Topology II - Final Exam

NORTHEASTERN UNIVERSITY
DEPARTMENT OF MATHEMATICS

Prof. Alex Suciu MTH 3107 - TOPOLOGY II Winter 1999

Take-Home Final Exam

Due Monday, March 22

Instructions: Do at least 5 of the following 6 problems. Give complete proofs or justifications for each statement you make. Show all your work.

Let X be a space. Show that:
- $\widetilde{H}_n(X;{\mathbb Z})=0$ for all n if and only if $\widetilde{H}_n(X;{\mathbb Q})=0$ and $\widetilde{H}_n(X;{\mathbb Z}_p)=0$ for all n and all primes p.
- $f: X\to Y$ induces isomorphisms in $H_*(-;{\mathbb Z})$ if and only if it induces isomorphisms in $H_*(-;{\mathbb Q})$ and $H_*(-;{\mathbb Z}_p)$ for all primes p.
Prove the following theorem of Borsuk: If $f:S^n\to S^n$ commutes with the antipodal map, then f has odd degree.
Remarks:
- There is a direct (and rather long) proof in Bredon's book (Theorem 20.6, pp. 244-245). Use instead the Borsuk-Ulam theorem (see hint to Problem 7, p. 245).
- Note that this is not a homotopy-theoretic result. Indeed, $fa\simeq af$ , for all f (show that!).
Let $X=K\times {\mathbb {RP}}^{3}$ be the product of the Klein bottle with the 3-dimensional projective space.
- Find a CW-decomposition of X.
- Determine the chain complex $(C_{\bullet}(X),d)$ associated to that cell decomposition.
- Compute the (cellular) homology groups H_*(X) and $H_{*}(X; {\mathbb Z}_2)$ .
Recall that, for a space X, and a short exact sequence $0\to G' \xrightarrow{\alpha} G\xrightarrow{\beta} G''\to 0$ of abelian groups, there is an associated long exact in homology,
$\begin{displaymath} \cdots \to H_i(X; G')\xrightarrow{\alpha_*} H_i(X;G)\xrightarrow{beta_*} H_i(X; G'')\xrightarrow{\partial} H_{i-1}(G')\to\cdots \end{displaymath}$
Compute explicitly this homology sequence (the terms and the maps), in case the coefficients sequence is $0\to {\mathbb Z}\xrightarrow{2} {\mathbb Z}\to {\mathbb Z}_2\to 0$ , and
- $X={\mathbb {RP}}^3$ .
- X= K, the Klein bottle.
Let A be an integral $3\times 3$ matrix, and let $f:T^3\to T^3$ be the induced self-map of the 3-torus. Compute the Lefschetz number L(f) in terms of the entries of A.
Let X be a finite simplicial complex, and let G be a finite group acting simplicially on X. Show that, for every $g\in G$ ,
$\begin{displaymath} L(g) = \chi(X^g)\end{displaymath}$
where $X^g=\{x\in X \mid gx=x\}$ .

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Created by Alexandru I. Suciu, Sunday March 14, 1999. Comments: alexsuciu@neu.edu

http://www.math.neu.edu/~suciu/mth3107/top2.final99/index.html