Topology III - Final Exam

NORTHEASTERN UNIVERSITY
DEPARTMENT OF MATHEMATICS

Prof. Alex Suciu MTH 3481 - TOPOLOGY 3 Spring 1998

Take-Home Final Exam

Due Monday, June 15, at 9 AM

Instructions: Do 6 of the following 7 problems. Give complete proofs or justifications for each statement you make. Show all your work.

Let and . Let , given by f(0)=(0,0) and .
- Show that $f_*:\pi_n(X)\to \pi_n(Y)$ is an isomorphism, for all $n\ge 0$ .
- Show that f is not a homotopy-equivalence.
- Does this contradict Whitehead's theorem? Why, or why not?
Let X be a connected, finite CW-complex, with having a non-trivial element of finite order. Let , where is the universal cover of X.
- Show that $\pi_n(X)\cong\pi_n(Y)$ , for all n > 1.
- Show that X is not homotopy-equivalent to Y.
- Does this contradict Whitehead's theorem? Why, or why not?
Let be the composite of the Hopf bundle map and the quotient map , which collapses the 2-skeleton of the 3-torus to a point.
- Show that $f_*=0:\pi_n(T^3)\to \pi_n(S^2)$ , for all $n\ge 0$ .
- Show that $f_*=0:H_n(T^3)\to H_n(S^2)$ , for all $n\ge 0$ .
- And yet f is not homotopic to a constant map.
Let X be a connected CW-complex, with $\pi_i(X)=0$ for 1 < i < n, for some n > 1. Let $h:\pi_n(X)\to H_n(X)$ be the Hurewicz homomorphism. Show that $H_n(X)/h(\pi_n(X)) \cong H_n(K(\pi_1(X),1)$ .
Let G be an abelian group.
- Show that H_n+1(K(G,n))=0, for n > 1.
- Show that there is a Moore space M(G,1) if and only if H₂(K(G,1))=0.
- For what values of n does there exist a Moore space of type $M({\mathbb Z}^n,1)$ ?
Let , with attaching map , and .
- Show that $M({\mathbb Z}_p,2)$ can be chosen so that X and Y have the same 3-skeleta.
- Show that $H^*(X;{\mathbb Z})\cong H^*(Y;{\mathbb Z})$ (as graded rings).
- Show that $H^*(X;{\mathbb Z}_p)\not\cong H^*(Y;{\mathbb Z}_p)$ (as graded rings).
Let G be a group, and let be a sequence of -modules.
- Construct a CW-complex X with $\pi_1(X)=G$ , and $\pi_n(X)=M_n$ (as ${\mathbb Z}G$ -modules).
- If $X=K(G,1)\times Y$ , where $\pi_1(Y)=0$ , show that $\pi_n(X)$ is trivial as a ${\mathbb Z}G$ -module, for all n>1.
- If $X={\mathbb {RP}}^n$ , show that $\pi_n(X)={\mathbb Z}$ is trivial as a ${\mathbb Z}{\mathbb Z}_2$ -module if and only if n is odd.

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Created by Alexandru I. Suciu, Wed Jun 10, 1998

alexsuciu@neu.edu

http://www.math.neu.edu/~suciu/mth3107/top3final/index.html