Eksamen august 2005

JMM

Matematik 3GT

Books, notes, calculators, and computers are allowed at this three hour written exam. You may write your answers in pencil. An unjustified answer counts as no answer.

Problem 1 (30%)

In R^{2} with the standard topology, let B = {x × y ∈ R^{2} | x^{2} + y^{2} < 1} be the open unit ball around 0 × 0. Consider a continuous path u: [0, 1] → R^{2} starting and ending at the point u(0) = 2 × 0 = u(1) outside B.

(1) Show that u^{−1}(B) is the union of at most countably many disjoint open intervals (a_{i}, b_{i}).

(2) Show that u^{−1}(0×0) is contained in the union of finitely many of the open intervals (a_{i}, b_{i}).

(3) Show that u maps the end-points, a_{i} and b_{i}, of the intervals (a_{i}, b_{i}) to the unit circle \overline{B} −B.

Problem 2 (20%) Let Δ ⊂ R^{3} be the subspace of R^{3} (with the standard topology) consisting of the four points (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1) and the straight line segments between these points. Let

[figure: •; •; •; •]

f: Δ → Δ be a homeomorphism. How many possibilities are there for f(0, 0, 0)? Can all these possibilities actually be realized by some homeomorphism?

Problem 3 (50%) The (standard) circle S^{1} is the subspace of R^{2} (with standard topology) given by

S^{1} = {x × y ∈ R^{2} | x^{2} + y^{2} = 1}

Let e: R → S^{1} be the continuous map given by e(t) = cos(2πt) × sin(2πt), t ∈ R.

The Warsaw circle is the subspace W of R^{2} (with standard topology) obtained as the union of the topologist’s sine curve

\overline{S} = (\{0\} × [−1, 1]) ∪ \{x × sin(1/x) | 0 < x ≤ 1/π\}

with the image of an injective continuous curve c: [0, 1] → R^{2} that is disjoint from \overline{S} except that c(0) = 0 × 0 and c(1) = 1/π × 0.

Figure 1. The Warsaw circle

(1) Explain that S^{1} and W are compact Hausdorff spaces.

(2) Explain that there is a compactification c_{1} : R → W with remainder W −c_{1}(R) homeomorphic to the interval [−1, 1] and a compactification c_{2} : R → S^{1} with remainder S^{1} − c_{2}(R) equal to a point.

(3) Find a quotient map q: W → S^{1} such that q ∘ c_{1} = c_{2} inducing a homeomorphism between

W/({0} × [−1, 1]) and S^{1}. \xymatrix{ & R \ar[dr]_{e} \\ W \ar@{.>}[ur]^{s_{1}} \ar[rr]_{q} &  & S^{1} \ar@{.>}[ul]_{s_{2}} \\  & R \ar[ul]^{c_{1}} \ar[ur]_{c_{2}}}

(4) Show that there does not exist a continuous map s_{2} : S^{1} → R such that e∘s_{2} is the identity map of S^{1}. (You may use without proof that S^{1} does not embed into R.)

(5) Does there exist a continuous map s_{1} : W → R such that e ∘ s_{1} = q?

(THE END)