. The orientation bundle 30 11.7. Local and global orientations 30 11.13. The oriented cover of a nonorientable manifold 32 12. Poincaré duality for compact manifolds 33 12.4. Connection with cup product 34 12.5. Applications: Borsuk–Ulam, division algebras, and the Hopf map 34 13. Colimits of modules 35 13.10. Cohomology with compact support 36 14. Poincaré duality for noncompact oriented manifolds 37 15. Alexander duality 38 15.8. Linking number 39 15.9. Invariance of Domain 40 References 42

Date: 2nd November 2004.

1. Singular chains

Let e_{0} = (1, 0, …, 0), …, e_{n} = (0, …, 0, 1) be the n + 1 standard basis vectors in R^{n+1}.

Definition 1.1. The standard n-simplex

Δ^{n} = conv{e_{0}, …, e_{n}} = {t_{0}e_{0} + ⋯ t_{n}e_{n} | ∑_{i_{0}}^{n} t_{i} = 1, t_{i} ≥ 0} ⊂ R^{n+1}

is the convex hull of the n + 1 standard basis vectors in R^{n+1}.

We can define linear maps out of simplices into vector spaces just by telling where the vertices go. If v_{0}, …, v_{n} are n + 1 points in some vector space V, define

[v_{0}, … v_{n}]: Δ^{n} → V , t_{0}e_{0} + ⋯ t_{n}e_{n} ↦ t_{0}v_{0} + ⋯ t_{n}v_{n}

In particular the ith face map

d^{i} = [e_{0}, …, ê_{i}, … e_{n}]: Δ^{n−1} → Δ^{n}, 0 ≤ i ≤ n,

embeds the (n − 1)-simplex in the n-simplex as the face opposite the ith vertex. The hat over the e_{i} means that e_{i} has been deleted form the list, so this linear map is given by

e_{0} → e_{0}, …, e_{i−1} → e_{i−1}, e_{i} → e_{i+1}, …, e_{n−1} → e_{n}

and it has the property that e_{i} is not in the image.

Example 1.2. There are 2 face maps

d^{0} = [e_{1}], d^{1} = [e_{0}]: Δ^{0} → Δ^{1},

3 face maps

d^{0} = [e_{1}, e_{2}], d^{1} = [e_{0}, e_{2}], d^{2} = [e_{0}, e_{1}]: Δ^{1} → Δ^{2}

and 4 face maps

d^{0} = [e_{1}, e_{2}, e_{3}], d^{1} = [e_{0}, e_{2}, e_{3}], d^{2} = [e_{0}, e_{1}, e_{3}], d^{3} = [e_{0}, e_{1}, e_{2}]: Δ^{2} → Δ^{3}

Draw pictures of these maps! Note that [e_{0}, e_{2}, e_{3}] ∘ [e_{0}, e_{2}] = [e_{0}, e_{3}] and [e_{0}, e_{2}, e_{3}] ∘ [e_{1}, e_{2}] = [e_{2}, e_{3}].

We need to know how these imbeddings behave under composition.

Lemma 1.3. For 0 ≤ j ≤ n and 0 ≤ i ≤ n + 1, the composition

\xymatrix{Δ^{n−1} \ar[r]^{d^{j}=[e_{0},...,ê_{j},...,e_{n}]} & Δ^{n} \ar[r]^{d^{i}=[e_{0},...,ê_{i},...,e_{n+1}]} & Δ^{n+1}}

is given by

d^{i}d^{j} = [e_{0}, …, ê_{i}, …, e_{n+1}]∘[e_{0}, …, ê_{j}, …, e_{n}] = \begin{cases}[e_{0}, …, ê_{j}, …, ê_{i}, …, e_{n+1}] & n + 1 ≥ i > j ≥ 0 \\ [e_{0}, …, ê_{i}, …, ê_{j+1}, …, e_{n+1}] & 0 ≤ i ≤ j ≤ n \end{cases}

and therefore

(1.4)  d^{i}d^{j−1} = [e_{0}, …, ê_{i}, …, e_{n+1}] ∘ [e_{0}, …, ê_{j}, …, e_{n}] = d^{j}d^{i},  i < j

Proof. Keep track of the n points e_{0}, …, e_{n−1} ∈ Δ^{n−1}. □

2. Singular chains

Let X be a topological space. An n-simplex in X is a (continuous) map σ: Δ^{n} → X of the n-simplex into X.

Definition 2.1. The nth chain group of X is the free abelian group C_{n}(X) generated by all simplices σ: Δ^{n} → X in X. The elements of C_{n}(X) are finite linear combinations ∑ n_{σ}σ of simplices in X. (By convention C_{n}(X) = 0 for n < 0.)

The n-th boundary map δ_{n}: C_{n}(X) → C_{n−1}(X) is the linear map with value

∂_{n}(σ) = ∑_{i=0}^{n}(−1)^{i}σ ∘ [e_{0}, …, ê_{i}, …, e_{n}] = ∑_{i=0}^{n}(−1)^{i}σd^{i}

on the n-simplex σ: Δ^{n} → X.

Proposition 2.2. The composition

C_{n+1} ⟶^{∂_{n+1}} C_{n}(X) ⟶^{∂_{n}} C_{n−1}(X)

of two boundary maps is trivial.

Proof. Let σ: Δ^{n+1} → X be any (n + 1)-simplex in X. We compute:

∂∂σ = ∂ (∑_{i=0}^{n+1}(−1)^{i}σd^{i}) = ∑_{i=0}^{n+1} ∑_{j=0}^{n}(−1)^{i+j}σd^{i}d^{j} = ∑_{i>j}(−1)^{i+j}σd^{i}d^{j} + ∑_{i≤j}(−1)^{i+j}σd^{i}d^{j}

= ∑_{i>j}(−1)^{i+j}σd^{i}d^{j} − ∑_{i<j}(−1)^{i+j}σd^{i}d^{j−1} = ∑_{i>j}(−1)^{i+j}σd^{i}d^{j} − ∑_{i<j}(−1)^{i+j}σd^{j}d^{i} = 0

since the two sums are identical as we see by switching the labels i and j in the second sum. □

The gadget (C_{∗}(X), ∂_{∗}), which is short for

⋯ → C_{n+1}(X) ⟶^{∂_{n+1}} C_{n}(X) ⟶^{∂_{n}} C_{n−1}(X) → ⋯ → C_{1}(X) ⟶^{∂_{1}} C_{0}(X) ⟶^{∂_{0}} 0

where ∂∂ = 0, is called the singular chain complex of X. Note that C_{∗}(−) is a functor from topological spaces to chain complexes.

Since ∂∂ = 0, the kernel, ker(∂_{n}), of the outgoing boundary operator at C_{n}(X) contains the image, im(∂_{n+1}), of the incoming boundary operator. Homology measures the difference between these two subgroups of C_{n}(X).

Definition 2.3. The nth homology group of X is the quotient

H_{n}(X) = ker(∂_{n})/ im(∂_{n+1})

of the n-cycles Z_{n}(X) = ker(∂_{n}) by the n-boundaries B_{n}(X) = im(∂_{n+1}). We let [z] ∈ H_{n}(X) be the homology class represented by the n-cycle z ∈ Z_{n}(X).

Any map f: X → Y induces a homomorphism H_{n}(f): H_{n}(X) → H_{n}(Y) given by H_{n}(f)[z] = [C_{n}(f)z] for any n-cycle z ∈ Z_{n}(X). In fact, H_{n}(−) is a (composite) functor from topological spaces to abelian groups.

3. The long exact sequence of a pair

Let (X, A) be a pair of spaces consisting of a topological space X with a subspace A ⊂ X. Define C_{n}(X, A) to be the quotient of C_{n}(X) by its subgroup C_{n}(A). Then we have the situation

\xymatrix{0 \ar[r] & C_{n}(A) \ar[r]^{i} \ar[d]_{∂_{n}} & C_{n}(X) \ar[r]^{j} \ar[d]_{∂_{n}} & C_{n}(X, A) \ar[r] \ar@{-->}[d]^{_{\overline{∂}_{n}}} & 0 \\ 0 \ar[r] & C_{n−1}(A) \ar[r]^{i} & C_{n−1}(X) \ar[r]^{j} & C_{n−1}(X, A) \ar[r] & 0}

where the rows are short exact sequences. Since the morphism j ∘ ∂_{n}: C_{n}(X) → C_{n−1}(X, A) vanishes on the subgroup C_{n}(A) of C_{n}(X) there will be a unique morphism \overline{∂}_{n}: C_{n}(X, A) → C_{n−1}(X, A) such that the diagram commutes. In fact (C_{∗}(X, A), \overline{∂}_{∗}) is a chain complex for \overline{∂}_{n−1} ∘ \overline{∂}_{n} = 0 since it is induced from ∂_{n−1} ∘ ∂_{n} = 0. Define the relative homology

H_{n}(X, A) = \frac{Z_{n}(X, A)}{B_{n}(X, A)} = \frac{∂^{−1}C_{n−1}(A)/C_{n}(A)}{B_{n}(X) + C_{n}(A)/C_{n}(A)} = \frac{∂^{−1}C_{n−1}(A)}{B_{n}(X) ∩ C_{n}(A)}

to be the degree n homology of this relative chain complex. The above commutative diagram can now be enlarged to a short exact sequence of chain complexes. The Fundamental Theorem of Homological Algebra tells us that there is an associated long exact sequence

\xymatrix{⋯ \ar[r] & H_{n}(A) \ar[r]^{i} & H_{n}(X) \ar[r]^{j} & H_{n}(X, A) \ar[r]^{∂} & H_{n−1}(A) \ar[r] & ⋯}

of homology groups. The connecting homomorphism

Z_{n+1}(X, A)/B_{n+1}(X, A) = H_{n+1}(X, A) ⟶^{∂} H_{n}(A) = Z_{n}(A)/B_{n}(A)

is induced by i^{−1}∂: ∂^{−1}C_{n}(A) → Z_{n}(A).

4. Homotopy invariance

Homology does not distinguish between homotopic maps.

Theorem 4.1. Homotopic maps induce identical maps on homology: If f ≃ g: X → Y, then H_{n}(f) = H_{n}(g): H_{n}(X) → H_{n}(Y).

The prism on a simplex Δ^{n} is the product space Δ^{n} ×I ⊂ R^{n+1} ×R of Δ^{n} and the unit interval I. The vertices of the prism are

v_{0} = (e_{0}, 0), …, v_{n} = (e_{n}, 0), w_{0} = (e_{0}, 1), …, w_{n} = (e_{n}, 1)

with the v_{i} at level 0 and the w_{i} at level 1. Define

[v_{0} ⋯ v_{i}w_{i} ⋯ w_{n}]: Δ^{n+1} → Δ^{n} × I, 0 ≤ i ≤ n,

to be the linear map that takes the n + 2 ordered vertices of Δ^{n+1} to the specified ordered n + 2 points of the prism. This gives a Δ-complex structure on Δ^{n} × I.

Lemma 4.2. For 0 ≤ j ≤ n + 1 and 0 ≤ i ≤ n, the composition

\xymatrix{Δ^{n} \ar[r]^{[e_{0}···ê_{j}···e_{n+1}]} & Δ^{n+1} \ar[r]^{[v_{0}···v_{i}w_{i}···w_{n}]} & Δ^{n} × I}

is given by

[v_{0} ⋯ v_{i}w_{i} ⋯ w_{n}] ∘ [e_{0} ⋯ ê_{j} ⋯ e_{n+1}] = \begin{cases}[v_{0} ⋯ v̂_{j} ⋯ v_{i}w_{i} ⋯ w_{n}] & j ≤ i \\ [v_{0} ⋯ v_{i}w_{i} ⋯ v̂_{j−1} ⋯ w_{n}] & j > i \end{cases}

For 0 ≤ k ≤ n − 1 and 0 ≤ h ≤ n the composition

\xymatrix{Δ^{n} \ar[r]^{[v_{0}···v_{k}w_{k}···w_{n−1}]} & Δ^{n−1} × I \ar[r]^{[e_{0}···ê_{h}···e_{n}]×id} & Δ^{n} × I}

is given by

([e_{0} ⋯ ê_{h} ⋯ e_{n}] × id) ∘ [v_{0} ⋯ v_{k}w_{k} ⋯ w_{n−1}] = \begin{cases}[v_{0} ⋯ v̂_{h} ⋯ v_{k+1}w_{k+1} ⋯ w_{n}] & h ≤ k \\ [v_{0} ⋯ v_{k}w_{k} ⋯ ŵ_{h} ⋯ w_{n}] & h > k \end{cases}

Proof. Keep track of the n + 1 vertices e_{0}, …, e_{n} ∈ Δ^{n}. □

Suppose now that F: X × I → Y is a homotopy between the maps f(x) = F (x, 0) and g(x) = F (x, 1). The prism operator associated to the homotopy F is the linear map P: C_{n}(X) → C_{n+1}(Y) whose value on the n-simplex σ: Δ^{n} → X

P (σ) = ∑_{i=0}^{n}(−1)^{i}F ∘ (σ × id) ∘ [v_{0} ⋯ v_{i}w_{i} ⋯ w_{n}]

is the alternating sum of the simplices

Δ^{n+1} ⟶^{[v_{0}···v_{i}w_{i}···w_{n}]} Δ^{n} × I ⟶^{σ×id} X × I ⟶^{F} Y

The boundary of the prism is the top, the bottom, and the prism on the boundary of the simplex:

FROM SINGULAR CHAINS TO ALEXANDER DUALITY 5

Lemma 4.3. ∂P (σ) = g ∘ σ − f ∘ σ − P (∂σ) Proof. Let σ: Δ^{n} → X be an n-simplex in X. Then

P σ = ∑_{i=0}^{n}(−1)^{i}F ∘ (σ × id) ∘ [v_{0} ⋯ v_{i}w_{i} ⋯ w_{n}] ∈ C_{n+1}(Y)

and

∂P σ = ∑_{i=0}^{n} ∑_{j=0}^{n+1}(−1)^{i+j}F ∘ (σ × id) ∘ [v_{0} ⋯ v_{i}w_{i} ⋯ w_{n}] ∘ [e_{0} ⋯ ê_{j} ⋯ e_{n+1}]

=^{4.2} ∑_{0≤j≤i≤n}(−1)^{i+j}F ∘ (σ × id)[v_{0} ⋯ v̂_{j} ⋯ v_{i}w_{i} ⋯ w_{n}] + ∑_{0≤i<j≤n+1}(−1)^{i+j}F ∘ (σ × id)[v_{0} ⋯ v_{i}w_{i}{ \atop w_{j−1}} ⋯ w_{n}]

= ∑_{0≤j≤i≤n}(−1)^{i+j}F ∘ (σ × id)[v_{0} ⋯ v̂_{j} ⋯ v_{i}w_{i} ⋯ w_{n}] − ∑_{0≤i≤j≤n}(−1)^{i+j}F ∘ (σ × id)[v_{0} ⋯ v_{i}w_{i}{ \atop w_{j−1}} ⋯ w_{n}]

The terms with i = j are

∑_{i=0}^{n} F ∘ (σ × id)[v_{0} ⋯ v_{i−1}v̂_{i}w_{i} ⋯ w_{n}] − ∑_{i=0}^{n} F ∘ (σ × id)[v_{0} ⋯ v_{i}ŵ_{i} ⋯ w_{n}]

= F ∘ (σ × id) ([w_{0} ⋯ w_{n}] + [v_{0}w_{1} ⋯ w_{n}] + ⋯ + [v_{0} ⋯ v_{n−1}w_{n}] − [v_{0}w_{1} ⋯ w_{n}] − ⋯ − [v_{0} ⋯ v_{n−1}w_{n}] − [v_{0} ⋯ v_{n}]) = F ∘ (σ × id)[w_{0} ⋯ w_{n}] − F ∘ (σ × id)[v_{0} ⋯ v_{n}] = g ∘ σ − f ∘ σ What’s left is

∑_{0≤j<i≤n}(−1)^{i+j}F (σ×id)[v_{0} ⋯ v̂_{j} ⋯ v_{i}w_{i} ⋯ w_{n}]+ ∑_{0≤i<j≤n}(−1)^{i+j+1}F (σ×id)[v_{0} ⋯ v_{i}w_{i} ⋯ ŵ_{j} ⋯ w_{n}] = (∗)

On the other hand

P (∂σ) = P (∑_{h=0}^{n}(−1)^{h}σ[e_{0} ⋯ ê_{h} ⋯ e_{n}])

= ∑_{h=0}^{n} ∑_{k=0}^{n−1}(−1)^{h+k}F (σ[e_{0} ⋯ ê_{h} ⋯ e_{n}] × id)[v_{0} ⋯ v_{k}w_{k} ⋯ w_{n−1}]

= ∑_{h=0}^{n} ∑_{k=0}^{n−1}(−1)^{h+k}F (σ × id)([e_{0} ⋯ ê_{h} ⋯ e_{n}] × id)[v_{0} ⋯ v_{k}w_{k} ⋯ w_{n−1}]

= ∑_{0≤h≤k<n}(−1)^{h+k}F (σ × id)[v_{0} ⋯ v̂_{h} ⋯ v_{k+1}w_{k+1} ⋯ w_{n}] + ∑_{0≤k<h≤n}(−1)^{h+k}F (σ × id)[v_{0} ⋯ v_{k}w_{k} ⋯ ŵ_{h} ⋯ w_{n}]

=^{k′=k+1} ∑_{0≤h<k≤n}(−1)^{h+k+1}F (σ × id)[v_{0} ⋯ v̂_{h} ⋯ v_{k}w_{k} ⋯ w_{n}] + ∑_{0≤k<h≤n}(−1)^{h+k}F (σ × id)[v_{0} ⋯ v_{k}w_{k} ⋯ ŵ_{h} ⋯ w_{n}]

which is −(∗). □ Proof of Theorem 4.1. Let [z] ∈ H_{n}(X) be the homology class represented by the n-cycle z ∈ C_{n}(X). Then

H_{n}(f)[z] = [C_{n}(f)z] =^{4.3} [−∂P z + C_{n}(g)z − P ∂z] = [C_{n}(g)z] = H_{n}(g)[z]

for ∂P z is a boundary and ∂z = 0 because z is a cycle. The homomorphisms involved make this

\xymatrix{C_{n−1}(X) \ar[dr]_{P} & C_{n}(X) \ar[dr]^{P} \ar[l]_{∂} \ar[d]^{f_{∗}} \ar[d]_{g_{∗}} \\  & C_{n}(Y) & C_{n+1}(Y) \ar[l]_{∂}}

diagram. □

The prism operator P, that satisfies the formula ∂P + P ∂ = g_{∗} − f_{∗} of Lemma 4.3, is said to be a chain homotopy between the chain maps f_{∗} and g_{∗}. We have showed that homotopic maps induce chain homotopic chain maps. Any two chain homotopic chain maps will induce identical maps on homology.

5. Excision

The excision theorem comes in two equivalent versions.

Theorem 5.1. (Excision) Let X be a topological space.

(1) Suppose that U ⊂ A ⊂ X. If cl(U) ⊂ int(A), then the inclusion map (X − U, A − U) → (X, A) induces an isomorphism H_{n}(X − U, A − U) ⟶^{≅} H_{n}(X, A) for all n ≥ 0.

(2) Suppose that X = A ∪ B. If X = int(A) ∪ int(B), then the inclusion map (B, A ∩ B) → (X, A) induces an isomorphism H_{n}(B, A ∩ B) ⟶^{≅} H_{n}(X, A) for all n ≥ 0.

The two versions are equivalent. The translation between version (1) to version (2) is obtained by letting X −U = B. Then A∩B = A∩(X −U) = A−U so that the pair (X −U, A−U) = (B, A∩B) and

cl(U) ⊂ int(A) ⟺ cl(X − B) ⊂ int(A) ⟺ X − int(B) ⊂ int(A) ⟺ X = int(A) ∪ int(B)

so that the two conditions are equivalent.

Since X = A ∪ B, the second version also reads: H_{n}(B, A ∩ B) ≅ H_{n}(A ∪ B, A).

The proof uses (iterated) subdivision which is a natural chain map Sd: C_{∗}(X) → C_{∗}(X) that decomposes any simplex into a sum of smaller simplices such that the subdivision of a cycle is a cycle in the same homology class. We shall first define subdivision in the special case where X = Δ^{n} is the standard n-simplex and then extend the definition by naturality.

5.2. Linear chains. A p-simplex in Δ^{n} is a map Δ^{p} → Δ^{n}. A linear p-simplex in Δ^{n} is a map of the special form [v_{0}v_{1} ⋯ v_{p}]: Δ^{p} → Δ^{n} given by

[v_{0}v_{1} ⋯ v_{p}] (∑ t_{i}e_{i}) = ∑ t_{i}v_{i}, t_{i} ≥ 0, ∑ t_{i} = 1,

where v_{0}, v_{1}, …, v_{p} are p + 1 points in Δ^{n}. A linear chain is any linear combination with Zcoefficients of finitely many linear simplices. Let L_{p}(Δ^{n}) be the abelian group of all linear chains. This is a subgroup of the abelian group C_{p}(Δ^{n}) of all singular chains. Since the boundary of a linear p-simplex

∂[v_{0}v_{1} ⋯ v_{p}] = ∑_{i=0}^{p}[v_{0} ⋯ v̂_{i} ⋯ v_{p}]

is a linear (p − 1)-chain, we have a (sub)chain complex

⋯ ⟶^{∂} L_{p}(X) ⟶^{∂} L_{p−1}(X) ⟶^{∂} ⋯ ⟶^{∂} L_{0}(Δ^{n}) ⟶^{∂} 0

of linear chains. Here, the boundary of any 0-simplex is defined to 0. It will be more convenient for us to work with the augmented linear chain complex

(5.3)  ⋯ ⟶^{∂} L_{p}(X) ⟶^{∂} L_{p−1}(X) ⟶^{∂} ⋯ ⟶^{∂} L_{0}(Δ^{n}) ⟶^{∂} L_{−1}(Δ^{n})

where L_{−1}(Δ^{n}) ≅ Z is generated by the (−1)-simplex [∅] with 0 vertices and the boundary of any 0-simplex ∂[v_{0}] = [∅].

5.4. Cones on linear chains. Fix a point b ∈ Δ^{n}. Define the cone on b to be the linear map

\begin{aligned}b: & L_{p}(Δ^{n}) → L_{p+1}(Δ^{n}), p ≥ −1, \\ & b[v_{0}v_{1} ⋯ v_{p}] = [bv_{0}v_{1} ⋯ v_{p}] \end{aligned}

that adds b as an extra vertex on any linear simplex. The cone operator raises degree by one as in the diagram

\xymatrix{L_{p+1}(Δ^{n}) \ar[r]^{∂} & L_{p}(Δ^{n}) \ar[dl]_{b} \ar[r]^{∂} \ar[d]_{id}^{0} & L_{p−1}(Δ^{n}) \ar[dl]_{b} \ar[r]^{∂} \ar[d]_{id}^{0} & ⋯ \ar[r]^{∂} & L_{0}(Δ^{n}) \ar[r]^{∂} \ar[dl]_{b} \ar[d]_{id}^{0} & L_{−1}(Δ^{n}) \ar[dl]_{b} \ar[d]_{id}^{0} \\ L_{p+1}(Δ^{n}) \ar[r]^{∂} & L_{p}(Δ^{n}) \ar[r]^{∂} & L_{p−1}(Δ^{n}) \ar[r]^{∂} & ⋯ \ar[r]^{∂} & L_{0}(Δ^{n}) \ar[r]^{∂} & L_{−1}(Δ^{n})}

with the augmented linear chain complex (5.3) in the two rows.

Since for instance

∂(b[v_{0}v_{1}]) = ∂[bv_{0}v_{1}] = [v_{0}v_{1}] − [bv_{1}] + [bv_{0}] = [v_{0}v_{1}] − b([v_{1}] − [v_{0}]) = [v_{0}v_{1}] − b∂[v_{0}v_{1}]

we guess that the boundary of the cone on a simplex is the base simplex minus the cone on the boundary of the simplex. This is indeed the case:

Lemma 5.5. ∂b + b∂ = id: L_{p}(Δ^{n}) → L_{p}(Δ^{n}) for all p ≥ −1.

Proof. It is enough to compute the boundary on the cone ∂b[v_{0} ⋯ v_{p}] of a linear simplex. For the (−1)-simplex, ∂b[∅] = ∂[b] = [∅] = (id − b∂)[∅]. For a 0-simplex, ∂b[v_{0}] = ∂[bv_{0}] = [v_{0}] − [b] = [v_{0}]−b[∅] = [v_{0}]−b∂[v_{0}] = (id−b∂)[v_{0}]. For p > 0 we take any linear p-simplex [v_{0} ⋯ v_{p}] ∈ L_{p}(Δ^{n}) and compute

∂b[v_{0} ⋯ v_{p}] = ∂[bv_{0} ⋯ v_{p}]

= [v_{0} ⋯ v_{p}] + ∑_{i=1}^{p+1}(−1)^{i}[bv_{0} ⋯ v̂_{i−1} ⋯ v_{p}]

= [v_{0} ⋯ v_{p}] − ∑_{i=0}^{p}(−1)^{i}[bv_{0} ⋯ v̂_{i} ⋯ v_{p}]

= [v_{0} ⋯ v_{p}] − b ∑_{i=0}^{p}(−1)^{i}[v_{0} ⋯ v̂_{i} ⋯ v_{p}]

= [v_{0} ⋯ v_{p}] − b∂[v_{0} ⋯ v_{p}]

= (id − b∂)[v_{0} ⋯ v_{p}]

to obtain our formula. □

This means that the augmented linear chain complex of Δ^{n} is contractible in the sense that the identity map and the zero map are chain homotopic. (Maybe not so surprising since Δ^{n} is contractible.) Any contractible chain complex is exact for the identity map and the zero map induce the same map on homology since they are chain homotopic.

5.6. Subdivision of linear chains. The barycentre of the linear simplex σ = [v_{0} ⋯ v_{p}] is the point

b(σ) = \frac{1}{p + 1}(v_{0} + ⋯ + v_{p}),

the center of gravity.

The subdivision operator is the chain map Sd: L_{p}(Δ^{n}) → L_{p}(Δ^{n}) defined recursively by

(5.7)  Sd(σ) = \begin{cases}σ & p = −1, 0 \\ b(σ) Sd(σ) & p > 0 \end{cases}

This means that the subdivision of a point is a point and the subdivision of a p-simplex for p > 0 is the cone with vertex in the barycentre on the subdivision of the boundary.

Lemma 5.8. Sd is a chain map: ∂ Sd = Sd ∂.

Proof. The claim is that subdivision Sd and boundary d commute. In degree −1 and 0, this is clear as Sd is the identity map there. Assume that Sd ∂ = ∂ Sd in all degrees < p and let σ be a linear p-simplex in Δ^{n}. Then

∂ Sd σ =^{def} ∂ (b Sd(∂σ)) =^{5.5} (id − b∂) Sd(∂σ) = Sd(∂σ) − b∂ Sd(∂σ)

=^{induction} Sd(∂σ) − b Sd(∂∂σ) =^{∂∂=0} Sd(∂σ)

since the induction hypothesis applies to ∂σ which has degree p − 1. □

Thus the subdivision of a cycle is a cycle and the subdivision of a boundary is a boundary. We now show that subdivision is chain homotopic to the identity map so that the it induces the identity map on homology.

8 J.M. MØLLER

Define T: L_{p}(Δ^{n}) → L_{p+1}(Δ^{n}) recursively by

(5.9)  T (σ) = \begin{cases}0 & p = −1 \\ b(σ)(σ − T ∂σ) & p ≥ 0 \end{cases}

Then T raises degrees by one as in the diagram

\xymatrix{L_{p+1}(Δ^{n}) \ar[r]^{∂} & L_{p}(Δ^{n}) \ar[dl]_{T} \ar[r]^{∂} \ar[d]^{Sd}_{id} & L_{p−1}(Δ^{n}) \ar[dl]_{T} \ar[r]^{∂} \ar[d]^{Sd}_{id} & ⋯ \ar[r]^{∂} & L_{0}(Δ^{n}) \ar[r]^{∂} \ar[dl]_{T} \ar[d]^{Sd}_{id} & L_{−1}(Δ^{n}) \ar[dl]_{T} \ar[d]^{Sd}_{id} \\ L_{p+1}(Δ^{n}) \ar[r]^{∂} & L_{p}(Δ^{n}) \ar[r]^{∂} & L_{p−1}(Δ^{n}) \ar[r]^{∂} & ⋯ \ar[r]^{∂} & L_{0}(Δ^{n}) \ar[r]^{∂} & L_{−1}(Δ^{n})}

with the augmented linear chain complex (5.3) in the two rows. Lemma 5.10. T ∂ + ∂T = id − Sd: L_{p}(Δ^{n}) → L_{p}(Δ^{n}) for all p ≥ −1. Proof. The formula holds in degree p = −1 where both sides are 0. The formula also holds in degree p = 0, since T [v_{0}] = v_{0}[v_{0}] = [v_{0}v_{0}] so that (T ∂+∂T)[v_{0}] = T [∅]+∂[v_{0}v_{0}] = 0+0 = 0 = (id−Sd)[v_{0}]. Assume now inductively that the formula holds in degrees < p and let σ be a linear p-simplex in Δ^{n}. The computation

∂T σ = ∂b(σ)(σ − T ∂σ)  by (5.9)
= (id − b(σ)∂)(σ − T ∂σ)  by (5.5)
= σ − T ∂σ − b(σ) (∂σ − ∂T ∂σ)
= σ − T ∂σ − b(σ) (∂σ − (∂σ − Sd ∂σ − T ∂∂σ)) by induction hypothesis
= σ − T ∂σ − b(σ) (∂σ − (∂σ − Sd ∂σ))  since ∂∂σ = 0
= σ − T ∂σ − b(σ) (Sd ∂σ)
= σ − T ∂σ − Sd σ  since Sd σ = b(σ) Sd ∂σ
= (id − T ∂ − Sd)(σ)

then shows that the formula also holds in degree p. □

This means that the subdivision of a cycle is a cycle in the same homology class. 5.7. Iterated subdivision of linear chains. For any k ≥ 0 let

Sd^{k} = \overbrace{Sd ∘ ⋯ ∘ Sd}^{k}: L_{p}(Δ^{n}) → L_{p}(Δ^{n})

be the kth fold iterate of the subdivision operator Sd. Since Sd is a chain map also Sd^{k} is a chain map, ∂ Sd = Sd ∂. Also Sd^{k} is chain homotopic to the identity map with

T_{k} = (1 + Sd + ⋯ + Sd^{k−1}) ∘ T

as chain homotopy. Lemma 5.12. T_{k}∂ + ∂T_{k} = id − Sd^{k}: L_{p}(Δ^{n}) → L_{p}(Δ^{n}) for all p ≥ −1. Proof. We simply compute the right hand side of the equation

id − Sd^{k} = (1 + Sd + ⋯ + Sd^{k−1}) ∘ (id − Sd)  by (5.10)
= (1 + Sd + ⋯ + Sd^{k−1}) ∘ (T ∂ + ∂T)  Sd is a chain map
= ((1 + Sd + ⋯ + Sd^{k−1}) ∘ T ) ∂ + ∂ ((1 + Sd + ⋯ + Sd^{k−1}) ∘ T ) by definition of T_{k}
= T_{k}∂ + ∂T_{k}

and get the left hand side. □

The diameter of a compact subspace of a metric space is the maximum distance between any two points of the subspace.

Lemma 5.13. Let σ = [v_{0} ⋯ v_{p}] be a linear simplex in Δ^{n}. Any simplex in the chain Sd^{k}(σ) has

diameter ≤ (\frac{n}{n+1})^{k} diam(σ).

I omit the proof. The important thing to notice is that iterated subdivision will produce simplices of arbitrarily small diameter because the fraction n/(n + 1) < 1.

5.14. Subdivision for general spaces. Let now X be an arbitrary topological space. For any singular n-simplex σ: Δ^{n} → X in X we define

Sd^{k}(σ) = σ_{∗}(Sd^{k} δ_{n}), T_{k}(σ) = σ_{∗}(T_{k}δ_{n})

where, as usual, σ_{∗}: L_{p}(Δ^{n}) → C_{p}(X) is the chain map induced by σ and δ_{n} ∈ L_{n}(Δ^{n}) is the n-simplex on the standard n-simplex Δ^{n} that is identity map δ_{n} = [e_{0} · e_{n}]: Δ^{n} → Δ^{n}.

Lemma 5.15. Sd^{k}: C_{n}(X) → C_{n}(X) is a natural chain map, ∂ Sd^{k} = Sd^{k} ∂, and T_{k}: C_{n}(X) → C_{n+1}(X) is a natural chain homotopy, ∂T_{k} + T_{k}∂ = id − Sd^{k}.

Proof. It is a completely formal matter to check this. It is best first to verify that subdivision is natural. Let σ: Δ^{n} → X be a singular n-simplex in X and let f: X → Y be a map. The computation

f_{∗} Sd^{k}(σ) =^{def} f_{∗}σ_{∗} Sd^{k}(δ^{n}) = (fσ)_{∗} Sd^{k}(δ^{n}) = =^{def} Sd^{k}(fσ) Sd^{k}(f_{∗}σ)

shows that subdivision is natural. Next we show that subdivision is a chain map:

∂(Sd^{k} σ) = ∂σ_{∗} Sd^{k}(δ^{n})  by definition
= σ_{∗}∂ Sd^{k}(δ_{n})  σ_{∗} is a chain map
= σ_{∗} Sd^{k}(∂δ_{n})  Sd^{k} is a chain map on linear chains
= Sd^{k} σ_{∗}(∂δ_{n})  Sd^{k} is natural
= Sd^{k} ∂σ_{∗}(δ_{n})  σ_{∗} is a chain map
= Sd^{k}(∂σ)

Similarly, we see that T_{k} is natural because

f_{∗}T_{k}σ = f_{∗}σ_{∗}T_{k}δ_{n} = (fσ)_{∗}T_{k}δ_{n} = T_{k}(fσ) = T_{k}(f_{∗}σ)

It then follows that

T_{k}∂σ = T_{k}∂σ_{∗}δ_{n} = T_{k}σ_{∗}∂δ_{n} =^{T_{k} natural} σ_{∗}T_{k}∂δ_{n}

and

∂T_{k}σ =^{def} ∂σ_{∗}T_{k}δ_{n} = σ_{∗}∂T_{k}δ_{n}

since σ_{∗} is a chain map. Finally,

(∂T_{k} + T_{k}∂)σ = σ_{∗}(∂T_{k} + T_{k}∂)δ_{n} = σ_{∗}(id − Sd^{k})δ_{n} = (id − Sd^{k})σ

finishes the proof. □

A cycle c ∈ C_{n}(X) and its iterated subdivision are in the same homology class since their difference c − Sd^{k} c = (∂T_{k} + T_{k}∂)c = ∂T_{k}c is a boundary.

5.16. A general form of excision. Let U = {U_{α}} be a covering of X. Define C_{n}^{U}(X) = ∑_{α} C_{n}(U_{α}), the U-small n-chains, to be the image of the addition homomorphism

⨁_{α} C_{n}(U_{α}) ⟶^{+} C_{n}(X)

A singular chain in X is thus U-small if it is a sum of singular simplices with support in one of the subspaces U_{α}. The boundary map on C_{n}(X) restricts to a boundary map on C_{n}^{U}(X) since the boundary operator is natural. This means that the C_{n}^{U}(X), n ≥ 0, constitute a chain complex. Let H_{n}^{U}(X) be its homology groups.

Lemma 5.17. Suppose that X = ⋃ int U_{α}. Let c ∈ C_{n}(X) be any singular chain in X. Then Sd^{k}(c) is U-small for k ≫ 0.

Proof. It is enough to show that Sd^{k}(σ) = σ_{∗}(Sd^{k}(δ_{n})) is U-small when k is big enough for any singular simplex σ: Δ^{n} → X. Choose k so big that the diameter of each simplex in the chain Sd^{k}(δ_{n}) is smaller than the Lebesgue number of the open covering {σ^{−1}(int U_{α})} of the compact metric space Δ^{n}. □

Theorem 5.18. Suppose that X = ⋃ int U_{α}. The inclusion chain map C_{n}^{U}(X) → C_{n}(X) induces an isomorphism on homology: H_{n}^{U}(X) ≅ H_{n}(X).

Proof. We shaw that the induced homomorphism H_{n}^{U}(X) → H_{n}(X) is surjective and injective. \underline{Surjective}: Let z be any n-cycle. For k ≫ 0, the subdivided chain Sd^{k}(z) is a U-small cycle in the same homology class as z (∂ Sd^{k} z = Sd^{k} ∂z = 0 and z − Sd^{k} z = (∂T_{k} + T_{k}∂)z = ∂T_{k}z). \underline{Injective}: Let z ∈ C_{n}^{U}(X) be a U-small n-cycle, ∂z = 0, which is a boundary in C_{n}(X), z = ∂c for some c ∈ C_{n+1}(X). The situation is sketched in the commutative diagram

\xymatrix{C_{n−1}^{U}(X) \ar[d] & C_{n}^{U}(X) \ar[d] \ar[l]_{∂} & C_{n+1}^{U}(X) \ar[l]_{∂} \ar[d] \\ C_{n−1}(X) & C_{n}(X) \ar[l]_{∂} & C_{n+1}(X) \ar[l]_{∂}}

where the rows are chain complexes and the down arrows are inclusions. The (n + 1)-chain c need not be U-small but its iterated subdivision Sd^{k}(c) is when k is big enough (5.17). The problem is that the boundary of Sd^{k}(c) need not equal z. The difference is

z − ∂ Sd^{k}(c) = ∂c − ∂ Sd^{k}(c) = ∂(c − Sd^{k}(c)) =^{5.12} ∂(T_{k}∂ + ∂T_{k})c = ∂T_{k}∂c = ∂T_{k}z

where T_{k}z ∈ C_{n+1}^{U}(X) = ∑ C_{n+1}(U_{α}) since z is U-small and T_{k} is natural (5.15). Hence z = ∂ Sd^{k} c + ∂T_{k}z = ∂(Sd^{k} c + T_{k}z) ∈ ∂C_{n+1}^{U}(X) is a boundary in the chain complex of U-small chains. □

There is also a relative version of this theorem. Let A ⊂ X be a subspace. Since Sd^{k} is natural there is an induced chain map on the relative chain complex for (X, A) as in the commutative diagram

\xymatrix{0 \ar[r] & C_{n}(A) \ar[r] \ar[d]_{_{Sd_{k}}} & C_{n}(X) \ar[r] \ar[d]_{_{Sd_{k}}} & C_{n}(X, A) \ar[r] \ar@{-->}[d] & 0 \\ 0 \ar[r] & C_{n}(A) \ar[r] & C_{n}(X) \ar[r] & C_{n}(X, A) \ar[r] & 0}

that we also denote Sd^{k}: C_{n}(X, A) → C_{n}(X, A). Similarly, the chain homotopy T_{k} is natural so it also comes in a relative version T_{k}: C_{n}(X, A) → C_{n+}(X, A). The formula ∂T_{k} + T_{k}∂ = id − Sd^{k} which is valid on C_{n}(A) and C_{n}(X) remains valid on C_{n}(X, A).

Let U ∩ A denote the covering {U_{α} ∩ A} of A. Define C_{n}^{U}(X, A) to be the quotient of C_{n}^{U}(X) by C_{n}^{U∩A}(A). Then there is an induced chain map as in the commutative diagram

\xymatrix{0 \ar[r] & C_{n}^{U∩A}(A) \ar[r] \ar[d] & C_{n}^{U}(X) \ar[r] \ar[d] & C_{n}^{U}(X, A) \ar[r] \ar@{-->}[d] & 0 \\ 0 \ar[r] & C_{n}(A) \ar[r] & C_{n}(X) \ar[r] & C_{n}(X, A) \ar[r] & 0}

with exact rows. Let H_{n}^{U}(X, A) be the homology groups of the chain complex C_{n}^{U}(X, A).

Theorem 5.19. Suppose that X = ⋃ int U_{α}. The inclusion chain map C_{n}^{U}(X, A) → C_{n}(X, A) induces an isomorphism on homology: H_{n}^{U}(X, A) ≅ H_{n}(X, A).

Proof. The above morphism between short exact sequences of chain complexes induces a morphism

\xymatrix{⋯ \ar[r] & H_{n+1}^{U}(X, A) \ar[r] \ar[d] & H_{n}^{U∩A}(A) \ar[r] \ar[d]^{≅} & H_{n}^{U}(X) \ar[r] \ar[d]^{≅} & H_{n}^{U}(X, A) \ar[r] \ar[d] & ⋯ \\ ⋯ \ar[r] & H_{n+1}(X, A) \ar[r] & H_{n}(A) \ar[r] & H_{n}(X) \ar[r] & H_{n}(X, A) \ar[r] & ⋯}

between induced long exact sequences in homology. Since two out of three of the vertical homomorphisms, H_{n}^{U}(X) → H_{n}(X) and H_{n}^{U}(A) → H_{n}(A) (⋃ int_{A}(A ∩ U_{α}) = ⋃ A ∩ int U_{α} = A ∩ ⋃ int U_{α} = A ∩ X = A), are isomorphisms by Theorem 5.18, the 5-lemma says that also the third vertical maps is an isomorphism. □

Proof of Theorem 5.1.( 2). Let U = {A, B} be the covering consisting of the two subsets A and B. Then U ∩ B = {B, A ∩ B}, so that C_{n}^{U∩B}(B) = C_{n}(B), and C_{n}^{U}(X) = C_{n}(A) + C_{n}(B) ⊂ C_{n}(X). Thus the U-small chain complex is

C_{n}^{U}(X, B) = \frac{C_{n}(A) + C_{n}(B)}{C_{n}(B)}

in this case.

The homomorphism H_{n}(A, A ∩ B) → H_{n}(X, B), claimed to be an isomorphism, is induced from the chain map C_{n}(A)/C_{n}(A ∩ B) → C_{n}(X)/C_{n}(B) which we may factor as

\frac{C_{n}(A)}{C_{n}(A ∩ B)} = \frac{C_{n}(A)}{C_{n}(A) ∩ C_{n}(B)} → \frac{C_{n}(A) + C_{n}(B)}{C_{n}(B)} = C_{n}^{U}(X, B) → \frac{C_{n}(X)}{C_{n}(B)}

where the first map is an isomorphism by one of the Noether isomorphism theorems and the second map induces an isomorphism on homology by Theorem 5.19. □

5.20. The long exact sequence of a quotient. A pair of spaces (X, A) is said to be a good pair if A is closed and A is the deformation retract of an open neighborhood.

Corollary 5.21. Let (X, A) be a good pair where A ≠ ∅. Then there is a long exact sequence

⋯ → H̃_{n}(A) → H̃_{n}(X) → H̃_{n}(X/A) → H̃_{n−1}(A) → ⋯

of reduced homology groups.

If we apply 5.21 to the good pair (D^{n}, S^{n−1}), n ≥ 1, we get that H̃_{n}(S^{n}) ≅ H̃_{n−1}(S^{n−1}) since the quotient space is S^{n} and D^{n} is contractible. Hence

H̃_{n}(S^{n}) ≅ H̃_{n−1}(S^{n−1}) ≅ ⋯ ≅ H̃_{0}(S^{0}) ≅ Z

for all n. This has an interesting consequence.

Theorem 5.22 (Brouwer’s fixed point theorem). Any continuous self map of the n-disc D^{n}, n > 0, has a fixed point.

Proof. If there existed a continuous map D^{n} → D^{n} without fixed points we could use it to construct a retraction of r: D^{n} → S^{n−1} of D^{n} onto its boundary. But then the identity map of S^{n−1} would factor through D^{n} and the identity map of H_{n−1}(S^{n−1}) ≠ 0 would factor through H_{n}(D^{n}) = 0. □

See [5] for more on fixed-point theory.

6. Δ-complexes [3, V.§7]

Definition 6.1. [6] A Δ-complex structure on a space X is a set S = {σ: Δ^{n} → X} of maps of standard simplices into X such that

(1) X is the disjoint union of the open cells σ(Δ^{n} − ∂Δ^{n}) and σ|Δ^{n} − ∂Δ^{n}: Δ^{n} − ∂Δ^{n} → X is injective for all σ ∈ S

(2) If σ: Δ^{n} → X is in the set S then also σd^{i}: Δ^{n−1} → X is in the set S for all face maps d^{i}: Δ^{n−1} → Δ^{n}, 0 ≤ i ≤ n.

(3) For all U ⊂ X: U is open ⟺ σ^{−1}(U) is open for all α

Let S_{n} ⊂ S be the subset of the maps whose domain is the standard n-simplex. Then

S = ⋃_{n≥0} S_{n}. Item (2) says that (Δ^{n−1} ⟶^{d^{i}} Δ^{n} ⟶^{σ} X) ∈ S_{n−1} whenever (Δ^{n} ⟶^{σ} X) ∈ S_{n}

for all face maps d^{i}: Δ^{n−1} → Δ^{n}, 0 ≤ i ≤ n − 1. Item (3) says that the evaluation map ev: ∐_{n≥0} S_{n} × Δ^{n} → X given by ev(σ, x) = σ(x) is a quotient map. By the universal property of quotient spaces, there is a commutative diagram

\xymatrix{∐_{n≥0} S_{n} × Δ^{n} \ar@{-->}[dr] \ar[rr]^{ev} &  & X \\  & ∐_{n≥0} S_{n} × Δ^{n}/ ∼ \ar[ur]_{\overline{ev}}}

where ∼ is the equivalence relation generated by the relation (σ, d^{i}x) ∼ (σd^{i}, x) for all (σ, x) ∈ S_{n} × Δ^{n−1}. Item (1) used inductively says that \overline{ev} is bijective. Since it also a quotient map, it is a homeomorphism.

If we write d_{i}: S_{n} → S_{n−1} for the map given by d_{i}σ = σd^{i} then these face maps satisfy the simplicial identities

(6.2)  d_{j−1}d_{i} = d_{i}d_{j},  i < j,

because d_{j−1}d_{i}σ = d_{j−1}(σd^{i}) = σd^{i}d^{j−1} =^{(1.4)} σd^{j}d^{i} = d_{i}(σd^{j}) = d_{i}d_{j}σ for all σ ∈ S_{n}.

Definition 6.3. A Δ-set^{1} is a a graded set S = ⋃_{n≥0} S_{n} with face maps d_{i}: S_{n} → S_{n−1}, 0 ≤ i ≤ n, n ≥ 0, satisfying the simplicial identities (6.2).

The realization of the Δ-set S is the space

(6.4)  |S| = ∐_{n≥0} S_{n} × Δ^{n}/ ∼

with ∼ the equivalence relation generated by (σ, d^{i}x) ∼ (d_{i}σ, x) for all (σ, x) ∈ S_{n} × Δ^{n−1}. (This simply means that the ith face of the simplex (σ, Δ^{n}) is glued to the simplex (d_{i}σ, Δ^{n−1}) by means of the ith face map d^{i}: Δ^{n−1} → Δ^{n}.) For each σ ∈ S_{n}, let σ: Δ^{n} → |S| be the map given by Δ^{n} ∋ x → (σ, x) ∈ |S|. The set of these maps is a Δ-complex structure on the realization |S| of the Δ-set S.

We have seen that the realization of a Δ-set has a Δ-complex structure and that a Δ-complex structure on a space X (6.1) is a homeomorphism |S| ⟶^{≃} X from the realization of a Δ-set S (6.3) to X.

6.5. The simplicial chain complex of a Δ-set. Let S = ⋃ S_{n} be a Δ-set 6.3. Write ZS_{n} for the free abelian group generated by the n-simplices of S. The simplicial chain complex (ZS_{∗}, ∂) of S is the chain complex

\xymatrix{0 & ZS_{0} \ar[l] & ZS_{1} \ar[l]_{∂} & ⋯ \ar[l]_{∂} & ZS_{n−1} \ar[l]_{∂} & ZS_{n} \ar[l]_{∂} & ⋯ \ar[l]_{∂}}

with boundary operator ∂σ = ∑(−1)^{i}d_{i}σ. The simplicial identities (6.2) imply that ∂∂ = 0 as in 2.2. The simplicial homology group H_{n}^{Δ}(S) is the nth homology group of the simplicial chain complex.

Note that there is an obvious chain map

(6.6)  (ZS_{∗}, ∂) → (C_{∗}(|S|), ∂)

from simplicial to singular chains given by S_{n} ∋ σ ↦ (Δ^{n} ∋ x → (σ, x) ∈ |S|). We could even let S′ ⊂ S be a sub-Δ-set of S and define a relative version (ZS_{∗}/ZS′_{∗}, ∂) → (C_{∗}(|S|)/C_{∗}(|S′|), ∂) of the above chain map.

Theorem 6.7. [6, 2.27] The (relative version of the) chain map ( 6.6) induces an isomorphism

H_{∗}^{Δ}(S, S′) ⟶^{≅} H_{∗}(|S|, |S′|)

between simplicial and singular homology groups. (The isomorphism is natural wrt simplicial maps [6, 2C].)

Computation of homology groups of a space with a Δ-complex structure is algorithmic.

1Some authors use the term “semi-simplicial set” instead since it is a “simplicial set” without degeneracies. See [10, Historical Remark 8.1.10]

Example 6.8. [2, §I.5, Exercise 3 p 19] [6, Example 1B.7] Let G be a group. Define EG = ⋃ EG_{n} to be the Δ-set with EG_{n} = G^{n+1} and with face maps d_{i}: G^{n+1} → G^{n} given by d_{i}[g_{0}, …, g_{n}] = [g_{0}, …, ĝ_{i}, …, g_{n}]. The realization |EG| is contractible by the homotopy h_{t}: |EG| → |EG| which on G^{n+1} × Δ^{n} is given by

([g_{0}, …, g_{n}], x) → ([e, g_{0}, …, g_{n}], (1 − t)d^{0}x + te_{0})

This homotopy is well-defined because

h_{t}(d_{j}[g_{0}, …, g_{n}], x) = ([e, g_{0}, …, ĝ_{j}, …, g_{n}], (1−t)d^{0}x+te_{0}) = (d_{j+1}[e, g_{0}, …, g_{n}], (1−t)d^{0}x+te_{0})

≃ ([e, g_{0}, …, g_{n}], (1−t)d^{j+1}d^{0}x+td^{j+1}e_{0}) = ([e, g_{0}, …, g_{n}], (1−t)d^{0}d^{j}x+te_{0}) = h_{t}([g_{0}, …, g_{n}], d^{j}x)

for any x ∈ Δ^{n−1} and all j ≥ 0. The start value of the homotopy is the identity map and the end value is

([e, g_{0}, …, g_{n}], e_{0}) = ([e, g_{0}, …, g_{n}], d^{n}e_{0}) ≃ ([e, g_{0}, …, g_{n−1}], e_{0}) ≃ ⋯ ≃ ([e], e_{0})

which is a single point. The simplicial chain complex ZEG_{∗}

\xymatrix{0 & ZEG_{0} \ar[l] & ZEG_{1} \ar[l]_{∂} & ⋯ \ar[l]_{∂} & ZEG_{n−1} \ar[l]_{∂} & ZEG_{n} \ar[l]_{∂} & ⋯ \ar[l]_{∂}}

has the homology of a point and so it is a free resolution over ZG of the trivial ZG-module Z, called the standard resolution. The space BG = G\|EG| is the space of G-orbits.

Example 6.9. Let X be any topological space. Define S(X) = ⋃ S_{n}(X) to be the Δ-set with S_{n}(X) equal to set of all singular n-simplexes σ: Δ^{n} → X and with face maps d_{i}σ = σd^{i}. By construction, the simplicial homology of the Δ-set S(X) is the singular homology of X so that the realization |S(X)| is a Δ-complex with the same (singular) homology as X.

7. The degree of a self-map of the sphere

Let f: S^{n} → S^{n} be a map of an n-sphere into itself and let f_{∗}: H_{n}(S^{n}) → H_{n}(S^{n}) be the induced map on the nth homology group. Choose a generator of H_{n}(S^{n}) ≅ Z and call it 1. Then

f_{∗}(1) = (deg f) · 1

for a unique integer deg f ∈ Z called the degree of f. (In case n = 0, use reduced homology.)

Suppose that the map f: S^{n} → S^{n} has the (ubiquitous) property that f^{−1}(y) is finite for some point y ∈ S^{n}, say f^{−1}(y) = {x_{1}, …, x_{m}}. Let V ⊂ S^{n} be an open neighborhood of y (eg V = S^{n}) and let U_{i} ⊂ S^{n} be disjoint open neighborhoods of x_{i} (eg small discs) such that f(U_{i}) ⊂ V for all i = 1, …, m. Then f maps U_{i} − x_{i} into V − y for x_{i} is the only point in U_{i} that hits y. Define deg f|x_{i}, the local degree of f at x_{i}, to be the integer that makes the diagram

\xymatrix{H_{n}(U_{i}, U_{i} − x_{i}) \ar[r]^{(f|U_{i})_{∗}} \ar[d]^{exc}_{≅} & H_{n}(V, V − y) \ar[d]_{exc}^{≅} \\ H_{n}(S^{n}, S^{n} − x_{i}) & H_{n}(S^{n}, S^{n} − y) \\ H_{n}(S^{n}) \ar[u]^{≅}_{j_{∗}} \ar[r]^{· deg f|x_{i}} & H_{n}(S^{n}) \ar[u]_{≅}^{j_{∗}}}

commute. This means that

(f|U_{i})_{∗}(1_{i}) = (deg f|x_{i}) · 1_{y}

where we write 1_{i} for the generator of the local homology group H_{n}(U_{i}, U_{i} − x_{i}) corresponding to 1 ∈ H_{n}(S^{n}) under the isomorphism of the left column and 1_{y} for the generator of the local homology group H_{n}(V, V − y) corresponding to 1 ∈ H_{n}(S^{n}) under the isomorphism of the right column.

Theorem 7.1. (Computation of degree) deg f = ∑ deg f|x_{i} Proof. Consider the commutative diagram

\xymatrix{⨁ H_{n}(U_{i}, U_{i} − x_{i}) \ar[rr]^{∑(f|U_{i})_{∗}} \ar[d]^{_{⨁ exc}}_{≅} \ar[r]^{^{∑ incl^{∗}}}_{≅} & H_{n}(∐ U_{i}, ∐(U_{i} − x_{i})) \ar[d]^{exc}_{≅} \ar[r]^{(f| ∐ U_{i})_{∗}} & H_{n}(V, V − y) \ar[d]_{exc}^{≅} \\ ⨁ H_{n}(S^{n}, S^{n} − x_{i}) & H_{n}(S^{n}, S^{n} − f^{−1}(y)) \ar[r]^{f_{∗}} \ar[l]_{(incl_{∗},...,incl_{∗})} & H_{n}(S^{n}, S^{n} − y) \\ ⨁ H_{n}(S^{n}) \ar[u]_{⨁ j_{∗}}^{≅} & H_{n}(S^{n}) \ar[l]_{(id_{∗},...,id_{∗})} \ar[u]_{j_{∗}} \ar[r]^{f_{∗}} & H_{n}(S^{n}) \ar[u]_{≅}^{j_{∗}}}

where all homomorphisms are induced either by inclusion maps or by the map f. Commutativity of the diagram says that

(deg f) · 1_{y} = ∑(f|U_{i})_{∗}(1_{i}) = ∑(deg f|x_{i}) · 1_{y} = (∑ deg f|x_{i}) · 1_{y}

and this identity proves the theorem. □

Example 7.2. Compute the degrees of some standard maps.

8. Cellular homology of CW-complexes

It is in principle easy to compute the homology of a Δ-complex from the simplicial chain complex (6.7). In practice, however, one often runs into the problem that it requires many simplices to build a Δ-complex. (Consider for instance the compact surface of a genus two.) CW-complexes are more flexible than Δ-complexes, but how can we compute the homology of a CW-complex?

8.1. Homology of an n-cellular extension. Let X be a space and Y = X ∪_{ϕ}∐ D_{α}^{n} an n-cellular extension of X with characteristic map Φ: ∐ D_{α}^{n} → Y and attaching map ϕ = Φ| ∐ S_{α}^{n−1} : ∐ S_{α}^{n−1} → X. This space sits in a commutative diagram

\xymatrix{∐ S_{α}^{n−1} \ar[r]^{ϕ} \ar@{^{(}->}[d] & X \ar@{^{(}->}[d] \\ ∐ D_{α}^{n} \ar[r]^{Φ} \ar[d]_{q} & Y \ar[d]^{q} \\ ⋁ D_{α}^{n}/S_{α}^{n−1} \ar[r]^{\overline{Φ}} & Y/X}

where the maps labeled q are quotient maps. The map \overline{Φ} between the quotients induced by the characteristic map is a homeomorphism which makes it very easy to compute relative homology.

If n = 0, Y is just X with some isolated points. Let us therefore now assume that n ≥ 1. Since the pair (D^{n}, S^{n−1}) is a good pair also (Y, X) is a a good pair and therefore

H_{k}(X, Y) ⟶_{≅}^{Φ_{∗}} H_{k} (∐ D_{α}^{n}, ∐ S_{α}^{n−1}) = \begin{cases}⨁ H_{n}(D_{α}^{n}, S_{α}^{n−1}) = ⨁ Z & k = n \\ 0 & k ≠ n \end{cases}

because the maps H_{k}(Y, X) ⟶_{≅}^{q_{∗}} H̃_{k}(Y/X) ⟵_{≅}^{\overline{Φ}_{∗}} H_{k}(⋁ D_{α}^{n}/S_{α}^{n−1}) ⟵_{≅}^{q_{∗}} H_{k} (∐ D_{α}^{n}, ∐ S_{α}^{n−1}) are iso-

morphisms. The unique nontrivial relative homology group sits in this segment

0 → H_{n}(X) → H_{n}(Y) → H_{n}(Y, X) ⟶^{∂_{n}} H_{n−1}(X) → H_{n−1}(Y) → 0

of the long exact sequence. The 0 to the left is H_{n+1}(Y, X) and the 0 to the right is H_{n−1}(Y, X).

Lemma 8.2. Let Y = X ∪_{ϕ} ∐ D_{α}^{n} be an n-cellular extension of X where n ≥ 1. Then

H_{k}(X, Y) ≅ \begin{cases}⨁ H_{n}(D_{α}^{n}, S_{α}^{n−1}) ≅ ⨁ Z & k = n \\ 0 & k ≠ n \end{cases}

and

H_{k}(Y) ≅ \begin{cases}H_{n}(X) ⊕ ker ∂_{n} & k = n \\ H_{n−1}(X)/ im ∂_{n} & k = n − 1 \\ H_{k}(X) & k ≠ n − 1, n \end{cases}

Attaching n-cells to a space introduces extra free generators in degree n, relations in degree n − 1, and has no effect in all other a degrees. The direct sum of the above formula is not natural.

8.3. The cellular chain complex. Let X be a CW-complex with skeletal filtration ∅ = X^{−1} ⊂ X^{0} ⊂ ⋯ ⊂ X^{n} ⊂ X^{n+1} ⊂ ⋯ ⊂ X.

Lemma 8.4. Let X be a CW-complex and X^{n} = X^{n−1} ∪_{ϕ} ∐ D_{α}^{n} the n-skeleton. Then

(1) H_{k}(X^{n}, X^{n−1}) = \begin{cases}⨁ H_{n}(D_{α}^{n}, S_{α}^{n−1}) & k = n \\ 0 & k ≠ n \end{cases}

(2) H_{k}(X^{n}) = 0 for k > n ≥ 0

(3) H_{k}(X^{n}) ≅ H_{k}(X) for 0 ≤ k < n.

Proof. (1) is just 8.2.

(2) The extensions X^{0} ⊂ ⋯ ⊂ X^{n} effect homology in degrees 0, 1, …, n but not in degrees k > n where 0 = H_{k}(X^{∘}) = H_{k}(X_{n}).

(3) The extensions X^{n} ⊂ ⋯ ⊂ X^{N} for N > n effect homology in degrees n, …, N but not in degrees k < n where H_{k}(X^{n}) ≅ H_{k}(X^{N}). A compactness argument using the notion of support of a singular chain now shows that H_{k}(X^{n}) = H_{k}(X) for k < n. □

The long exact sequence for the pair (X^{n}, X^{n−1}) contains the segment

(8.5)  \xymatrix{0 \ar[r] & H_{n}(X^{n}) \ar[r]^{j_{n}} & H_{n}(X^{n}, X^{n−1}) \ar[r]^{∂_{n}} & H_{n−1}(X^{n−1}) \ar[r] & H_{n−1}(X) \ar[r] & 0}

with 0 = H_{n}(X^{n−1}) to the left and 0 = H_{n−1}(X^{n}, X^{n−1}) to the right. We have also used that H_{n−1}(X^{n}) = H_{n−1}(X). Combine the long exact sequences (8.5) for the two pairs (X^{n+1}, X^{n}) and (X^{n}, X^{n−1})

[figure: H_{n+1}(X^{n+1}); j_{n+1}; H_{n+1}(X^{n+1}, X^{n}); ∂_{n+1}; H_{n}(X^{n}); H_{n}(X); H_{n}(X^{n}); j_{n}; H_{n}(X^{n}, X^{n−1}); ∂_{n}; H_{n−1}(X^{n−1}); H_{n−1}(X)]

and extract the cellular chain complex

(8.6)  ⋯ → H_{n+1}(X^{n+1}, X^{n}) ⟶^{d_{n+1}} H_{n}(X^{n}, X^{n−1}) ⟶^{d_{n}} H_{n−1}(X^{n−1}, X^{n−2}) → ⋯

where d_{n} = j_{n−1} ∘ ∂_{n}. This is really a chain complex because d_{n} ∘ d_{n+1} = j_{n−1} ∘ ∂_{n} ∘ j_{n} ∘ ∂_{n+1} = 0 since ∂_{n} ∘ j_{n} = 0 by exactness of (8.5). Define

H_{n}^{CW}(X) = ker d_{n}/ im d_{n+1}

to be the degree n homology of this chain complex.

Theorem 8.7. H_{n}^{CW}(X) ≅ H_{n}(X) (the isomorphism is natural wrt cellular maps)

Proof. H_{n}(X) ≅ H_{n}(X^{n})/ im ∂_{n+1} = j_{n}H_{n}(X^{n})/j_{n} im ∂_{n+1} = ker ∂_{n}/ im d_{n+1} = ker j_{n−1}∂_{n}/ im d_{n+1} = ker d_{n}/ im d_{n+1} = H_{n}^{CW}(X). □

Definition 8.8. A map f: X → Y between CW-complexes is cellular if it respects the skeletal filtrations in the sense that f(X^{k}) ⊂ Y^{k} for all k.

Note that the cellular chain complex (8.6) and the isomorphism between cellular and singular homology (8.7) are natural with respect to cellular maps.

Let Φ: (∐ D_{α}^{n}, ∐ S_{α}^{n−1}) → (X^{n}, X^{n−1}) be the characteristic map and ϕ = Φ| ∐ S_{α}^{n−1} the attaching map for the n-cells of X. Choose generators (orientations) [D_{α}^{n}] ∈ H_{n}(D_{α}^{n}, S_{α}^{n−1}) and let e_{α}^{n} = Φ_{∗}([D_{α}^{n}]) ∈ H_{n}(X^{n}, X^{n−1}). The set {e_{α}^{n}} is then a basis for the free abelian group H_{n}(X_{n}, X^{n−1}) = Z{e_{α}^{n}}. As generators for H_{n−1}(S_{α}^{n−1}) and H_{n}(D_{α}^{n}/S_{α}^{n−1}) we shall use the images of [D_{α}^{n}] under the isomorphisms

\xymatrix{H_{n−1}(S_{α}^{n−1}) & H_{n}(D_{α}^{n}) \ar[l]_{∂} \ar[r]^{q_{∗}} & H_{n}(D_{α}^{n}/S_{α}^{n−1})} , n > 1,

of homology groups.

We next compute the matrix for the cellular boundary map

(8.9)  d_{n}: H_{n}(X^{n}, X^{n−1}) = Z{e_{α}^{n}} → H_{n−1}(X^{n−1}, X^{n−2}) = Z{e_{β}^{n−1}}.

When n = 1 the cellular boundary map is simply the singular boundary map

\xymatrix{⨁ H_{1}(D_{α}^{1}, S_{α}^{0}) \ar[d]_{Φ_{∗}} \ar[r]^{∂} & ⨁ H_{0}(S_{α}^{0}) = ⨁ Z\{S^{0}\} \ar[d]^{ϕ_{∗}} \\ H_{1}(X^{1}, X^{0}) \ar[r]^{d_{1}=∂_{1}} & H_{0}(X^{0}) = Z\{X^{0}\}}

given by d_{1}e_{α}^{1} = ϕ_{∗}(∂[D_{α}^{1}]). When n > 1, the commutative diagram

\xymatrix{⨁ H_{n}(D_{α}^{n}, S_{α}^{n−1}) \ar[d]_{Φ_{∗}}^{≅} \ar[r]^{∂}_{≅} & ⨁ H_{n−1}(S_{α}^{n−1}) \ar[d]^{ϕ_{∗}} \\ H_{n}(X^{n}, X^{n−1}) \ar[r]^{∂_{n}}_{≅} & H_{n−1}(X^{n−1}) \ar[r]^{j_{n−1}} & H_{n−1}(X^{n−1}, X^{n−2}) \ar[d]^{q_{∗}}_{≅} & ⨁ H_{n−1}(D_{β}^{n−1}, S_{β}^{n−2}) \ar[l]_{Φ_{∗}}^{≅} \ar[d]^{q_{∗}}_{≅} \\  &  & H_{n−1}(X^{n−1}/X^{n−2}) & ⨁ H_{n}(D_{β}^{n−1}/S_{β}^{n−2}) \ar[l]_{_{\overline{Φ}_{∗}}}^{≅}}

shows:

Theorem 8.10. (Cellular boundary formula) When n > 1, the cellular boundary map ( 8.9) is given by

d_{n}(e_{α}^{n}) = ∑ d_{αβ}e_{β}^{n−1}

where the integer d_{αβ} is the degree of the map

\xymatrix{S_{α}^{n−1} \ar[r]^{i_{α}} & ∐ S_{α}^{n−1} \ar[r]^{ϕ} & X^{n−1} \ar[r]^{q} & X^{n−1}/X^{n−2} & ⋁ D_{β}^{n−1}/S_{β}^{n−2} \ar[l]_{Φ}^{≃} \ar[r]^{q_{β}} & D_{β}^{n−1}/S_{β}^{n−2}}

where i_{α} is an inclusion map and q and q_{β} are quotient maps.

The degree d_{αβ} must be computed relative to the chosen orientations for H_{n−1}(S_{α}^{n−1}) and H_{n−1}(D_{β}^{n−1}/S_{β}^{n−2}).

8.11. Compact surfaces. We compute the homology groups of the compact surfaces.

Definition 8.12. The compact orientable surface of genus g ≥ 1 is the 2-dimensional CW-complex

M_{g} = ⋁_{i=1}^{g} S_{a_{i}}^{1} ∨ S_{b_{i}}^{1} ∪_{∏[a_{i},b_{i}]} D^{2}

where the attaching map for the 2-cell is ∏[a_{i}, b_{i}] = a_{1}b_{1}a_{1}^{−1}b_{1}^{−1} ⋯ a_{g}b_{g}a_{g}^{−1}b_{g}^{−1}.

In particular, M_{1} = T is a torus. In general M_{g} = T # ⋯ #T is the connected sum of g copies of a torus [9, Figure 74.8]. The cellular chain complex of M_{g} has the form

\xymatrix{0 & Z\{e^{0}\} \ar[l] & Z\{a_{1}, b_{1}, …, a_{g}, b_{g}\} \ar[l]_{d_{1}} & Z\{e^{2}\} \ar[l]_{d_{2}} & 0 \ar[l]}

The only problem is d_{2} for the boundary map d_{1} = 0 as H_{0}(M_{g}) = Z. The coefficient in d_{2}e^{2} of, for instance, a_{1} is the degree of the map

\xymatrix{S^{1} \ar[r]^{∏[a_{i},b_{i}]} & ⋁_{i=1}^{g} S_{a_{i}}^{1} ∨ S_{b_{i}}^{1} \ar[r]^{q_{a_{1}}} & S_{a_{1}}^{1}}

which is homotopic to the map a_{1}a_{1}^{−1} : S^{1} → S^{1} of degree 0. In this way we see that also d_{2} = 0. We conclude that

H_{k}(M_{g}) = H_{k}^{CW}(M_{g}) = \begin{cases}Z & k = 0 \\ Z^{2g} & k = 1 \\ Z & k = 2 \\ 0 & k > 2 \end{cases}

Note the symmetry in the homology groups.

Definition 8.13. The compact nonorientable surface of genus g ≥ 1 is the CW-complex

N_{g} = ⋁_{i=1}^{g} S_{a_{i}}^{1} ∪_{∏ a_{i}^{2}} D^{2}

where the attaching map for the 2-cell is ∏ a_{i}^{2} = a_{1}^{2} ⋯ a_{g}^{2}.

In particular, N_{1} = S^{1}∪_{2}D^{2} = RP^{2} is the real projective plane. In general N_{g} = RP^{2}# ⋯ #RP^{2} is the connected sum of g copies of RP^{2} [9, Figure 74.10]. The cellular chain complex of N_{g} has the form

\xymatrix{0 & Z\{e^{0}\} \ar[l] & Z\{a_{1}, …, a_{g}\} \ar[l]_{d_{1}} & Z\{e^{2}\} \ar[l]_{d_{2}} & 0 \ar[l]}

The only problem is d_{2} for the boundary map d_{1} = 0 as H_{0}(N_{g}) = Z. The coefficient of, for instance, a_{1} in d_{2}e^{2} is the degree of the map

\xymatrix{S^{1} \ar[r]^{∏ a_{i}^{2}} & ⋁_{i=1}^{g} S_{a_{i}}^{1} \ar[r]^{q_{a_{1}}} & S_{a_{1}}^{1}}

which is homotopic to the map a_{1}a_{1} : S^{1} → S^{1} of degree 2. In this way we see that d_{2}(e^{2}) = 2(a_{1} + ⋯ + a_{g}). We conclude that

H_{k}(N_{g}) = H_{k}^{CW}(N_{g}) = \begin{cases}Z & k = 0 \\ Z^{g−1} ⊕ Z/2 & k = 1 \\ 0 & k ≥ 2 \end{cases}

The Classification theorem for compact surfaces [9, Thm 77.5] says that any compact surface is homeomorphic to precisely one of the model surfaces M_{g}, g ≥ 0, or N_{g}, g ≥ 1. Thus two compact surfaces are homeomorphic iff they have isomorphic first homology groups.

Figure 1. The compact orientable surface M_{2} of genus 2 8.14. Real projective space. [3, V.§6.Exmp 6.13, Ex 4]

Let S^{k} = {(x_{0}, …, x_{k}) ∈ R^{k+1} | |x_{0}|^{2} + ⋯ + |x_{0}|^{2} = 1} ⊂ R^{k+1} be the unite sphere in R^{k+1}. Real projective k-space is the quotient space

RP^{k} = S^{k}/S^{0}

obtained by identifying two real k + 1-tuples if they are proportional by a real number (necessarily of absolute value 1). The orbit of the point (x_{0}, …, x_{k}) ∈ S^{k} is traditionally denoted [x_{0} : ⋯: z_{k}] ∈ RP^{k}. Then [x_{0} : ⋯: x_{k}] = [λx_{0} : ⋯: λz_{k}] for λ ∈ S^{0} = {±1}. Note that RP^{k−1} may be considered to be the subspace

RP^{k−1} = {[x_{0} : ⋯: x_{k}] ∈ RP^{k} | x_{k} = 0} ⊂ RP^{k}

consisting of points with the last homogeneous coordinate equal to 0. Any point of RP^{k} − RP^{k−1}, any point with x_{k} ≠ 0, is in fact equivalent to a point with last coordinate x_{k} positive since

[x_{0} : ⋯: x_{k}] = [λx_{0} : ⋯: λx_{k−1} : |x_{k}|], λ = x_{k}/|x_{k}|

when x_{k} ≠ 0. Let D^{k} = {(x_{0}, …, x_{k−1}) | |x_{0}|^{2} + ⋯ + |x_{k−1}|^{2} ≤ 1} be the unit disc in R^{k} and Φ: D^{k} → S^{k} the map Φ(x) = (x, \sqrt{1 − |x|^{2}}) extending the projection map ϕ: S^{k−1} → RP^{k−1}. The push-out diagram

\xymatrix{S^{k−1} \ar[r]^{ϕ} \ar@{^{(}->}[d] & RP^{k−1} \ar@{^{(}->}[d] \ar[ddr] \\ D^{k} \ar[r] \ar[drr]_{ϕΦ} & RP^{k−1} ∪_{ϕ} D^{k} \ar@{-->}[dr]^{≃} \\  &  & RP^{k}}

shows that RP^{k} is the k-cellular extension of RP^{k−1} with characteristic map ϕΦ. Thus real projective n-space RP^{n} has a CW-complex structure with k-skeleton RP^{k} = RP^{k−1} ∪_{ϕ_{k−1}} D_{+}^{k}. The cellular chain complex of RP^{n}

\xymatrix{0 & Z \ar[l]_{d_{0}} & Z \ar[l]_{d_{1}} & ⋯ \ar[l] & Z \ar[l]_{d_{k−1}} & Z \ar[l]_{d_{k}} & ⋯ \ar[l]_{d_{k+1}} & Z \ar[l]_{d_{n}} & 0 \ar[l]}

has one copy of Z in each degree 0 ≤ k ≤ n. The cellular boundary map d_{k} is multiplication with the degree of the map

\xymatrix{S^{k−1} \ar[r]^{ϕ_{k−1}} & RP^{k−1} \ar[r]^{q} & RP^{k−1}/RP^{k−2} & D_{+}^{k−1}/S^{k−2} \ar[l]_{Φ}^{≃}}

given by

S^{k−1} ∋ x ⟶^{Φ^{−1}∘q∘ϕ_{k−1}} \begin{cases}x ∈ D_{+}^{k−1}/S^{k−2} & x_{0} ≥ 0 \\ a(x) ∈ D_{+}^{k−1}/S^{k−2} & x_{0} ≤ 0 \end{cases}

where a is the antipodal map a(x) = −x. We use local degrees to compute the degree of this map. Let N = (1, 0, …, 0) be the North and S = (0, …, 0, 1) the South pole. The preimage of N ∈ D_{+}^{k}/S^{k−1} consists of N and S. Let U be the interior of D_{+}^{k}. The commutative diagram

\xymatrix{H_{k−1}(aU, aU − S) \ar[d]_{≅} \ar[r]^{a_{∗}} & H_{k−1}(U, U − N) \ar[d]_{≅} \ar[r]^{(Φ^{−1}∘q∘ϕ_{k−1}|U)_{∗}} & H_{k−1}(U, U − N) \ar[d]_{≅} \\ H_{k−1}(S^{k−1}, S^{k−1} − S) \ar[r]^{a_{∗}} & H_{k−1}(S^{k−1}, S^{k−1} − N) & H_{k−1}(D_{+}^{k−1}/S^{k−2}, D_{+}^{k−1}/S^{k−2} − N) \\ H_{k−1}(S^{k−1}) \ar[u]^{≅}_{j} \ar[r]^{a_{∗}=·(−1)^{k}} & H_{k−1}(S^{k−1}) \ar[u]_{≅}^{j} \ar[r]^{d_{k−1}|N} & H_{k−1}(D_{+}^{k−1}/S^{k−1}) \ar[u]_{≅}^{j}}

shows that the local degree d_{k}|S = deg a · d_{k}|N = (−1)^{k} · d_{k}|N. Hence (7.1)

d_{k} = d_{k}|N + d_{k}|S = 1 + (−1)^{k} = \begin{cases}0 & k odd \\ 2 & k even \end{cases}

It follows that for instance

H_{k}(RP^{3}) = \begin{cases}Z & k = 0 \\ Z/2 & k = 1 \\ 0 & k = 2 \\ Z & k = 3 \\ 0 & k > 3 \end{cases} H_{k}(RP^{4}) = \begin{cases}Z & k = 0 \\ Z/2 & k = 1 \\ 0 & k = 2 \\ Z/2 & k = 3 \\ 0 & k ≥ 4 \end{cases}

and in general, H_{k}(RP^{n}) is either 0, Z, or Z/2.

Alternatively, one may use a CW-decomposition of S^{n} with skeleta S^{k} and with two cells in each dimension permuted by the antipodal map a: S^{n} → S^{n}. This makes the antipodal map cellular and we write τ = a_{∗}: H_{k}(S^{k}, S^{k−1}) → H_{k}(S^{k}, S^{k−1}) for the induced map on the cellular chain complex; τ is an isomorphism of H_{k}(S^{k}, S^{k−1}) with ττ = id.

As 0-cells we use e^{0} = (1, 0, …) and τe^{0} = ae^{0}. As k-cells for k ≥ 1 we use the two cells with characteristic maps Φ and aΦ where Φ: D^{k} → S^{k} given by Φ(x) = (x, \sqrt{1 − |x|^{2}}). The claim is that it is possible to choose orientations [D^{k}] ∈ H_{k}(D^{k}, S^{k−1}), 0 ≤ k ≤ n, recursively such that

d_{k}e^{k} = \begin{cases}(1 − τ)e^{k−1} & k odd \\ (1 + τ)e^{k−1} & k even \end{cases}

where e^{k} = Φ_{∗}[D^{k}] ∈ H_{k}(S^{k}, S^{k−1}). This is clearly possible when k = 1. Then e^{1} +τe^{1} = (1+τ)e^{1} is a cellular cycle because

d_{1}(1 + τ)e^{1} = (1 + τ)d^{1}e^{1} = (1 + τ)(1 − τ)e^{0} = 0

since the cellular boundary map commutes with τ (induced from a cellular map). If n = 1 we stop here. If not, the cellular chain complex is exact in degree 1 (the homology is H_{1}(S^{n}) = 0) so the commutative diagram

[figure: H_{2}(D^{2}, S^{1}); Φ_{∗}; ∂; ≅; H_{1}(S_{1}); ϕ_{∗}; H_{2}(S^{2}, S^{1}); ∂_{2}; d_{2}; H_{1}(S^{1}); j_{1}; H_{1}(S^{1}, S^{0}) ∋ (1 + τ)e^{1}; d_{1}; 0]

shows that there is an orientation [D^{2}] ∈ H_{k}(D^{2}, S^{1}) such that (1+τ)e^{1} = j_{1}ϕ_{∗}∂[D_{2}] = d_{2}Φ_{∗}[D^{2}] = d_{2}e^{2}. Then (1 − τ)e^{2} is a cellular cycle because

d_{2}(1 − τ)e^{2} = (1 − τ)d^{2}e^{2} = (1 − τ)(1 + τ)e^{1} = 0

If n = 2 we stop here. If not we can find, as above, an orientation such that d_{3}e^{3} = (1 − τ)e^{2} etc etc.

Since the projection map RP^{n} → S^{n} is cellular there is an induced map

\xymatrix{⋯ \ar[r] & H_{k}(S^{k}, S^{k−1}) = Z\{e^{k}, τe^{k}\} \ar[r]^{d_{k}} \ar[d]_{ϕ_{∗}} & H_{k−1}(S^{k−1}, S^{k−2}) = Z\{e^{k−1}, τe^{k−1}\} \ar[r]^{d_{k−1}} \ar[d]_{ϕ_{∗}} & ⋯ \\ ⋯ \ar[r] & H_{k}(RP^{k}, RP^{k−1}) = Z\{ϕ_{∗}(e^{k})\} \ar[r]^{d_{k}} & H_{k−1}(RP^{k−1}, RP^{k−2}) = Z\{ϕ_{∗}(e^{k−1})\} \ar[r]^{d_{k−1}} & ⋯}

between the cellular chain complexes. We conclude that the cellular boundary map for RP^{n} for 0 ≤ k ≤ n is given by

d_{k}(ϕ_{∗}e^{k}) = \begin{cases}0 & k odd \\ 2ϕ_{∗}e^{k−1} & k even \end{cases}

because ϕ_{∗}τ = ϕ_{∗}a_{∗} = (ϕa)_{∗} = ϕ_{∗} so that ϕ_{∗}(1 + τ)e^{k−1} = 2ϕ_{∗}e^{k−1} and ϕ_{∗}(1 − τ)e^{k−1} = 0. The cellular chain complex for the infinite projective space RP^{∞} is

\xymatrix{0 & Z \ar[l] & Z \ar[l]_{0} & Z \ar[l]_{2} & Z \ar[l]_{0} & Z \ar[l]_{2} & ⋯ \ar[l]}

so that the reduced homology groups

H̃_{k}(RP^{∞}) = \begin{cases}0 & k even \\ Z/2 & k odd \end{cases}

alternate between 0 and Z/2.

Let C_{2} be the group of order 2. A homological algebraist will say that this cellular chain complex C = {H_{k}(S^{k}, S^{k−1})} of S^{∞} is a resolution of the trivial ZC_{2}-module Z by free ZC_{2}-modules, that C ⊗_{ZC_{2}} Z is the cellular chain complex of RP^{∞} [2, Prop II.2.4], and that H_{k}RP^{∞} = Tor_{k}^{ZC_{2}}(Z, Z).

8.15. Complex projective space. Let S^{2n+1} = {(z_{0}, …, z_{n}) ∈ C^{n+1} | |z_{0}|^{2} + ⋯ + |z_{n}|^{2} = 1} be the unit sphere in C^{n+1} = R^{2n+2}. Complex projective n-space is the quotient space

CP^{n} = S^{2n+1}/S^{1}

obtained by identifying two (n + 1)-tuples of S^{2n+1} if they are proportional by a complex number (necessarily of absolute value 1). The quotient map ϕ: S^{2n+1} → CP^{n} is called the Hopf map (in particular for n = 1 when the map is a map S^{3} → S^{2}). The orbit of the point (z_{0}, …, z_{n}) ∈ S^{2n+1} is traditionally denoted [z_{0} : ⋯: z_{n}] ∈ CP^{n}. Then [z_{0} : ⋯: z_{n}] = [λz_{0} : ⋯: λz_{n}] for all complex numbers λ ∈ S^{1} of absolute value |λ| = 1. Note that CP^{n−1} may be considered to be the subspace

CP^{n−1} = {[z_{0} : ⋯: z_{n}] ∈ CP^{n} | z_{n} = 0} ⊂ CP^{n}

consisting of points with the last homogeneous coordinate equal to 0. Any point of CP^{n} −CP^{n−1}, any point with x_{n} ≠ 0, is in fact equivalent to a point with last coordinate x_{n} positive since

[x_{0} : ⋯: x_{n}] = [λx_{0} : ⋯: λx_{n−1} : |x_{n}|], λ = \overline{x}_{n}/|x_{n}|

when x_{n} ≠ 0. Let D^{2n} = {(z_{0}, …, z_{n−1}) ∈ C^{n} | |z_{0}|^{2} + ⋯ + |z_{n−1}|^{2} ≤ 1} be the unit disc in C^{n} = R^{2n} and Φ: D^{2n} → CP^{2n} the map Φ(z) = [z : \sqrt{1 − |z|^{2}}] extending the Hopf map ϕ: S^{2n−1} → CP^{n−1}. The push-out property

\xymatrix{S^{2n−1} \ar[r]^{ϕ} \ar@{^{(}->}[d] & CP^{n−1} \ar@{^{(}->}[d] \ar[ddr] \\ D^{2n} \ar[r] \ar[drr]_{ϕΦ} & CP^{n−1} ∪_{ϕ} D^{2n} \ar@{-->}[dr]^{≃} \\  &  & CP^{n}}

shows that CP^{n} is the n-cellular extension of CP^{n−1} with characteristic map Φ. Thus CP^{n} is a CW-complex with 2k-skeleton CP^{k} and with one cell in each even dimension ≤ 2n. The cellular chain complex immediately shows that the homology

H_{k}(CP^{n}) = \begin{cases}Z & k even and 0 ≤ k ≤ n \\ 0 & otherwise \end{cases}

is concentrated in even degrees.

Similarly, the quaternion projective space HP^{n} is a CW-complex with one cell in dimensions 4k, 0 ≤ k ≤ n, and with homology concentrated in these dimensions. There is a Hopf map S^{7} → S^{4}.

8.16. Lens spaces. [3, V.§3.Ex 3, V.§7.Ex 5] Let S^{2n+1} = {(z_{0}, …, z_{n}) ∈ C^{n+1} | |z_{0}|^{2} + ⋯ + |z_{n}|^{2} = 1} be the unit sphere in C^{n+1} = R^{2n+2} and m ≥ 2 an integer. The lens space is the quotient space

L^{2n+1}(m) = S^{2n+1}/C_{m}

for the action on S^{2n+1} of the group \sqrt[m]{1} = C_{m} = ⟨ω_{m}⟩ ⊂ S^{1} of the mth roots of unity generated by the ω_{m} = e^{2πi/m}. (If m = 2, L^{2n+1}(2) = RP^{2n+1}.) As in the case of RP^{n} we find a CWstructure on S^{2n+1} such that the group action is cellular. There will be m cells in each dimension k, 0 ≤ k ≤ 2n + 1. As 0-cells we use e^{0} = (1, 0, …) and its m translates ω_{m}^{j}e^{0}, 0 ≤ j < m.

(The 0-skeleton is C_{m}.) As 1-cells we use e^{iθ}, θ ∈ [0, 2π/m] and its m translates. In general the characteristic maps are

Φ: D^{2k} → S^{2k+1}  Ψ: D^{2k+1} =D^{2k} × [0, 2π/m] → S^{2k+1}
z → (z, \sqrt{1 − |z|^{2}})  (z, θ) → (z, e^{iθ}\sqrt{1 − |z|^{2}})

and their m translates ω_{m}^{j}Φ and ω_{m}^{j}Ψ, 0 ≤ j < m. It can be of help to draw pictures of these 2m cells in case n = 0 where we obtain a CW-decomposition of S^{1} such that the cyclic group C_{m} acts cellularly. With this CW-decomposition the cellular chain complex of S^{2n+1} consists of a free group on m generators in each dimension. Since the action by the generator e^{2π/m} is cellular, there is an induced action of C_{m} on the chain complex. Let τ denote the map induced by e^{2π/m}. We claim that there are orientations [D^{k} ∈ H_{k}(D^{k}, S^{k−1}) such that the elements e^{k} = Φ_{∗}[D^{k}] if k is even and e^{k} = Ψ_{∗}[D^{k}] if k is odd satisfy

d_{k}e^{k} = \begin{cases}(1 − τ)e^{k−1} & k odd \\ (1 + τ + ⋯ + τ^{m−1})e^{k−1} & k even \end{cases}

The orientations are constructed recursively. It is clearly possible to choose the orientation such that d_{1}e^{1} = e^{0} − τe^{0}. Then the element (1 + τ + ⋯ + τ^{m−1})e^{1} is a cellular cycle for

d_{1}(1 + τ + ⋯ + τ^{m−1})e^{1} = (1 + τ + ⋯ + τ^{m−1})d_{1}e^{1} = (1 + τ + ⋯ + τ^{m−1})(1 + τ)e^{0} = 0

and as all cellular cycles are cellular boundaries we can as in the case of RP^{n} find an orientation [D^{2}] ∈ H_{2}(D^{2}, S^{1}) such that (1 + τ + ⋯ + τ^{m−1})e^{1} = d_{2}Φ_{∗}[D^{2}] = d_{2}e^{2}. Proceed in this way.

This CW-structure on S^{2n+1} induces a CW-structure on the quotient space L^{2n+1}(m) such that quotient map q: S^{2n+1} → L^{2n+1}(m) is cellular. The lense space will have one cell in each dimension k, 0 ≤ k ≤ 2n + 1, and the cellular boundary map

d_{k}(q_{∗}e^{k}) = \begin{cases}0 & k odd \\ mq_{∗}e^{k−1} & k even \end{cases}

alternates between 0 and multiplication by m. We conclude that the cellular chain complex of the infinite lense space L^{∞}(m) is

\xymatrix{0 & Z \ar[l] & Z \ar[l]_{0} & Z \ar[l]_{m} & Z \ar[l]_{0} & Z \ar[l]_{m} & ⋯ \ar[l]}

so that the reduced homology groups

H̃_{k}(L^{∞}(m)) = \begin{cases}Z/m & k odd \\ 0 & k even \end{cases}

alternate between Z/m and 0.

A homological algebraist will say that this cellular chain complex C of S^{∞} is a resolution of the trivial ZC_{m}-module Z by free ZC_{m}-modules, that C ⊗_{ZC_{m}} Z is the cellular chain complex of L^{∞}(m) [2, Prop II.2.4], and that H_{k}L^{∞}(m) = Tor_{k}^{ZC_{m}}(Z, Z).

9. The Eilenberg–MacLane Complexes K(G, 1) [2, Chp II] [6, Chp 1.B]

Let G be a group.

Definition 9.1. A K(G, 1) is a connected CW-complex with fundamental group isomorphic to G whose universal covering space is contractible.

The circle S^{1} is a K(Z, 1) because the universal covering space R is contractible.

The infinite projective space RP^{∞} is a K(C_{2}, 1) and the infinite lense space L^{∞}(m) is a K(C_{m}, 1) because the infinite sphere S^{∞} is contractible [6, Example 1B.3].

Knot complements are K(G, 1)s [6, Example 1B6].

The orientable surfaces M_{g} of genus g ≥ 1 and the nonorientable surfaces N_{g} of genus g ≥ 2 are K(G, 1)s where G is the fundamental group [2, §II.4] [6, Example 1B2]. It is clear that the torus S^{1} × S^{1} is a K(Z × Z, 1). One now proceeds by induction. There is a theorem that says that the push-out of a diagram K(G_{1}, 1) ← K(H, 1) → K(G_{2}, 1) of K(G, 1)s and maps that are injective on π_{1} is again a K(G, 1) [2, Thm II.7.3] [6, Thm 1B.11]. This theorem can be used here [6, Example 1B.14]. Alternatively, see [6, Exercise 4.2.16]. (The orientable surface S^{2} of genus 0 and the nonorientable surface RP^{2} of genus 1 are not K(G, 1)s for the universal covering space S^{2} is not contractible as H_{2}(S^{2}) is nontrivial.) Also noncompact surfaces and surfaces with boundary are K(G, 1) [2, §II.4, Examples].

K(G, 1) × K(H, 1) = K(G × H, 1) and K(G, 1) ∨ K(H, 1) = K(G ∗ H, 1) by the theorem [2, Thm II.7.3] [6, Thm 1B.11] mentioned above.

The double mapping cylinder X_{mn} of the degree m and the degree n self-map of the circle is a K(G, 1) [6, 1B.12].

For any group G there is a K(G, 1), namely the simplicial complex BG (6.8), and all K(G, 1)s are homotopy equivalent [6, Thm 1B.8]; they represent the same homotopy type. We define the kth group homology of G to be H_{k}(K(G, 1)), often denoted simply H_{k}(G). Thus we have computed the group homology of all cyclic groups. In the language of homological algebra,

H_{k}(G) = Tor_{k}^{ZG}(Z, Z)

since the simplicial chain complex for BG is Δ_{∗}(BG) = EG_{∗} ⊗_{ZG} Z where EG_{∗} is a free resolution of Z over ZG.

10. Cohomology

The singular cochain complex of the space X with coefficients in the abelian group G is the dual

\xymatrix{0 \ar[r] & C^{0}(X; G) \ar[r]^{δ} & C^{1}(X; G) \ar[r]^{δ} & ⋯ \ar[r]^{δ} & C^{k}(X; G) \ar[r]^{δ} & C^{k+1}(X; G) \ar[r]^{δ} & ⋯}

where C^{k}(X : G) = Hom_{Z}(C_{k}(X), G) = G⟨S_{k}(X)⟩ is the abelian group of all functions from the set S_{k}(X) of singular k-simplices in X to G. The coboundary map, given by δϕ = ϕ∂, is the dual of the boundary map.

10.1. Evaluation. Cochains act (most naturally from the right) on chains by the evaluation map

C_{k}(X; G) × C^{k}(X; G) ⟶^{⟨ , ⟩} G, ⟨c, ϕ⟩ = ϕ(c)

Since ⟨∂c, ϕ⟩ = ⟨c, δϕ⟩ (for c ∈ C_{k+1}(X; G) and ϕ ∈ C^{k}(X; G)) we have that ⟨B_{k}, Z^{k}⟩ = 0 = ⟨Z_{k}, B^{k}⟩ so there is an induced bilinear evaluation map

H_{k}(X; G) × H^{k}(X; G) ⟶^{⟨ , ⟩} G, ⟨[z], [ϕ]⟩ = ϕ(z)

on homology. We may view this bilinear map as a linear map

H^{k}(X; G) ⟶^{h} Hom_{Z}(H_{k}(X; G), G), h([ϕ])([z]) = ⟨[z], [ϕ]⟩ = ϕ(z)

from cohomology to the dual of homology. Is h an isomorphism?

10.1. Cup and cap products. In the following we use coefficients in some ring R. The product in R allows us to define pairings, the cup and the cap product, which are the bilinear maps

C^{k}(X; R) × C^{ℓ}(X; R) ⟶^{∪} C^{k+ℓ}(X; R), C_{k+ℓ}(X; R) × C^{k}(X; R) ⟶^{∩} C_{ℓ}(X; R)

given by the formulas

(ϕ ∪ ψ)(σ) = ⟨σ|[v_{0} ⋯ v_{k}]), ϕ⟩ · ⟨σ|[v_{k} ⋯ v_{k+ℓ}], ψ⟩, σ ∩ ϕ = ⟨σ|[v_{0} ⋯ v_{k}], ϕ⟩σ|[v_{k} ⋯ v_{k+ℓ}]

where σ ∈ C_{k+ℓ}(X; R) is any singular simplex. These products are dual to each other in the sense that

⟨c, ϕ ∪ ψ⟩ = ⟨c ∩ ϕ, ψ⟩, c ∈ C_{k+ℓ}(X; R), ϕ ∈ C^{k}(X; R), ψ ∈ C^{ℓ}(X; R)

If f: X → Y is a map and σ ∈ C_{k+ℓ}(X; R) and now ϕ ∈ C^{k}(Y; R), ψ ∈ C^{ℓ}(Y; R) then

(10.2)  f^{∗}ϕ ∪ f^{∗}ψ(σ) = ϕ ∪ ψ(f_{∗}σ) = f^{∗}(ϕ ∪ ψ)(σ),  f_{∗}σ ∩ ϕ = f_{∗}(σ ∩ f^{∗}ϕ)

Cup and cap products behave nicely under the (co)boundary operator.

Lemma 10.3. δ(ϕ ∪ ψ) = δϕ ∪ ψ + (−1)^{k}ϕ ∪ δψ and ∂(σ ∩ ϕ) = (−1)^{k}(∂σ ∩ ϕ − σ ∩ δϕ).

Proof. The first formula can be verified by direct computation. Since cup and cap products are dual to each other, the second formula follows from the first one. □

These boundary formulas imply that there are induced pairings on homology

H^{k}(X; R) × H^{ℓ}(X; R) ⟶^{∪} H^{k+ℓ}(X; R),  H_{k+ℓ}(X; R) × H^{k}(X; R) ⟶^{∩} H_{ℓ}(X; R)
[ϕ] ∪ [ψ] = [ϕ ∪ ψ]  [z] ∩ [ϕ] = [z ∩ ϕ]

such that ⟨[z], [ϕ] ∪ [ψ]⟩ = ⟨[z] ∩ [ϕ], [ψ]⟩.

If we let C^{∗}(X; R) = ⨁ C^{k}(X; R) and H^{∗}(X; R) = ⨁ H^{k}(X; R), then (C^{∗}(X; R), δ, ∪) is a differential graded ring and (H^{∗}(X; R), ∪) a graded ring. The graded homology group H_{∗}(X; R) = ⨁ H_{k}(X; R) is a graded module over the graded ring H^{∗}(X; R). If f: X → Y is any map then (10.2) the induced map f^{∗}: H^{∗}(Y; R) → H^{∗}(X; R) is a ring homomorphism and the induced map f_{∗}: H_{∗}(X; R) → H_{∗}(Y; R) is an H^{∗}(Y; R)-module homomorphism.

If X is path-connected, we have identifications H_{0}(X; R) ⟶^{≃} R and R ⟶^{≃} H^{0}(X; R) under which the cup product R × H^{ℓ}(X; R) ⟶^{∪} H^{ℓ}(X; R) and the cap product H_{ℓ}(X; R) × R ⟶^{∩} H_{ℓ}(X; R) are simply scalar multiplication and the cap product H_{k}(X; R) × H^{k}(X; R) → R is evaluation ⟨ , ⟩.

Theorem 10.4 (The cohomology ring is graded commutative). If R is a commutative ring then H^{∗}(X; R) is graded commutative in the sense that

α ∪ β = (−1)^{|α||β|}β ∪ α

for homogeneous elements α and β.

Proof. The proof is surprisingly complicated. Let ϵ_{n} denote the sign (−1)^{\frac{1}{2} n(n+1)} (the determinant of the linear map R^{n+1} → R^{n+1} : (x_{0}, …, x_{n}) → (x_{n}, …, x_{0})). Then ϵ_{k+ℓ} = (−1)^{kℓ}ϵ_{k}ϵ_{ℓ}.

Define a linear map

C_{n}(X) ⟶^{ρ} C_{n}(X), ρ(σ) = ϵ_{n}\overline{σ}

where \overline{σ}(v_{i}) = v_{n−i} is the simplex σ with vertices in the reverse order.

It turns out that ρ is a chain map chain homotopic to the identity. One considers the prism Δ^{n} × I, puts σ at the bottom Δ^{n} and ρ(σ) at the top Δ^{n} and writes down a chain homotopy using the Δ-complex structure on Δ^{n} × I.

The dual cochain map ρ^{∗}: C^{n}(X) → C^{n}(X) is also chain homotopic to the identity so that it induces the identity map on cohomology. Direct computation shows that

ϵ_{k}ϵ_{ℓ}(ρ^{∗}ϕ ∪ ρ^{∗}ψ) = ϵ_{k+ℓ}ρ^{∗}(ψ ∪ ϕ)

and this proves the theorem. □

10.5. Relative cup and cap products. Suppose that A ⊂ X. The same formula that we used above for the cup product also defines bilinear maps

C^{k}(X; R) × C^{ℓ}(X, A; R) ⟶^{∪} C^{k+ℓ}(X, A; R), C^{k}(X, A; R) × C^{ℓ}(X, A; R) ⟶^{∪} C^{k+ℓ}(X, A; R)

For if ϕ ∈ C^{k}(X; R) is any cochain, and ψ ∈ C^{ℓ}(X; R) = Hom_{R}(C_{ℓ}(X; R), R) is a cochain that vanishes on C_{ℓ}(A, R), then the cochain ϕ ∪ ψ ∈ Hom_{R}(C_{k+ℓ}(X; R), R) vanishes on C_{k+ℓ}(A, R). Similarly, the same formula that we used above for the cap product defines bilinear maps

C_{k+ℓ}(X, A; R) × C^{k}(X; R) ⟶^{∩} C_{ℓ}(X, A; R), C_{k+ℓ}(X, A; R) × C^{k}(X, A; R) ⟶^{∩} C_{ℓ}(X; R)

This is because, first, C_{k+ℓ}(A; R) ∩ ϕ ⊂ C_{ℓ}(A; R) for all ϕ ∈ C^{k}(X; R), and, second, C_{k+ℓ}(A; R) ∩ ϕ = 0 if ϕ vanishes on C_{k}(A; R).

Since the boundary formulas (10.3) still hold there are induced bilinear maps

H^{k}(X; R) × H^{ℓ}(X, A; R) ⟶^{∪} H^{k+ℓ}(X, A; R), H^{k}(X, A; R) × H^{ℓ}(X, A; R) ⟶^{∪} H^{k+ℓ}(X, A; R)
H_{k+ℓ}(X, A; R) × H^{k}(X; R) ⟶^{∩} H_{ℓ}(X, A; R) H_{k+ℓ}(X, A; R) × H^{k}(X, A; R) ⟶^{∩} H_{ℓ}(X; R)

There are also cap products

H_{k+ℓ}(X; R) × H^{k}(X; R) ⟶^{∩} H_{ℓ}(X, A; R), H_{k+ℓ}(X; R) × H^{k}(X, A; R) ⟶^{∩} H_{ℓ}(X; R)

obtained by composing the above cap products with a map induced from an inclusion. Thus H^{∗}(X; R) and H^{∗}(X, A; R) are graded commutative R-algebras with H_{∗}(X; R) and H_{∗}(X, A; R) as graded modules. Let f: (X, A) → (Y, B) be any map. The induced maps f^{∗}: H^{∗}(Y; R) → H^{∗}(X; R) and f^{∗}: H^{∗}(Y, B; R) → H^{∗}(X, A; R) on cohomology are algebra maps and the induced maps f_{∗}: H_{∗}(X; R) → H_{∗}(Y; R) and f_{∗}: H_{∗}(X, A; R) → H_{∗}(Y, B; R) on homology are homomorphisms of modules (10.2).

If X is path-connected, we have identifications H_{0}(X; R) ⟶^{≃} R and R ⟶^{≃} H^{0}(X; R) under which the cup product R×H^{ℓ}(X, A; R) ⟶^{∪} H^{ℓ}(X, A; R) and the cap product H_{ℓ}(X, A; R)×R ⟶^{∩} H_{ℓ}(X; R) are simply scalar multiplication and the cap product H_{k}(X, A; R)×H^{k}(X, A; R) → R is evaluation ⟨ , ⟩ for relative (co)homology groups.

10.6. UCT. No, h is not always an isomorphism but it is always surjective and we have control over the kernel. The Universal Coefficient Theorem expresses to what extent the two functors homology and dualizing commute.

Theorem 10.6 (UCT). Let (C_{∗}, ∂) be a chain complex of free abelian groups. Form the dual cochain complex (Hom(C_{∗}; G), δ) of homomorphisms of the chain complex into some abelian group G. Then there is a natural short exact sequence

\xymatrix{0 \ar[r] & Ext_{Z}(H_{k−1}(C_{∗}), G) \ar[r] & H^{k}(Hom(C_{∗}, G)) \ar[r]^{h} & Hom_{Z}(H_{k}(C_{∗}), G) \ar[r] & 0}

The sequence always splits but not naturally.

Note that the UCT applies to the singular chain complex of a space or a pair of spaces, the cellular chain complex of a CW-complex, and the simplicial chain complex of a Δ-complex.

What makes the proof of the UCT work is the fact (which we have used several times) that any subgroup of a free abelian group is free.

Proposition 10.7. Any subgroup of a free abelian groups is free.

Proof. Let us first consider finitely generated free abelian groups. We use induction over the cardinality of a basis. To start the induction, let Z be free of rank 1. Any subgroup of the abelian group Z is free for it is of the form mZ for some m ∈ Z and mZ is isomorphic to Z or to 0. Let now Z^{n+1} = Z^{n} ⊕ Z be free of rank n + 1. Let G be a subgroup. There is a similar splitting G = G_{n} ⊕G_{1} of G. We know that G_{1} and G_{n} are free by induction. Thus G is free. Use transfinite induction for the general case. □

Now this proof works equally well for any PID R since the submodules of the module R, the ideals in the ring R, are of the form mR for some m ∈ R. So we have the more general

Proposition 10.8. Any submodule of a free module over a PID is free.

Thus there is a more general form of the UCT for chain complexes of free modules over a PID R. Just replace Z by R in 10.6. In particular, we could let R = k be a field. Over a field there is not even an Ext-term since all modules, vector spaces, are free themselves.

Corollary 10.9. Let k be a field and let (C_{∗}, ∂) be a chain complex of vector spaces over k. Then there is an isomorphism

H^{i}(C_{∗}; V) = Hom_{k}(H_{i}(C_{∗}), V)

for any k-vector space V.

In particular, H^{i}(X; k) ≅ Hom_{k}(H_{i}(X), k); over a field, cohomology is the dual of homology.

10.10. The simplicial cochain complex of a Δ-set. The above constructions for singular (co)chains can be repeated for simplicial (co)chains.

Let S = ⋃ S_{n} be a Δ-set (6.3) and G an abelian group. We associate an invariant, a graded group, H_{Δ}^{∗}(S; G), to S.

Write G⟨S_{n}⟩ = {S_{n} ⟶^{ϕ} G} for the abelian group consisting of all functions of the set S_{n} of n-simplices into the abelian group G. The simplicial cochain of S complex with coefficients in G, (G⟨S_{∗}⟩, δ) or (Δ^{∗}(|S|, δ), is the cochain complex

\xymatrix{0 \ar[r]^{δ} & G⟨S_{0}⟩ \ar[r]^{δ} & G⟨S_{1}⟩ \ar[r]^{δ} & ⋯ \ar[r]^{δ} & G⟨S_{n−1}⟩ \ar[r]^{δ} & G⟨S_{n}⟩ \ar[r]^{δ} & ⋯}

with coboundary operator δ(ϕ)(σ) = ϕ(∂σ) = ∑(−1)^{i}ϕ(d_{i}σ) for all ϕ: S_{n−1} → G and all σ ∈ S_{n}. Since the simplicial coboundary δ is the dual of the simplicial boundary ∂ we have δδ = 0 so that this is really a cochain complex. The simplicial cohomology group H_{Δ}^{n}(S) is the nth cohomology group of the simplicial cochain complex.

If the coefficient group G = R is a ring then there are simplicial cup and cap products

R⟨S_{k}⟩ × R⟨S_{ℓ}⟩ ⟶^{∪} R⟨S_{k+ℓ}⟩, R[S_{k+ℓ}] × R⟨S_{k}⟩ ⟶^{∩} R[S_{ℓ}]

given by the same formulas as before (in disguise)

ϕ ∪ ψ(σ) = ⟨\underbrace{d_{k+1} ⋯ d_{k+1}}_{ℓ} σ, ϕ⟩ · ⟨\underbrace{d_{0} ⋯ d_{0}}_{k} σ, ψ⟩, σ ∩ ϕ = ⟨\underbrace{d_{k+1} ⋯ d_{k+1}}_{ℓ} σ, ϕ⟩ \underbrace{d_{0} ⋯ d_{0}}_{k} σ

where σ ∈ S_{n+ℓ} is a simplex in S and ϕ: S_{k} → R, ψ: S_{ℓ} → R.

The simplicial cochain complex of S is a quotient of the singular cochain complex of |S| by the projection map

(10.11)  (G⟨S_{∗}⟩, δ) ← (C^{∗}(|S|), δ)

dual to the inclusion map (6.6) of the simplicial chain complex to the singular chain complex.

Theorem 10.12. (R⟨S_{∗}⟩, δ, ∪) is a differential graded ring and ( 10.11) is a quotient morphism of differential graded rings inducing an isomorphism (natural wrt simplicial maps)

H_{Δ}^{∗}(S; R) ⟵^{≅} H^{∗}(|S|, R)

of graded rings.

Proof. The first two statements are immediate since the simplicial cup product and the singular cup product are in fact defined by the same formula. Since we already know that we have an isomorphism on homology (6.7) the UCT (10.6) and the 5-lemma imply that we also have an isomorphism on cohomology. □

This theorem shows that there is a computer program that computes the cohomology ring of any finite Δ-complex. (Has anybody actually written such a program?)

Example 10.13. (The cohomology ring H^{∗}(RP^{2}; F_{2})) Using the Δ-complex structure on RP^{2}

[figure: •; x_{0}; •; x_{1}; •; x_{1}; b_{1}; b_{2}; a; a; c_{1}; c_{2}]

indicated by the picture we get simplicial chain and cochain complexes of the form

\xymatrix{0 & F_{2}\{x_{0}, x_{1}\} \ar[l] & F_{2}\{a, b_{1}, b_{2}\} \ar[l]_{∂_{1}} & F_{2}\{c_{1}, c_{2}\} \ar[l]_{∂_{2}} & 0 \ar[l] \\ 0 \ar[r] & F_{2}\{x_{0}^{t}, x_{1}^{t}\} \ar[r]^{∂_{1}^{t}} & F_{2}\{a^{t}, b_{1}^{t}, b_{2}^{t}\} \ar[r]^{∂_{2}^{t}} & F_{2}\{c_{1}^{t}, c_{2}^{t}\} \ar[r] & 0}

where y^{t} is the homomorphism that is dual to y and

∂_{1} = (\begin{matrix}0 & 1 & 1 \\ 0 & 1 & 1 \end{matrix})  and  ∂_{2} = (\begin{matrix}1 & 1 \\ 1 & 1 \\ 1 & 1 \end{matrix})

and ∂_{1}^{t} and ∂_{2}^{t} are the transposed matrices. We read off the homology groups

H_{1}^{Δ}(RP^{2}; F_{2}) = Z_{1}/B_{1} = F_{2}{a, b_{1} + b_{2}}/F_{2}{a + b_{1} + b_{2}} ≅ F_{2}{a}
H_{2}^{Δ}(RP^{2}; F_{2}) = Z_{2} = F_{2}{c_{1} + c_{2}}

The homology class [RP^{2}] ∈ H_{2}^{Δ}(RP^{2}; F_{2}) represented by 2-cycle RP^{2} = c_{2} + c_{2} is called the orientation class. We also read off the cohomology groups

H_{Δ}^{1}(RP^{2}; F_{2}) = Z^{1}/B^{1} = F_{2}{a^{t} + b_{1}^{t}, a^{t} + b_{2}^{t}}/F_{2}{b_{1}^{t} + b_{2}^{t}} ≅ F_{2}{[a^{t} + b_{1}^{t}]}
H_{Δ}^{2}(RP^{2}; F_{2}) = Δ^{2}/B^{2} = F_{2}{c_{1}^{t}, c_{2}^{t}}/F_{2}{c_{1}^{t} + c_{2}^{t}} ≅ F_{2}{[c_{1}^{t}]}

There is just one interesting cup product namely the square of the cohomology class represented by the 1-cocycle α = a^{t} + b_{1}^{t}. The cup product α ∪ α is the 2-cocycle with values

(a^{t} + b_{1}^{t}) ∪ (a^{t} + b_{1}^{t})(c_{1}) = (a^{t} + b_{1}^{t})(b_{1}) · (a^{t} + b_{1}^{t})(a) = 1 · 1 = 1
(a^{t} + b_{1}^{t}) ∪ (a^{t} + b_{1}^{t})(c_{2}) = (a^{t} + b_{1}^{t})(b_{2}) · (a^{t} + b_{1}^{t})(a) = 0

on the basis {c_{1}, c_{2}} for the 2-chains Δ_{2}(RP^{2}; F_{2}). This means that that α ∪ α = c_{1}^{t} in the DGR Δ^{∗}(RP^{2}; F_{2}) and that [α] ∪ [α] = [c_{1}^{t}] in the graded ring H_{Δ}^{∗}(RP^{2}; F_{2}). Note that the cohomology class [α] of α is the dual of the homology class under the UCT isomorphism H^{1}(RP^{2}; F_{2}) ≅ Hom_{F_{2}}(H_{1}(RP^{2}; F_{2}), F_{2}) of the homology class [a]. We conclude that H_{Δ}^{∗}(RP^{2}; F_{2}) ≅ F_{2}[α]/α^{3} is a truncated polynomial algebra on the nonzero class α = [a]^{t} in degree 1. We also note that cap product with the orientation class

[RP^{2}] ∩ −: H^{k}(RP^{2}; F_{2}) → H_{2−k}(RP^{2}; F_{2}), 0 ≤ k ≤ 2,

is an isomorphism in that [RP^{2}] ∩ [α] = [a] because ⟨[RP^{2}] ∩ [α], [α]⟩ = ⟨[RP^{2}], [α] ∪ [α]⟩ = c_{1}^{t}(c_{1} + c_{2}) = 1.

Example 10.14. (The cohomology ring H^{∗}(N_{2}; F_{2})) Using the Δ-complex structure on N_{2} = RP^{2}#RP^{2} (8.13) indicated by the picture we get a simplicial chain complex Δ_{∗}(N_{2}; F_{2}) and a

[figure: b_{1}; b_{2}; b_{4}; b_{4}; a_{1}; a_{1}; a_{2}; a_{2}; c_{1}; c_{3}; c_{2}; c_{4}]

simplicial cochain complex Δ^{∗}(N_{2}; F_{2}) of the form

\xymatrix{0 & F_{2}\{x_{0}, x_{1}\} \ar[l] & F_{2}\{a_{1}, a_{2}, b_{1}, b_{2}, b_{3}, b_{4}\} \ar[l]_{∂_{1}} & F_{2}\{c_{1}, c_{2}, c_{3}, c_{4}\} \ar[l]_{∂_{2}} & 0 \ar[l] \\ 0 \ar[r] & F_{2}\{x_{0}^{t}, x_{1}^{t}\} \ar[r]^{∂_{1}^{t}} & F_{2}\{a_{1}^{t}, a_{2}^{t}, b_{1}^{t}, b_{2}^{t}, b_{3}^{t}, b_{4}^{t}\} \ar[r]^{∂_{2}^{t}} & F_{2}\{c_{1}^{t}, c_{2}^{t}, c_{3}^{t}, c_{4}^{t}\} \ar[r] & 0}

where

∂_{1} = (\begin{matrix}0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 1 \end{matrix})  and  ∂_{2} = (\begin{matrix}1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{matrix})

and ∂_{1}^{t} and ∂_{2}^{t} are the transposed matrices of ∂_{1} and ∂_{2}. We read off the homology groups

H_{1}^{Δ}(N_{2}; F_{2}) = \frac{F_{2}\{a_{1}, a_{2}, b_{1} + b_{2}, b_{2} + b_{3}, b_{3} + b_{4}\}}{F_{2}\{a_{1} + b_{1} + b_{2}, a_{1} + b_{2} + b_{3}, a_{2} + b_{3} + b_{4}\}} ≅ F_{2}\{[a_{1}], [a_{2}]\}

H_{2}^{Δ}(N_{2}; F_{2}) = Z_{2} = F_{2}{[c_{1} + c_{2} + c_{3} + c_{4}]}

The homology class represented by the 2-cycle N_{2} = c_{1} + c_{2} + c_{3} + c_{4} is the orientation class. We also read off the cohomology groups

H_{Δ}^{1}(N_{2}; F_{2}) = \frac{F_{2}\{a_{1}^{t} + b_{2}^{t}, a_{2}^{t} + b_{4}^{t}, b_{1}^{t} + b_{2}^{t} + b_{3}^{t} + b_{4}^{t}\}}{F_{2}\{b_{1}^{t} + b_{2}^{t} + b_{3}^{t} + b_{4}^{t}\}} ≅ F_{2}\{[a_{1}^{t} + b_{2}^{t}], [a_{2}^{t} + b_{4}^{t}]\}

H_{Δ}^{2}(N_{2}; F_{2}) = Δ^{2}/B^{2} = F_{2}{c_{1}^{t}, c_{2}^{t}, c_{3}^{t}, c_{4}^{t}}/F_{2}{c_{1}^{t} + c_{2}^{t}, c_{3}^{t} + c_{4}^{t}, c_{1}^{t} + c_{4}^{t}} ≅ F_{2}{[c_{1}^{t}]}

The 1-cocycles α_{1} = a_{1}^{t} + b_{2}^{t} and α_{2} = a_{2}^{t} + b_{4}^{t} are dual to the 1-cycles a_{1} and b_{1} in the sense that a_{1} ∩ α_{1} = x_{1} and b_{1} ∩ β_{1} = x_{1}. The 2-cocycle c_{1}^{t} is dual to the orientation class as RP^{2} ∩ c_{1}^{t} = x_{1}. The only interesting cup products are the products of the cohomology classes in degree 1. Using the table

\begin{matrix}σ & σ|[v_{0}, v_{1}] & σ|[v_{1}, v_{2}] \\ c_{1} & b_{1} & a_{1} \\ c_{2} & b_{2} & a_{1} \\ c_{3} & b_{3} & a_{2} \\ c_{4} & b_{4} & a_{2} \end{matrix}

listing the front and back faces of the four 2-simplices of T, we find that

α_{1} ∪ α_{1}(c_{i}) = \begin{cases}1 & i = 2 \\ 0 & i ≠ 2 \end{cases} and α_{2} ∪ α_{2}(c_{i}) = \begin{cases}1 & i = 4 \\ 0 & i ≠ 4 \end{cases}

and therefore α_{1} ∪ α_{1} = c_{2}^{t} ∼ c_{1}^{t} and α_{2} ∪ α_{2} = c_{4}^{t} ∼ c_{1}^{t} in the DGR Δ^{∗}(N_{2}; F_{2}). Similarly, α_{1}∪α_{2} = 0 = α_{2}∪α_{1}. We conclude that the cup product H_{Δ}^{1}(N_{2}; F_{2})×H_{Δ}^{1}(N_{2}; F_{2}) ⟶^{∪} H_{Δ}^{2}(N_{2}; F_{2}) is a nondegenerate bilinear form with matrix

(\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix})

with respect to the basis {α_{1}, α_{2}} for the 2-dimensional F_{2}-vector space H_{Δ}^{1}(N_{2}; F_{2}). Alternatively, we see that cap product with the orientation class

[N_{2}] ∩ −: H^{k}(N_{2}; F_{2}) → H_{2−k}(N_{2}; F_{2}), 0 ≤ k ≤ 2,

is an isomorphism in that N_{2} ∩ α_{1} = a_{1} and N_{1} ∩ α_{2} = a_{2} because

⟨N_{2} ∩ α_{1}, α_{1}⟩ = ⟨N_{2}, α_{1} ∪ α_{1}⟩ = 1  ⟨N_{2} ∩ α_{2}, α_{1}⟩ = ⟨N_{2}, α_{2} ∪ α_{1}⟩ = 0
⟨N_{2} ∩ α_{1}, α_{2}⟩ = ⟨N_{2}, α_{1} ∪ α_{2}⟩ = 0  ⟨N_{2} ∩ α_{2}, α_{2}⟩ = ⟨N_{2}, α_{2} ∪ α_{2}⟩ = 1

We have now computed the F_{2}-cohomology rings for the nonorientable surfaces N_{g} of genus g = 1, 2. Can you guess what is the cohomology ring for N_{g} in general?

Example 10.15. (The integer cohomology ring H^{∗}(T; Z) of the torus) Using the Δ-complex structure on the torus T indicated by the picture we get a simplicial chain complex Δ_{∗}(T; Z) and

[figure: c_{4}; c_{1}; c_{2}; c_{3}; a_{1}; b_{1}; a_{1}; b_{1}; d_{1}; d_{3}; d_{2}; d_{4}]

a simplicial cochain complex Δ^{∗}(T; Z) of the form

\xymatrix{0 & Z\{x_{0}, x_{1}\} \ar[l] & Z\{a_{1}, b_{1}, c_{1}, c_{2}, c_{3}, c_{4}\} \ar[l]_{∂_{1}} & Z\{d_{1}, d_{2}, d_{3}, d_{4}\} \ar[l]_{∂_{2}} & 0 \ar[l] \\ 0 \ar[r] & Z\{x_{0}^{t}, x_{1}^{t}\} \ar[r]^{∂_{1}^{t}} & Z\{a_{1}^{t}, b_{1}^{t}, c_{1}^{t}, c_{2}^{t}, c_{3}^{t}, c_{4}^{t}\} \ar[r]^{∂_{2}^{t}} & Z\{d_{1}^{t}, d_{2}^{t}, d_{3}^{t}, d_{4}^{t}\} \ar[r] & 0}

where

∂_{1} = (\begin{matrix}0 & 0 & −1 & −1 & −1 & −1 \\ 0 & 0 & −1 & −1 & −1 & −1 \end{matrix}) and ∂_{2} = (\begin{matrix}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ −1 & 1 & 0 & 0 \\ 0 & −1 & −1 & 0 \\ 0 & 0 & 1 & −1 \\ 1 & 0 & 0 & 1 \end{matrix})

and ∂_{1}^{t} and ∂_{2}^{t} are the transposed of these matrices. We read off the homology groups

H_{1}^{Δ}(T; Z) = \frac{Z\{a_{1}, b_{1}, c_{1} − c_{4}, c_{2} − c_{4}, c_{3} − c_{4}\}}{Z\{a_{1} − c_{2} + c_{3}, b_{1} − c_{3} + c_{4}, c_{1} − c_{2} + c_{3} − c_{4}\}} ≅ Z\{[a_{1}], [b_{1}]\}

H_{2}^{Δ}(T; Z) = Z^{2} = Z{[d_{1} + d_{2} − d_{3} − d_{4}]} The homology class represented by the 2-cycle T = d_{1} + d_{2} − d_{3} − d_{4} is called the orientation class of the manifold T. We also read off the cohomology groups

H_{Δ}^{1}(T; Z) = \frac{Z\{a_{1}^{t} − c_{3}^{t} − c_{4}^{t}, b_{1}^{t} + c_{2}^{t} + c_{3}^{t}, c_{1}^{t} + c_{2}^{t} + c_{3}^{t} + c_{4}^{t}\}}{Z\{c_{1}^{t} + c_{2}^{t} + c_{3}^{t} + c_{4}^{t}\}} ≅ Z\{[a_{1}^{t} − c_{3}^{t} − c_{4}^{t}], [b_{1}^{t} + c_{2}^{t} + c_{3}^{t}]\}

H_{Δ}^{2}(T; Z) = \frac{Z\{d_{1}^{t}, d_{2}^{t}, d_{3}^{t}, d_{4}^{t}\}}{Z\{d_{1}^{t} + d_{4}^{t}, d_{2}^{t} + d_{4}^{t}, d_{3}^{t} − d_{4}^{t}\}} ≅ Z\{[d_{1}^{t}]\}

Since all homology groups are free abelian groups, evaluation H_{k}(T; Z) × H_{k}(T; Z) → Z is a nondegenerate pairing in this case. The 1-cocycles α_{1} = a_{1}^{t} − c_{3}^{t} − c_{4}^{t} and β_{1} = b_{1}^{t} + c_{2}^{t} + c_{3}^{t} represent cohomology classes dual to the homology classes [a_{1}], [b_{1}] ∈ H^{1}(T; Z) and the 2-cocycle d_{1}^{t} represents a cohomology class dual to the orientation class [T].

For the computation of cup product is is convenient to make a table

\begin{matrix}σ & σ|[v_{0}, v_{1}] & σ|[v_{1}, v_{2}] \\ d_{1} & c_{4} & a_{1} \\ d_{2} & c_{1} & b_{1} \\ d_{3} & c_{3} & a_{1} \\ d_{4} & c_{4} & b_{1} \end{matrix}

of the front and back faces of the four 2-simplices of T. Computing in the ring Δ^{∗}(T; Z) we find that

α_{1}∪β_{1} = (a_{1}^{t}−c_{3}^{t}−c_{4}^{t})∪(b_{1}^{t}+c_{2}^{t}+c_{3}^{t}) = −d_{4}^{t} ∼ d_{1}^{t}, β_{1}∪α_{1} = (b_{1}^{t}+c_{2}^{t}+c_{3}^{t})∪(a_{1}^{t}−c_{3}^{t}−c_{4}^{t}) = d_{3}^{t} ∼ −d_{1}^{t}

while the other cup products of the two 1-cocycles α_{1}, β_{1} are 0. We conclude that the cup product H_{Δ}^{1}(T; Z) × H_{Δ}^{1}(T; Z) ⟶^{∪} H_{Δ}^{2}(T; Z) is a nondegenerate bilinear form with matrix

(\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix})

with respect to the basis {α_{1}, β_{1}} for the rank 2 free abelian group H_{Δ}^{1}(T; Z). Alternatively, we see that cap product with the orientation class

[T] ∩ −: H^{k}(T; Z) → H_{2−k}(T; Z), 0 ≤ k ≤ 2,

is an isomorphism in that T ∩ α_{1} = b_{1} and T ∩ β_{1} = −a_{1} because

⟨T ∩ α_{1}, α_{1}⟩ = ⟨T, α_{1} ∪ α_{1}⟩ = 0  ⟨T ∩ β_{1}, α_{1}⟩ = ⟨T, β_{1} ∪ α_{1}⟩ = −1
⟨T ∩ α_{1}, β_{1}⟩ = ⟨T, α_{1} ∪ β_{1}⟩ = 1  ⟨T ∩ β_{1}, β_{1}⟩ = ⟨T, β_{1} ∪ β_{1}⟩ = 0

Example 10.16. The torus T and the wedge sum X = S^{1} ∨ S^{1} ∨ S^{2} have isomorphic homology and cohomology groups (for any choice of coefficients) but they do not have the same cup product structure (nor the same fundamental group).

Example 10.17. The Moore space M(Z/m, 1) = S^{1} ∪_{m} D^{2} has an obvious Δ-complex structure [6, Example 3.9]. The simplicial chain and cochain complexes with coefficients in Z/m are

\xymatrix{0 & Z/m\{v_{0}, v_{1}\} \ar[l] & Z/m\{e, e_{0}, …, e_{m−1}\} \ar[l]_{∂_{1}} & Z/m\{T_{0}, …, T_{m−1}\} \ar[l]_{∂_{2}} & 0 \ar[l] \\ 0 \ar[r] & Z/m\{v_{0}^{t}, v_{1}^{t}\} \ar[r]^{∂_{1}^{t}} & Z/m\{e^{t}, e_{0}^{t}, …, e_{m−1}^{t}\} \ar[r]^{∂_{2}^{t}} & Z/m\{T_{0}^{t}, …, T_{m−1}^{t}\} \ar[r] & 0}

where (for m = 4)

∂_{1} = (\begin{matrix}0 & −1 & −1 & −1 & −1 \\ 0 & 1 & 1 & 1 & 1 \end{matrix}) ∂_{2} = (\begin{matrix}1 & 1 & 1 & 1 \\ 1 & 0 & 0 & −1 \\ −1 & 1 & 0 & 0 \\ 0 & −1 & 1 & 0 \\ 0 & 0 & −1 & 1 \end{matrix})

We read off the homology

H_{1}^{Δ}(M; Z/m) = \frac{ker ∂_{1}}{im ∂_{2}} ≅ Z/m\{[e]\}

H_{2}^{Δ}(M; Z/m) = ker ∂_{2} = Z/m{[T_{0} + T_{1} + T_{2} + T_{3}]}

and the cohomology groups

H_{Δ}^{1}(M; Z/m) = \frac{ker ∂_{2}^{t}}{im ∂_{1}^{t}} = Z/m\{[e^{t} + e_{1}^{t} + 2e_{2}^{t} + 3e_{3}^{t}]\}

H_{Δ}^{2}(M; Z/m) = \frac{Z/m\{T_{0}^{t}, T_{1}^{t}, T_{2}^{t}, T_{3}^{t}\}}{Z/m\{T_{0}^{t} − T_{3}^{t}, T_{1}^{t} − T_{3}^{t}, T_{2}^{t} − T_{3}^{\}}} ≅ Z/m\{[T_{0}^{t}]\}

Let α = e^{t} +e_{1}^{t} +2e_{2}^{t} +3e_{3}^{t} be the generating cocycle in degree 1 and β = T_{0}^{t} the generating cocycle in degree 2. From the table

\begin{matrix}σ & σ|[v_{0}, v_{1}] & σ|[v_{1}, v_{2}] & α ∪ α(σ) \\ T_{0} & e_{0} & e & 0 \\ T_{1} & e_{1} & e & 1 \\ T_{2} & e_{2} & e & 2 \\ T_{3} & e_{3} & e & 3 \end{matrix}

we conclude that α ∪ α = T_{1}^{t} + 2T_{2}^{t} + 3T_{3}^{t} ∼ (1 + 2 + 3)T_{0}^{t} = 2β when m = 4. In general we get that the cup product is given by

[α] ∪ [α] = (1 + 2 + ⋯ + (m − 1)) = \begin{cases}0 & m is odd \\ \frac{m}{2}[β] & m is even \end{cases}

because the terms k + (m − k) = m = 0 cancel.

10.18. Cellular cohomology of CW-complexes. The groups H^{k}(X^{k}, X^{k−1}; G) form the cellular cochain complex. Define H_{CW}^{k} (X; G) to be its cohomology.

Theorem 10.19. The cellular cochain complex is dual to the cellular chain complex and the cellular cohomology group H_{CW}^{k} (X; G) is isomorphic to singular cohomology group H^{k}(X; G).

We can now easily compute the cohomology of CW-complexes such as the compact surfaces and projective spaces. It is, however, not so easy to compute cup and cap products in this way.

11. Orientation of manifolds

What does it mean that a manifold is orientable?

11.1. Local homology groups. Let X be a space, A ⊂ X a subspace, and R a commutative ring. The local homology group is

H_{k}(X|A; R) = H_{k}(X, X − A; R)

and if A_{1} ⊂ A_{2}, the restriction homomorphism r_{A_{1}}^{A_{2}} : H_{k}(X|A_{2}; R) → H_{k}(X|A_{1}; R) is the homomorphism (in the opposite direction of the inclusion) induced from the inclusion (X, X − A_{2}) ⊂ (X, X−A_{1}). In particular, there is a restriction homomorphism r_{A}^{X}: H_{k}(X) = H_{k}(X|X) → H_{k}(X|A) for all subspaces A of X. The most important coefficients rings will be R = Z and R = F_{2}.

The groups are said to be local because they only depend on a neighborhood of A.

Lemma 11.2. Let A ⊂ X be a pair of spaces and R a commutative ring.

(1) H_{k}(U|A; R) ≅ H_{k}(X|A; R) for any open neighborhood U of cl A.

(2) If f: X → Y is injective on some open neighborhood of A then there is an induced map f_{∗}: H_{k}(X|A; R) → H_{k}(Y |B; R) where f(A) = B.

Proof. (1) Since X −U = cl(X −U) ⊂ int(X −A) = X −cl A we can excise X −U from (X, X −A).

(2) Define f_{∗}: H_{k}(X|A) → H_{k}(Y |B) as \xymatrix{H_{k}(X|A) & H_{k}(U|A) \ar[l]_{≅} \ar[r]^{f_{∗}} & H_{k}(Y |B)} for any open neighborhood U of cl A such that f|U is injective. Let U_{1} and U_{2} be open neighborhoods of cl A such that the restrictions of f to U_{1} and U_{2} are injective. From the commutative diagram

\xymatrix{H_{k}(U_{1}|A) \ar[d]_{≅} \ar[drr]^{f_{∗}} \\ H_{k}(X|A) & H_{k}(U_{1} ∩ U_{2}|A) \ar[l]_{≅} \ar[r]^{f_{∗}} & H_{k}(Y |B) \\ H_{k}(U_{2}|A) \ar[u]^{≅} \ar[urr]_{f_{∗}}}

we see that the definition is unambiguous. □

Let now M be a topological n-manifold.

Lemma 11.3 (Local continuation). Suppose that x ∈ B ⊂ R^{n} ⊂ M where R^{n} is coordinate neighborhood of x and B is an open disc (ball). Then the restriction homomorphism

r_{x}^{B}: H_{n}(M|B; R) → H_{n}(M|x; R)

is an isomorphism and both homology groups are isomorphic to R.

Proof. By locality we can assume that M = R^{n} for H_{n}(M|B) ≅ H_{n}(R^{n}|B) and H_{n}(M|x) ≅ H_{n}(R^{n}|x) (11.3). In that case, since R^{n} is contractible,

H_{n}(R^{n}|B) = H_{n}(R^{n}, R^{n} − B) ≅ H̃_{n−1}(R^{n} − B)
H_{n}(R^{n}|x) = H_{n}(R^{n}, R^{n} − x) ≅ H̃_{n−1}(R^{n} − x)

where R^{n} − x contains R^{n} − B contains S^{n−1} as a deformation retract. □

11.4. The orientation bundle. We may construct a covering space by placing the local homology group above each point (similar to the construction of the tangent bundle of a smooth manifold). Define the set

M_{R} = ⋃_{x∈M} H_{n}(M|x; R),

to be the union of the local homology groups and define p: M_{R} → M to be the map that sends H_{n}(M|x; R) ⊂ M_{R} to x ∈ M. The set M_{R} has a local product structure that we use to define a toplogy on M_{R}. For each open disc B ⊂ R^{n} ⊂ M there is a bijection (11.3)

(11.5)  B × H_{n}(M|B; R) ⟶^{r} p^{−1}(B) = M_{R}|B = ⋃_{y∈B} H_{n}(M|y; R),  r(y, μ) = r_{y}^{B}μ

that we declare to be a homeomorphism (where H_{n}(M|B; R) has the discrete topology). This makes p: M_{R} → M a covering map by design. It is clear that M_{R} is again a manifold since any local homeomorphism R^{n} ↪ M lifts to a unique local homeomorphism

(11.6)  \xymatrix{ & M_{R} \ar[d]^{p} \\ R^{n} \ar[ur]^{h̃} \ar@{^{(}->}[r] & M}

once the value h̃(x) is specified. (Any covering space of a manifold is a manifold.) Orientability is the question of whether M_{R} globally is a product.

11.7. Local and global orientations. An element μ of an R-module H is a generator of H if the submodule Rμ is all of H. If H ≅ R then there is a bijection between the unit group R^{×} and the set of generators for H.

Definition 11.7. A local R-orientation at x ∈ M is a generator μ_{x} for H_{n}(M|x; R). An Rorientation for M is a section μ: M → M_{R} such that μ(x) is a local R-orientation at all points x ∈ M. A manifold is R-orientable if it has an R-orientation.

This means that an R-orientation for M is a function x → μ_{x} that assigns a local R-orientation to each point x ∈ M such that for all open discs B ⊂ R^{n} all points y ∈ B have the same continuation to a generator μ_{B} ∈ H_{n}(M|B; R). (We say that the choice of local orientations is consistent.)

What is the π_{1}(X, x)-action given by the R-orientation bundle? For each path homotopy class ω: x_{0} → x_{1} let ω_{∗}: H_{n}(M|x_{0}; R) → H_{n}(M|x_{1}; R) be the map defined by unique path lifting.

Lemma 11.8. ω_{∗}: H_{n}(M|x_{0}; R) → H_{n}(M|x_{1}; R) is an R-module isomorphism.

Proof. We have ω_{∗}(μ + ν) = ω_{∗}(μ) + ω_{∗}(ν) because fibrewise addition of lifts of ω is a lift of ω and we have ω_{∗}(rμ) = rω_{∗}(μ) because scalar multiplication by r of a lift of ω is a lift of ω. □

Thus the action of π_{1}(X, x) acts through multiplication with units in R. When R = Z the only units are ±1 so the monodromy action in this case is the homomorphism

θ_{Z} : π_{1}(M, x) → {±1} = Z^{×}

given by μ_{x} · ω = θ(ω)μ_{x} where μ_{x} is a local orientation. (ie θ(ω) = 1 if the loop ω preserves local orientation and θ(ω) = −1 if it reverses local orientation.) In the general case the monodromy action is the homomorphism

θ_{R}: π_{1}(M, x) ⟶^{θ_{Z}} Z^{×} → R^{×}

given, as before, by multiplication by ±1. To see this, use that the unit Z → R induces a covering map M_{Z} → M_{R} corresponding to a π_{1}(X, x)-map H_{n}(M|x; Z) → H_{n}(M|x; R). Since ω is Rlinear and acts as multiplication by ±1 on H_{n}(M|x; Z) is also acts as multiplication by ±1 on H_{n}(M|x; R).

Can you compute the monodromy action for M = MB, the (open) Möbius band (imagine a person walking head up along the core circle of the band), and M = RP^{2}?

When the ring is not specified it is understood that R = Z. For instance, “M is orientable” means “M is Z-orientable”.

Proposition 11.9. Assume that the manifold M is connected. Then the following are equivalent: (1) M is R-orientable

(2) The monodromy action θ_{R} is the trivial homomorphism

(3) The orientation bundle M_{R} is isomorphic to the trivial R-module bundle M × R If the fundamental group π_{1}(M) contains no index 2 subgroups (eg if M is simply connected) then M is orientable. If M is orientable then it has exactly two orientations. If M is orientable then M is R-orientable for all R, and if M is nonorientable then M is R orientable iff −1 = 1 in R. Any open submanifold of an oriented manifold is oriented.

Proof. If μ: M → M_{R} is an orientation for M then the map M ×R → M_{R} given by (x, r) → rμ(x) is a trivialization. If the fundamental group π_{1}(M) contains no index 2 subgroups then θ_{Z} is trivial since otherwise the kernel would be an index 2-subgroup. If U ⊂ M is an open submanifold then U_{R} = M_{R}|U is trivial if M_{R} = R × M is trivial. □

For instance, any surface that contains an open Möbius band is nonorientable.

Theorem 11.10 (Orientations along compact subspaces). Let M be a an n-manifold and K ⊂ M a compact subspace. Then

(1) H_{>n}(M|K; R) = 0

(2) For any section α: M → M_{R} there exists

\xymatrix{ & M_{R} \ar[d] \\ K \ar@{-->}[ur]^{α|K} \ar@{^{(}->}[r] & M \ar[u]_{α}}

a unique class α_{K} ∈ H_{n}(M|K; R) such that (α|K)(x) = r_{x}^{K}α_{K} for all x ∈ K.

The proof of this important theorem is divided into several steps. The main ingredient is the use of an Mayer-Vietoris argument.

Lemma 11.11. If Theorem 11.10 is true for the compact subsets K, L and K ∩ L of M then it also true for K ∪ L.

Proof. Use the relative Mayer–Vietoris sequence [6, p 237]. □

Proof of Theorem 11.10. [6, Lemma 3.27]. (1) Suppose that M = R^{n} and that K ⊂ R^{n} is compact and convex. Then H_{n}(M|K; R) ≅ H_{n−1}(S^{n−1}; R) ≅ R and there is an isomorphism K × H_{n}(M|K; R) ≅ M_{R}|K. □

Corollary 11.12. Let M be a connected compact n-manifold and x a point in M. Then (1) H_{>n}(M; R) = 0

(2) Restriction H_{n}(M; R) → H_{n}(M|x; R) is injective and the image is H_{n}(M|x; R)^{π_{1}(M,x)} Consequently, the top homology group is

H_{n}(M; R) ≅ \begin{cases}R & M is R-orientable \\ _{2}R & M is not R-orientable \end{cases}

Proof. (1) Take K = M in the lemma and remember that H_{k}(M|M; R) = H_{k}(M; R).

(2) The lemma with K = M says that there is a bijection between the set of sections of M_{R} → M and H_{n}(M; R). We can thus identify the restriction map r_{x}^{M}: H_{n}(M; R) → H_{n}(M|x; R) with the evaluation map s → s(x) of sections. From covering space theory we know that this map is injective and that the image is the invariant submodule H_{n}(M|x; R)^{π_{1}(X,x)} (consisting of all elements of H_{n}(M|x; R) if M is orientable and all elements of order 2 if M is nonorientable). □

The orientable surface M_{g} is orientable because H_{2}(M_{g}; Z) ≅ Z and the nonorientable surface N_{g} is nonorientable because H_{2}(M_{g}; Z) = 0. Real projective n-space RP^{n}, n > 0, is orientable iff n is odd.

For a compact R-oriented manifold there is an R-orientation class μ_{M} ∈ H_{n}(M; R). For a noncompact R-oriented manifold there is not a single R-orientation class but rather a system of R-orientation classes μ_{K} ∈ H_{n}(M|K; R) along the compact subsets of M (and agreeing under the restriction homomorphisms).

11.13. The oriented cover of a nonorientable manifold. If M is nonorientable then the orbits for the π_{1}(X, x) action on H_{n}(M|x; Z) ≅ Z are {0} and {±r}, r ≠ 0. The one point orbit correspond to the 0-section. The 2-point orbits correspond to a double covering map M̃_{Z} → M called the orientation covering. This covering is realizable as the subcovering of M_{Z} consisting of the two generators in each fibre.

Note that any map f: M → N of manifolds that is locally a homeomorphism lifts to a map of covering spaces M_{Z} → N_{Z}. The corresponding monodromy actions are related by the commutative diagram

\xymatrix{π_{1}(M, x) \ar[r]^{f_{∗}} \ar[d]_{θ} & π_{1}(N, f(x)) \ar[d]^{θ} \\ Aut_{R}(H_{n}(M|x; R)) \ar[r]^{f_{∗}}_{≅} & Aut_{R}(H_{n}(N|f(x); R))}

where the bottom isomorphism is induced from the isomorphism f_{∗}: H_{n}(M|x; R) → H_{n}(N|f(x); R) of 11.2.(2).

Proposition 11.13. The manifold M̃_{Z} is orientable.

Proof. The fundamental group π_{1}(M̃_{Z}, μ_{x}) = ker θ as it is isomorphic to the subgroup of π_{1}(M, x) consisting of the loops ω at x ∈ M that lift to loops at μ_{x} ∈ M̃_{Z}, that is μ_{x} · ω = μ_{x}. □

The orientation cover of the Möbius band is the band annulus S^{1} × (0, 1), the orientation cover of RP^{2n} is S^{2n}, and the orientation cover of the nonorientable surface N_{g+1} is M_{g}. In each case the reason is that whenever M̃ → M is an orientable connected double cover of a nonorientable manifold M then it must be the orientation cover (because π_{1}(M) has index 2 in π_{1}(M) and is contained in ker θ so these two subgroups are identical).

The orientation cover M_{g} → N_{g+1} arises just as RP^{2} comes from S^{2}: Embed M_{g} symmetrically around (0, 0, 0) in R^{3} such that there is an antipodal ±1-action on M_{g} ⊂ R^{3}. The quotient space is a surface; it contains a Möbius band so it is nonorientable, and the Euler characteristic is \frac{1}{2}χ(M_{g}) = \frac{1}{2}(2 − 2g) = 1 − g = 2 − (1 + g) = χ(N_{g+1}). We conclude that the orbit space M_{g}/{±1} = N_{g+1}.

12. Poincaré duality for compact manifolds

We here state the Poincaré duality theorem for closed (compact with no boundary) manifolds. We later state and prove Poincaré for not necessarily compact manifolds.

Theorem 12.1. Let M be a compact, connected R-oriented n-dimensional manifold. Capping with the orientation class [M] ∈ H_{n}(M; R)

PD: H^{k}(M; R) → H_{n−k}(M; R), PD(α) = [M] ∩ α

is an isomorphism.

Theorem 12.2. The cohomology algebras for finite dimensional projective spaces are

H^{∗}(RP^{n}; F_{2}) ≅ F_{2}(α)/(α^{n+1}),  |α| = 1
H^{∗}(CP^{n}; Z) ≅ Z(α)/(α^{n+1}),  |α| = 2
H^{∗}(HP^{n}; Z) ≅ Z(α)/(α^{n+1}),  |α| = 4

The cohomology of the infinite projective spaces are the corresponding polynomial algebas.

Proof. The manifolds CP^{n} and HP^{n} (8.15) are orientable (they are simply connected) and the manifold RP^{n} (8.14) is F_{2}-orientable. We shall here take the case of CP^{n} (the two other cases are similar). Let α ∈ H^{2}(CP^{n}; Z) be a generator. The claim is that α^{i} generates H^{2i}(CP^{n}; Z) when 1 ≤ i ≤ n. It is enough to prove that α^{n} generates H^{2n}(CP^{n}; Z) (for if α^{i} is divisible by some natural number k > 1 then α^{n} is also divisible by k). Assume, inductively, that corollary holds for CP^{n−1}. We evaluate α^{n} on [CP^{n}] and find that

⟨[CP^{n}], α^{n}⟩ = ⟨[CP^{n}] ∩ α, α^{n−1}⟩ = ±1

for, by Poincaré duality (12.1), [CP^{n}] ∩ α generates H_{2n−2}(CP^{n}) and by induction hypothesis α^{n−1} generates H^{2n−2}(CP^{n−1}) = H^{2n−2}(CP^{n}) = Hom(H_{2n−2}(CP^{n}), Z). But then α^{n} must be a generator of H^{2n}(CP^{n}) = Hom(H_{2n}(CP^{n}), Z). □

Theorem 12.3. The cohomology algebra for the lens space L^{2n+1}(m) ( 8.16) with coefficients in the ring Z/m is

H^{∗}(L^{2n+1}(m); Z/m) = \begin{cases}Z/m[α, β]/(α^{2}, β^{n+1}) & m is odd \\ Z/m[α, β]/(α^{2} − \frac{m}{2}β, β^{n+1}) & m is even \end{cases}

where the generators have degrees |α| = 1 and |β| = 2.

Proof. The cellular cochain complex with Z/m-coefficients (10.18, 8.16) for the lense space tells us that H^{i}(L^{2n+1}(m); Z/m) = Z/m for 0 ≤ i ≤ m. Let α ∈ H^{1}(L^{2n+1}(m); Z/m) be 1-dimensional and β ∈ H^{2}(L^{2n+1}(m); Z/m) generator. From simplicial computations (10.17) in the 2-skeleton M(Z/m, 1) = S^{1} ∪_{m} D^{2} of L^{2n+1}(m) we know that we may choose β such that α^{2} = \frac{m}{2}β when m is even. When m is odd α^{2} = 0 by graded commutativity.(10.4). If course, β^{n+1} = 0 for dimensional reasons. We claim that β^{i}α^{j} generates H^{2i+j}(L^{2n+1}(m); Z/m) when 2i + j ≤ 2n + 1 and j = 0, 1. It is enough to show that β^{n}α generates H^{2n+1}(L^{2n+1}(m); Z/m). The lense space L^{2n+1}(m) is orientable (11.12), hence Z/m-orientable (11.9), since H_{2n+1}(L^{2n+1}(m)) = Z (8.16). We evaluate this cohomology class on the orientation class and find that

⟨[L], β^{n}α⟩ = ⟨[L] ∩ β, β^{n−1}α⟩

is a unit in Z/m: By Poincaré duality (12.1), [L] ∩ β generates H_{2n−1}(L^{2n+1}(m); Z/m), and by induction hypothesis αβ^{n−1} generates H^{2n−1}(L^{2n−1}(m); Z/m) ≅ H^{2n−1}(L^{2n+1}(m); Z/m) which is isomorphic to the dual group Hom(H_{2n−1}(L^{2n+1}(m); Z/m), Z/m). But then β^{n}α must generate H^{2n+1}(L^{2n+1}(m); Z/m). □

When p is a prime number

H^{∗}(L^{∞}(p); F_{p}) = \begin{cases}F_{2}[α] & p = 2 \\ F_{p}[α, β] & p > 2 \end{cases}

where |α| = 1 and |β| = 2.

12.4. Connection with cup product. When M is a compact, connected, R-oriented n-manifold there is a commutative diagram

\xymatrix{H^{i}(M; R) × H^{n−i}(M; R) \ar[d]_{PD×id}^{≅} \ar[r]^{∪} & H^{n}(M; R) \ar[d]^{PD}_{≅} \\ H_{n−i}(M; R) × H^{n−i}(M; R) \ar[r]^{∩} & H_{0}(M; R)}

which says that Poincaré duality translates the cup product of two classes, α ∈ H^{i}(M; R) and β ∈ H^{n−i}(M; R), in complementary dimensions, into the evaluation homomorphism

H_{n−i}(M; R) × H^{n−i}(M; R) ⟶^{∩} H_{0}(X; R) ⟶^{≅} R

of the bottom line. So when this evaluation pairing is a duality pairing, also cup product is an evaluation pairing. The reason is simply that

PD(α ∪ β) = [M] ∩ (α ∪ β) = ([M] ∩ α) ∩ β = PD(α) ∩ β

because H_{∗}(M; R) is a right H^{∗}(M; R)-module (10.1).

We implicitly used this connection with cup product when calculating the cohomology algebras of projective and lense spaces.

The signature of a compact, connected, R-oriented manifold of dimension 4k is the signature of the symmetric bilinear cup product form on H^{2k}(M; R).

12.5. Applications: Borsuk–Ulam, division algebras, and the Hopf map. Here is a standard application of the cohomology ring for the real projective spaces.

Theorem 12.6 (Borsuk–Ulam). For any continuous map f: S^{n} → R^{n}, n ≥ 1, there is a point x ∈ S^{n} such that f(−x) = f(x).

Proof. Suppose that f: S^{n} → R^{n} is a map such that f(x) and f(−x) are unequal for all x ∈ S^{n}. We can use f to construct a map g: S^{n} → S^{n−1}, namely g(x) = \frac{f(x)−f(−x)}{|f(x)−f(−x)|}, such that g(−x) = −g(x) for all x ∈ S^{n} (g is an odd map). This contradicts connectedness of S^{1} when n = 1 so we may assume that n ≥ 2 and even n > 2 since the case n = 2 is known already. Since g respects the equivalence relation it induces a map \overline{g} of quotient spaces such that

\xymatrix{S^{n} \ar[d] \ar[r]^{g} & S^{n−1} \ar[d] \\ RP^{n} \ar[r]^{\overline{g}} & RP^{n−1}}

commutes. Since the fundamental group π_{1}(RP^{n}) is cyclic of order 2 and generated by the image of any path in S^{n} from a point to its antipode [6, 1.43], this implies that the induced map \overline{g}_{∗}: π_{1}(RP^{n}) → π_{1}(RP^{n−1}) is an isomorphism (cf the proof of [6, 3.20]). Using the connection between H^{1} and π_{1} we see that also H^{1}(\overline{g}): H^{1}(RP^{n−1}; F_{2}) → H^{1}(RP^{n}; F_{2}) is an isomorphism. But this contradicts the structure of the cohomology algebras for 0 = \overline{g}^{∗}(0) = \overline{g}^{∗}(α^{n}) = \overline{g}^{∗}(α)^{n} ≠ 0 where α is a generator of H^{1}(RP^{n−1}; F_{2}). □

Theorem 12.7. If R^{n} is a real division algebra then n is a power of 2.

Proof. See [6, 3.20] - well, you need to know the Künneth theorem. □

Theorem 12.8 (The Hopf map). The Hopf map S^{3} → S^{2} is not nullhomotopic.

Proof. The Hopf map is the attaching map for the 4-cell in CP^{2}. The cohomology rings show that CP^{2} and S^{2} ∨ S^{4} are not homotopy equivalent. So the attaching map in CP^{2} is not homotopic to the constant map. □

13. Colimits of modules

Definition 13.1. A (right) directed set is a partially ordered nonempty set I with the property that for any pair of elements i, j ∈ I there is a third element k ∈ I such that both i ≤ k and j ≤ k. A directed system of R-modules over the directed set I is a functor F from I into the category of R-modules. A subset J of a directed set I is cofinal if for any element i ∈ I there is an element j ∈ J such that i ≤ j.

Definition 13.2. The colimit of a directed system F of R-modules is an R-module L with Rmodule homomorphisms f_{i}: F_{i} → L such that

\xymatrix{F_{i} \ar[dr]_{f_{i}} \ar[rr]^{F_{i<j}} &  & F_{j} \ar[dl]^{f_{j}} \\  & L}

commutes and such that any other R-module A with homomorphisms a_{i}: F_{i} → A such that a_{j}F_{i<j} = a_{i} exists a unique R-module homomorphism L → A such that

(13.3)  \xymatrix{F_{i} \ar[dr]_{f_{i}} \ar[ddr]_{a_{i}} \ar[rr]^{F_{i<j}} &  & F_{j} \ar[dl]^{f_{j}} \ar[ddl]^{a_{j}} \\  & L \ar@{-->}[d] \\  & A}

commutes.

The colimit of the directed system F is the universal example of a constant system with a map from F. In particular, the colimit of the constant functor is the value of the functor.

Theorem 13.4 (Existence and uniqueness of colim). Any directed system F of R-modules has a colimit, unique up to isomorphism.

Proof. Let L be the quotient of ⨁_{i∈I} F_{i} by the the submodule generated by all elements of the form x_{i} − F_{i<j}x_{i} for i ≤ j and x_{i} ∈ F_{i}. One can verify that this works. □

The colimit of the system F is denoted colim F. From the above explicit construction of the colimit we get:

Corollary 13.5 (Recognition principle). The map colim F → A arising from the universal property ( 13.3) is an isomorphism if and only if

(1) Every a ∈ A we can find i ∈ I and x_{i} ∈ F_{i} such that a = f_{i}x_{i}

(2) If a_{i}x_{i} = 0 for some i ∈ I and x_{i} ∈ F_{i} then F_{i<j}x_{i} = 0 for some j > i.

If J is a sub-directed set of I then the universal property for the colimit gives a map colim_{J} F |J → colim_{I} F from the colimit of the small set to the colimit over the large set.

Corollary 13.6. If J is cofinal then J is directed and colim_{J} F |J → colim_{I} F is an isomorphism for any directed system F.

In particular, if I has a maximal element m (with i ≤ m for all i ∈ I), then F_{m} → colim_{I} F is an isomorphism.

Theorem 13.7. [1] The functor colim is exact.

This means that if F, G, and H are directed systems over the same directed set I and if for each i ∈ I we have exact sequences F_{i} → G_{i} → H_{i}, natural wrt i ∈ I, then in the limit we also have an exact sequence colim F_{i} → colim G_{i} → colim H_{i}. This property implies that if C_{i} is a directed system of chain complexes then the map colim H_{q}(C_{i}) ⟶^{≅} H_{q}(colim C_{i}), induced from C_{i} → colim C_{i}, is an isomorphism, that is: homology commutes with direct limits. Here is an immediate application to topology:

Corollary 13.8. Suppose that the space X = ⋃_{i∈I} X_{i} is the union of subspaces X_{i}, indexed by some directed set I. Suppose that any compact set in X is contained in X_{i} for some i ∈ I. Then the map colim H_{q}(X_{i}) ⟶^{≅} H_{q}(X) is an isomorphism.

Proof. The assumption on compact subsets, applied to supports of singular chains in X, and the recognition principle, show that colim C_{q}(X_{i}) → C_{q}(X) is an isomorphism. This isomorphism survives to homology by exactness. □

Example 13.9. The commutative diagram

[figure: 0; Z; p^{n}; Z; p; Z/p^{n}; 0; 0; Z; p^{n+1}; Z; Z/p^{n+1}; 0]

shows a short exact sequences of direct systems of N. The colimit of the first, constant, system is Z, the colimit of the middle system is Z[1/p] with maps f_{n}: Z → Z[1/p] given by f_{n}(1) = p^{−n}. Let Z/p^{∞} denote the limit of the third system. The short exact sequence

0 → Z → Z[1/p] → Z/p^{∞} → 0

shows that Z/p^{∞} = Z[1/p]/Z.

13.10. Cohomology with compact support. Let M be a manifold and R a ring. As a notational convention we write H^{q}(M|A; R) for H^{q}(M, M − A; R), just as we did for homology (11.1). When A_{1} ⊂ A_{2}, there is an R-module extension homomorphism e_{A_{2}}^{A_{1}} : H^{q}(M|A_{1}) → H^{q}(M|A_{2}) (in the same direction as the inclusion) induced from the inclusion (M, M − A_{2}) ⊂ (M, M − A_{1}).

The set K of compact subspaces of M, ordered by inclusion, is a directed set and the functor that takes K ∈ K to H^{q}(M|K; R) is a directed system of R-modules. We define cohomology with compact support

H_{c}^{q}(M; R) = colim_{K∈K} H^{q}(M|K)

as the colimit of the local cohomology groups with support in compact subsets. If M is compact H_{c}^{q}(M) = H^{q}(M) because M is maximal in K.

For any open submanifold U of M define H_{c}^{q}(U) → H_{c}^{q}(M) (note the direction of the arrow) using the maps (13.11)

H_{c}^{q}(U) = colim_{K⊂U} H^{q}(U|K) ⟵^{≅} colim_{K⊂U} H^{q}(M|K) → colim_{K⊂M} H^{q}(M|K) = H_{c}^{q}(M)

where an excision isomorphism occurs (cf 11.2).

Lemma 13.12. Let U and V be open submanifolds of the manifold M. Then there is an exact Mayer-Vietoris sequence

⋯ → H_{c}^{q}(U ∩ V) → H_{c}^{q}(U) ⊕ H_{c}^{q}(V) → H_{c}^{q}(U ∪ V) → H_{c}^{q+1}(U ∩ V) → ⋯

in cohomology with compact support.

Proof. Consider the case M = U ∪ V. Suppose that K ⊂ U and L ⊂ V are compact subsets. Then there is relative Mayer-Vietoris sequence

[figure: ⋯; H^{q}(M|K ∩ L); H^{q}(M|K) ⊕ H^{q}(M|L); H^{q}(M|K ∪ L); ⋯; H^{q}(U ∩ V |K ∩ L); ≅; H^{q}(U|K) ⊕ H^{q}(V |L); ≅; H^{q}(U ∪ V |K ∪ L)]

where the vertical arrows are excision isomorphisms. Pass to the limit (over the directed set K_{U} × K_{V} and use local compactness to see that any compact subset of U ∪ V is of the form K ∪ L) using that colim is exact. □

A slight variation of a construction above gives a map H_{c}^{q}(U) → H̃^{q}(M), U ≠ M, defined as

H_{c}^{q}(U) = colim_{K⊂U} H^{q}(U|K) ⟵^{≅} colim_{K⊂U} H^{q}(M|K) → colim_{K⊂U} H^{q}(M|M − x) = H^{q}(M, x)

if x is a point outside U.

Example 13.13. (Cohomology with compact support of R^{n}) When R^{n} = U ⊂ S^{n} = R^{n} ∪ {∞} and K is replaced by the cofinal system of discs D^{n}(0, r), the above map is an isomorphism

H_{c}^{q}(R^{n}; R) ≅ H^{q}(S^{n}, ∞; R) ≅ \begin{cases}R & q = n \\ 0 & q ≠ n \end{cases}

because H^{q}(S^{n}|D^{n}(0, r)) ⟶^{≅} H^{q}(S^{n}|S^{n} − {∞}) for all r > 0. Alternatively, the natural transformation H^{q}(R^{n}|0) → H^{q}(R^{n}|D^{n}(0, r)) from the constant functor implies that H^{q}(R^{n}|0) → H_{c}^{q}(R^{n}) is an isomorphism where H^{q}(R^{n}|0) ≅ Hom(H_{q}(R^{n}|0), R) by UCT.

14. Poincaré duality for noncompact oriented manifolds

Let M be any R-oriented manifold. For every compact subset K of M we have (11.10) an R-orientation class μ_{K} ∈ H_{n}(M|K; R) along K. Naturality of relative cap products (10.5) with these orientation classes show that

\xymatrix{H^{q}(M|K; R) \ar[dr]_{μ_{K}∩−} \ar[rr]^{e_{L}^{K}} &  & H^{q}(M|L; R) \ar[dl]^{μ_{L}∩−} \\  & H_{n−q}(M; R)}

commutes: Suppose that K ⊂ L so that (M, M − K) ⊃ (M, M − L). Let i be the inclusion. Then id = i_{∗}: H_{∗}(M) → H_{∗}(M) is an H^{∗}(M|K)-module homomorphism so that μ_{L} ∩ϕ = i_{∗}(μ_{L} ∩i^{∗}ϕ) = i_{∗}(μ_{L}) ∩ ϕ = μ_{K} ∩ ϕ for all ϕ ∈ H^{∗}(M|K). By the universal property for colimits (13.3) there is an R-module homomorphism PD: H_{c}^{q}(M; R) → H_{n−q}(M; R).

Theorem 14.1 (Poincaré duality). Let M be an oriented n-manifold. The R-module homomorphism

PD: H_{c}^{q}(M; R) → H_{n−q}(M; R)

is an isomorphism.

The proof is divided into several steps. The main ingredient a Mayer-Vietoris argument.

Lemma 14.2. If Theorem 14.1 is true for the open subsets U, V and U ∩ V of M then it also true for U ∪ V.

Proof. Consider the diagram

\xymatrix{⋯ \ar[r] & H_{c}^{q}(U ∩ V) \ar[r] \ar[d]^{PD} & H_{c}^{q}(U) ⊕ H_{c}^{q}(V) \ar[r] \ar[d]^{PD} & H_{c}^{q}(U ∪ V) \ar[r] \ar[d]^{PD} & H_{c}^{q+1}(U ∩ V) \ar[r] \ar[d]^{PD} & ⋯ \\ ⋯ \ar[r] & H_{n−q}(U ∩ V) \ar[r] & H_{n−q}(U) ⊕ H_{n−q}(V) \ar[r] & H_{n−q}(U ∩ V) \ar[r] & H_{n−q−1}(U ∩ V) \ar[r] & ⋯}

where the top row is the exact sequence from (13.12), the bottom row is the Mayer-Vietoris sequence for (U∪V, U, V), and the vertical maps are Poincaré duality maps. A longer and nontrivial argument shows that this diagram commutes up to sign. Now use the 5-lemma. □

Proof of Theorem 14.1. (1) Let (U_{i})_{i∈I} be a collection of open sets U_{i} ⊂ M linearly ordered by inclusion. Assume that (14.1) is true for each of the open sets U_{i}. The colimit of the isomorphisms PD: H_{c}^{q}(U_{i}) → H_{n−q}(U_{i}) is an isomorphism colim H_{c}^{q}(U_{i}) ⟶^{≅} colim H_{n−q}(U_{i}). It is not hard to see that colim H_{c}^{q}(U_{i}) ≅ H_{c}^{q}(⋃ U_{i}) and that colim H_{n−q}(U_{i}) ≅ H_{n−q}(⋃ U_{i}). This is Poincaré duality for ⋃ U_{i}.

(2) The theorem is true for any open and convex subset of R^{n} for such an open set is homeomorphic to R^{n} and the observations of (13.13) apply. By (14.2) it is also true for any open subset of R^{n}.

(3) Consider the partially ordered collection of all opens subsets of M that enjoy Poincaré duality. By (2) this is a nonempty collection and by (1) any linearly ordered subset has an upper bound. Zorn’s lemma now says that there are maximal elements. Such a maximal element must equal M for by (2) and (14.2) we can always enlarge any open proper open subset of M with Poincaré duality to a larger open subset with Poincaré duality. □

15. Alexander duality

Let M be a manifold and A a closed subset of M. Order the open neighborhoods of A be reverse inclusion so that U_{0} ≤ U_{1} means that U_{0} ⊃ U_{1} ⊃ A. There is then a commuative diagram

\xymatrix{H^{q}(U_{0}; R) \ar[rr]^{i^{∗}} \ar[dr] &  & H^{q}(U_{1}; R) \ar[dl] \\  & H^{q}(A; R)}

and U → H^{q}(U) is a direct system over the directed set of open neighborhoods of A. The Alexander–Čech cohomology group of A ⊂ M is

Ȟ^{q}(A; R) = colim_{U⊃A} H^{q}(U; R)

where the colimit is taken over the directed set of all open neighborhoods U of A. There are homomorphisms

(15.1)  H^{q}(M; R) → Ȟ^{q}(A; R) → H^{q}(A; R)

because M is an open neighborhood of A and because of the universal property (13.3) of the colimit.

In some cases the value of this cohomology group depends only on A and not on M (nor the embedding of A into M).

Proposition 15.2. If A and M are compact manifolds (more generally, ANRs [7, pp 25–32] [4]) then Ȟ^{q}(A; R) → H^{q}(A; R) is an isomorphism.

We shall now embed H^{q}(M) → Ȟ^{q}(A) into a long exact sequence when M is compact. Let U ⊃ A be any open neighborhood of A. Consider the connecting homomorphism δ: H^{q}(U) → H^{q}(M, U). The codomain is H^{q+1}(M, U) = H^{q+1}(M|M − U) = H^{q+1}(M − A|M − U) (11.2). Now note that the direct systems of open neighborhoods of A (ordered by reverse inclusion) and compact subsets of the complement M − A are isomorphic by the map U → M − U. So the colimit of these connecting homomorphisms is a homomorphism of degree +1

δ: Ȟ^{q}(A) = colim_{U⊃A} H^{q}(U) → colim_{U⊃A} H^{q+1}(M − A|M − U) = H_{c}^{q+1}(M − A)

from Alexander–Čech cohomology of A to cohomology with compact support of the complement.

Proposition 15.3. Let M be a compact manifold and A a closed subset of M. Then the sequence

⋯ → H_{c}^{q}(M − A) ⟶^{(13.11)} H^{q}(M) ⟶^{(15.1)} Ȟ^{q}(A) ⟶^{δ} H_{c}^{q+1}(M − A) → ⋯

is exact (with coefficiants in any ring R).

If M is both compact and R-oriented let μ_{M} ∈ H_{n}(M; R) be the orientation class (11.10). Let A ⊂ M be a closed, hence compact, subset. For any open neighborhood U of A, there is cap product (10.5)

H_{n}(U|A) × H^{q}(U) ⟶^{∩} H_{n−q}(U|A)

Under the excision isomorphism H_{∗}(U|A) ≅ H_{∗}(M|A) this is a cap product

H_{n}(M|A) × H^{q}(U) ⟶^{∩} H_{n−q}(M|A)

Capping with with r_{A}^{M}μ_{M} in H_{n}(M|A) gives an R-linear homomorphism μ_{M} ∩ −: H^{q}(U) → H_{n−q}(M|A). When U_{0} ⊃ U_{1} ⊃ A, naturality of the cap product makes the diagram

\xymatrix{H^{q}(U_{0}; R) \ar[rr]^{i^{∗}} \ar[dr]_{μ_{M} ∩−} &  & H^{q}(U_{1}; R) \ar[dl]^{μ_{M} ∩−} \\  & H_{n−q}(M|A; R)}

commute so by the universal property for colimits (13.3) these cap products determine an R-module homomorphism AD: Ȟ^{q}(A; R) → H_{n−q}(M|A; R).

Theorem 15.4 (Alexander duality). Let M be a compact R-oriented n-manifold and A ⊂ M a closed subset. Then the R-module homomorphism

AD: Ȟ^{q}(A; R) → H_{n−q}(M|A; R)

is an isomorphism.

Proof. The diagram

\xymatrix{⋯ \ar[r] & H_{c}^{q}(M − A) \ar[r] \ar[d]^{PD}_{≅} & H^{q}(M) \ar[r] \ar[d]^{PD}_{≅} & Ȟ^{q}(A; R) \ar[r] \ar[d]^{AD} & ⋯ \\ ⋯ \ar[r] & H_{n−q}(M − A) \ar[r] & H_{n−q}(M) \ar[r] & H_{n−q}(M|A; R) \ar[r] & ⋯}

commutes up to sign. The top row is the exact sequence of (15.3) and the bottom row is exact sequence of the pair (M, M − A). The 5-lemma implies that AD is an isomorphism. □

Corollary 15.5. Let A ⊂ R^{n+1} be a compact n-manifold (or ANR) embedded in R^{n+1}. Then

H^{q}(A; R) ≅ H̃_{n−q}(R^{n+1} − A; R)

for all commutative rings R.

Proof. Since A is a compact manifold, Ȟ^{q}(A) ≅ H^{q}(A). Apply Alexander duality (15.4) to the embedding A ⊂ R^{n+1} ⊂ S^{n+1} and get

Ȟ^{q}(A) ≅^{AD} H_{n+1−q}(S^{n+1}|A) ≅^{exc} H_{n+1−q}(R^{n+1}|A) ≅ H̃_{n−q}(R^{n+1} − A; R)

where we use that R^{n+1} is contractible for the last isomorphism. □

We can now show a vast generalization of (part of) the Jordan curve theorem.

Theorem 15.6 (General Separation Theorem). Let A ⊂ R^{n+1} be a compact n-manifold embedded in R^{n+1}. If A has k components then the complement R^{n+1} − A has k + 1 components.

Proof. Using both Poincaré (14.1) and Alexander Duality (15.4) with F_{2}-coefficients we get

H_{0}(A; F_{2}) ≅^{PD} H^{n}(A; F_{2}) ≅^{AD} H̃_{0}(R^{n+1} − A; F_{2})

We use that any manifold is F_{2}-orientable. □

Corollary 15.7. A compact nonorientable n-manifold cannot embed in R^{n+1}.

Proof. If the compact manifold A embeds in R^{n+1} then

H^{n}(A; Z) ≅^{AD} H̃_{0}(R^{n+1} − A; Z)

is a free nontrivial (15.6) abelian group. On the other hand, if A is nonorientable then H_{n}(A; Z) = 0 by (11.12). The UCT (10.6) then implies that Ext_{Z}(H_{n−1}(A; Z), Z) ≅ H^{n}(A; Z) (where H_{n−1}(A; Z) is a finitely generated abelian group) is a torsion group. This is a contradiction. □

In particular, the nonorientable compact surfaces cannot embed in R^{3}. Do they embed in R^{4}?

15.8. Linking number. Suppose that A_{1} and A_{2} are disjoint compact submanifolds of R^{n}. The linking number is the bilinear map

L: H̃_{p}(A_{1}; Z) × H_{n−p−1}(A_{2}; Z) → H̃_{p}(R^{n} − A_{2}; Z) × H_{n−p−1}(A_{2}; Z)

⟵_{≅}^{(15.5)×id} H^{n−p−1}(A_{2}; Z) × H_{n−p−1}(A_{2}; Z) ⟶^{⟨·,·⟩} Z

For instance, A_{1} and A_{2} could be knots in R^{3}.

15.9. Invariance of Domain. We show a classical result about open subsets of R^{n}. We first show that seen through the eyes of homology any embedding of an n-disc or an (n − 1)-sphere in S^{n} looks like the standard embeddings.

Lemma 15.10. Let S^{r−1} ⊂ S^{n} and D^{r} ⊂ S^{n} where n ≥ 1 and r > 1. Then (1) H̃_{∗}(S^{n} − D^{r}) = 0

(2) H̃_{∗}(S^{n} − S^{r−1}) = H̃_{∗}(S^{r})

Proof. We shall only prove the first case since the second case is similar (at least when n ≥ 2). Since the r-disc D^{r} is an ANR (even an AR by Tietze) Alexander duality, H^{q}(D^{r}) ≅ H_{n−q}(S^{n}|D^{r}), says that H_{n}(S^{n}|D^{r}) ≅ Z is the only nontrivial homology group for the pair (S^{n}, S^{n} − D^{r}). In the exact sequence for the pair (S^{n}, S^{n} − D^{r}),

H_{n+1}(S^{n}|D^{r}) = 0 → H̃_{n}(S^{n} − D^{r}) → H̃_{n}(S^{n}) → H̃_{n}(S^{n}|D^{r}) → H_{n−1}(S^{n} − D^{r}) → 0 = H̃_{n−1}(S^{n})

the homomorphism H_{n}(S^{n}) → H_{n}(S^{n}|D^{r}) is an isomorphism (since H_{n}(S^{n}) → H_{n}(S^{n}|D^{r}) → H_{n}(S^{n}|x), x ∉ D^{r} is an isomorphism since S^{n} is orientable). Thus H_{i}(S^{n}−D^{r}) = 0 for i = n−1, n. The rest of the homology groups in positive degrees are also 0 by exactness. □

For instance, all knot complements S^{3} − S^{1} have the homology of S^{1} (but not the same fundamental group). The complement S^{n} −S^{n−1} of any copy of an (n−1)-sphere in an n-sphere has two (path-)components, each with the homology of a point. One might guess that these components are discs. This is indeed true when n = 2 (Schoenflies) but not when n = 3 (Alexander horn sphere).

Corollary 15.11. Let f: D^{n} → S^{n} or (f: D^{n} → R^{n}) be an embedding of an n-disc into the nsphere (or Euclidean n-space). Then f(D^{n} − S^{n−1}) is open.

Proof. It is enough to prove the corollary in case of S^{n} since R^{n} is an open subset of S^{n}. According to (15.10) the complement S^{n} − S^{n−1} has two path-components, U_{0} and U_{1}. Since the open subset S^{n} − S^{n−1} is locally path-connected, these path-components are open [8, §19]. We may write S^{n} − S^{n−1} in two ways

U_{0} ∪ U_{1} = S^{n} − S^{n−1} = (S^{n} − D^{n}) ∪ (D^{n} − S^{n−1})

as a disjoimt union. According to (15.10) again S^{n} − D^{n} is path-connected, and also D^{n} − S^{n−1} is path-connected by general topology. But then S^{n}−D^{n} is contained in one of the path-components, say U_{0}. The path-connected space D^{n}−S^{n−1} is contained in either U_{0} or U_{1}. It cannot be contained in U_{0} for then U_{1} would be empty. Thus S^{n} − D^{n} ⊂ U_{0} and D^{n} − S^{n−1} ⊂ U_{1}. In fact, these two inclusions are equalities since S^{n} − D^{n} and D^{n} − S^{n−1} together fill up S^{n} − S^{n−1}. □

Theorem 15.12 (Invariance of Domain). Let U be an open subset of R^{n} and f: U → R^{n} an injective continuous map. Then f(U) is open and f: U → f(U) is a homeomorphism.

Proof. The open space U is a union of (scaled) copies of open discs D^{n}−S^{n−1}. Since the restriction of f to each compact D^{n} is an embedding, the image f(D^{n} − S^{n−1}) is open in R^{n} by (15.11). Thus f(U) is a union of open sets so it is open and f is an open map. □

This shows that, for subsets of R^{n}, the property of being open in R^{n} is a topological invariant:

\xymatrix{U \ar@{^{(}->}[r] \ar[dd]_{f}^{≃} & R^{n} \\  &  & U open ⟹ V open \\ V \ar@{^{(}->}[r] & R^{n}}

This is not true in general [11, §18].

Corollary 15.13. Let f: M → N be an embedding of a compact n-manifold M into a connected n-manifold N. Then f is a homeomorphism.

Proof. Since M is compact and N is Hausdorff the image f(M) is closed. By an argument as above, f(M) is open. Since N is connected, f(M) = N. Thus f is a bijection and hence a homeomorphism since M is compact and N is Hausdorff. □

FROM SINGULAR CHAINS TO ALEXANDER DUALITY 41

See [6, 2B.4] for an application of (15.13) to real division algebras. Corollary 15.14. A compact n-manifold cannot embed in R^{n}. Proof. If the compact n-manifold M embeds in the connected manifold R^{n} then M = R^{n} by (15.13). But this is absurd since M is compact and R^{n} noncompact. □

For example, S^{n} does not embed in R^{n}. It follows that R^{n} cannot contain a subspace homeomorphic to R^{m} for m > n for then it would also contain a copy of S^{n} ⊂ R^{m}.

References

[1] N. Bourbaki, Algèbre, Chp. 2, 3rd ed., Hermann, Paris, 1962.

[2] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1982. MR 83k:20002

[3] Albrecht Dold, Lectures on algebraic topology, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 200, Springer-Verlag, Berlin, 1980. MR 82c:55001

[4] James Dugundji, Topology, Allyn and Bacon Inc., Boston, Mass., 1966. MR 33 #1824

[5] Andrzej Granas and James Dugundji, Fixed point theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR 2004d:58012

[6] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 2002k:55001

[7] Sze-tsen Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York, 1959. MR 21 #5186

[8] Jesper M. Møller, General topology, http://www.math.ku.dk/ moller/e03/3gt/notes/gtnotes.dvi.

[9] James R. Munkres, Topology. Second edition, Prentice-Hall Inc., Englewood Cliffs, N.J., 2000. MR 57 #4063 [10] Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics,

vol. 38, Cambridge University Press, Cambridge, 1994. MR 95f:18001

[11] Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 41 #9173

Matematisk Institut, Universitetsparken 5, DK–2100 København

  E-mail address: moller@math.ku.dk