Algebraic Topology Spring Semester 2004

Practical information from SIS. The course will be taught in English.

Notes

Homotopy theory for beginners

Classification of covering maps (pdf)

From singular chains to Alexander duality (pdf)

Latest news

June 1: I have finished grading your exercises.

April 22:Q: Given a (noncontractible) space X does there always exist a (nontrivial) group G that annihilates X in the sense that H_*(X;G)=G.
A: No, if X=S^n is the n-sphere, then H_n(S^n;G)=G for all G (Morten)
Q: What if one allows chains as coefficients? A: Answer (Morten)

April 20: The dead-line for the homology exercises has been moved to April 29.

March 26: I would like to cancel the lecture Tuesday April 6 and instead hold a lecture Tuesday April 13.

March 24: Computational Topology

March 10: Here is a hint for 1.3.21: The space X is the union along a circle, S¹, of a Möbius band, MB, and a torus, T. The fundamental group of X has two generators, a and c, and one relation, a and c^2 commute. The universal covering space of X can be built from (several copies of) the universal covering spaces of its two pieces joined along the universal covering space of their intersection. It is a good strategy first to think about these questions: What is universal covering space of the Möbius band? What is the lift of S¹ as a loop in the Möbius band? What is the universal covering space of the torus? What is the lift of S¹ as a loop in T? How can we possibly put these two universal covering spaces together?

March 9: You will get 100 points for all exercises minus 1.3.21 and you will get an extra 25 points if you do 1.3.21.

Feb 19: Tuesday and Thursday afternoons are office hours. I will be in my office and you are invited to drop by for a shorter or longer topology chat.

Feb 18: The book is again available from the bookstore.

Jan 20: Since there are now 30 students that have signed up for this course I urged the institute to hire a teaching assistant. The application was turned down.

Textbook

Our textbook will be

Allen Hatcher: Algebraic Topology. Cambridge University Press.

The book is available from Universitetsbogladen for DKK 335 (minus student discount) or directly from Allen Hatcher. The author also maintains a site with updates.

Course Plan

The plan is to read Chp 0 - 3 of Hatcher's book

The fundamental group (and introduction) (5 weeks)

Date	Lecture	Topics	Exercises
03.02	Chp 0	Abstract homotopy theory Homotopy, deformation retraction, CW-complex. Comments (dvi, pdf) to Chp 0	0.17, 0.20
05.02	Chp 0	Homotopy extension property (HEP)
10.02	1.1 - 1.8	Paths, homotopy of paths, loops. The fundamental group of a space Brouwer's fixed point thm (Munkres §55)	1.1.6, 1.1.9, 1.1.16 (e) and (f), 1.1.18
12.02	1.8 - 1.16	Covering spaces: Unique path lifting and unique HLP (cf. Munkres §54) The fundamental group of the circle Fundamental thm of algebra (Munkres §56)
17.02	1.14, 1.17 - 1.20	Borsuk- Ulam thm (Munkres §57) The fundamental group of the n-sphere Free coproducts of groups
19.02	1.21 - 1.29	van Kampen (cf. Munkres p. 426) The double mapping cylinder Torus knot complements (see Rolfsen, Knots and Links, p 52, p 327, Chp 6)	1.2.14 1.2.16 (The surface is the union of a nested sequence of subspaces)
24.02	1.30 - 1.38	The fundamental group of a CW-complex A lifting criterion The universal covering space	1.3.12 (The group G(a,b\|a²,b²,(ab)^4) is dihedral of order 8) 1.3.21, 1.3.30
26.02	1.36 - 1.44	Classification of covering spaces Deck transformations. Another note on covering spaces.
02.03	1.3 -1.4	Classification of covering spaces (continued)
04.03	1.3 - 1.4	The universal covering space Cayley graphs and complexes

Thursday March 11: Hand in your answers to the exercises in covering spaces - You will get 100 points for all exercises minus 1.3.21 and you will get extra 25 points if you do 1.3.21.

Singular homology (5 weeks)

Date	Lecture	Topics	Exercises
09.03	2.1 - 2.5	$\Delta$ -complexes (revision) Simplicial homology Singular homology of general spaces	2.1.1, 2.1.6 Hint: See Hartley, Hawkes, Rings, Modules and Linear Algebra, Thm 7.10 and pp 112-113
11.03	2.5 - 2.12	Homotopy invariance	2.1.11 Submit your fundamental group and covering space exercises!
16.03	2.13 - 2.17	Discussion of exercises Relative homology	2.1.9
18.03	2.18 - 2.22	Discussion of exercises Relative homology	2.1.17 (compare with 1.2.9)
23.03	2.27	Excision (see Comments to Chp 2 and Chp 3) Subdivision of linear chains	2.1.26
25.03	2.2	Excision (continued) Homology of quotient spaces Homology of spheres	2.2.9
30.03	2.22 - 2.26	The equivalence of simplicial and singular homology	2.2.11
01.04	2.28 - 2.33	The degree of a self-map of a sphere Computation of degree from local degrees	2.2.23
06.04	No lecture	Fall Break
08.04	No lecture	Fall Break
13.04	Chp 2	Cellular homology Homology of compact orientable surfaces Euler characteristic Homology with coefficients	2.2.28
15.04	Chp 2	Homology of compact nonorientable surfaces Homology of projective spaces Moore spaces The Mayer-Vietoris sequence	2.2.41

Thursday April 29: Hand in your answers to the exercises in homology.

Singular cohomology (5 weeks)

Date	Lecture	Topics	Exercises
20.04	Chp 3	Cohomology UCT
22.04	3.1 - 3.3	Singular, cellular, and simplicial cochains and cohomology groups	3.1.3
27.04	3.3 - 3.5	Mayer-Vietoris	3.1.11 (Cellular maps of CW-complexes induce (co)chain homomorphisms of cellular (co)chain complexes)
29.04		Cup and cap products
04.05		Some cohomology rings Commutativity of the cup product
06.05	3.6 - 3.11	Orientation of manifolds	3.2.1 (See Example 3.13; local degree 2.30 is helpful)
11.05		Poincaré duality for compact manifolds Cohomology of projective spaces
13.05		Cohomology with compact support Poincaré duality for noncompact manifolds
18.05		Alexander duality, Alexander's horned sphere Generalized Jordan- Brouwer theorem Invariance of Domain	3.2.2 (Lysternik-Schnirelmann category)
20.05	No lecture	Ascension Day	3.2.3, 3.2.7

Tuesday June 01: Hand in your answers to the exercises in cohomology.

Prerequisites

You should know basic general topology: Topological spaces, continuous maps, (locally) connected topological space, (locally) compact topological space, quotient space, manifold. You may use my notes (pdf, dvi) (based on Munkres' book) as a reference. You should also know very basic algebra: Group, ring, vector space, module. You can get an idea of the required prerequisites by leafing through Hatcher's book.

Credits

The course is worth 10 ECTS. To earn these credits I expect you to actively participate in the course. There will be N written assignments during the course. You are (more than) welcome to work on and discuss the assignments with other students in the course but you should hand in your own answer. If you have successfully answered 50% of the assignments, you have passed the course.

Books

Bott and Tu: Differential forms in algebraic topology.
Bredon: Geometry and Topology.
Dold: Lectures on Algebraic Topology.
Greenberg and Harper: Algebraic Topology.
Massey: A basic course in algebraic topology.
Jiri Matousek. Using the Borsuk-Ulam Theorem; Lectures on Topological Methods in Combinatorics and Geometry (Springer 2002).
May: A concise course in algebraic topology.
Rotman: An introduction to algebraic topology.
Spanier: Algebraic topology.
Switzer: Algebraic topology - homology and homotopy.
Whitehead. Elements of homotopy theory.
George K. Francis: A topological Picture Book.

Jesper Michael Møller

Last modified: Tue Feb 12 08:42:22 CET 2008