Topology II - Final Exam

NORTHEASTERN UNIVERSITY
DEPARTMENT OF MATHEMATICS

Prof. Alex Suciu MTH 3107 - TOPOLOGY II Winter 1998

Take-Home Final Exam

Due Tuesday, March 17

Instructions: Do at least 6 of the following 10 problems. Give complete proofs or justifications for each statement you make. Show all your work.

Let X=[0,1] and $A=\{1,\frac{1}{2},\frac{1}{3},\dots ,\frac{1}{n}, \dots\}\cup \{0\}$ . Is H₁(X,A) isomorphic to $\widetilde{H}_1(X/A)$ ? Explain why, or why not.
- Let X and Y be two finite CW complexes. Show that $\chi(X\times Y)=\chi(X)\chi(Y)$ .
- Let A and B be two subcomplexes of X such that $X=A\cup B$ . Show that $\chi(X)=\chi(A)+\chi(B)-\chi(A\cap B)$ .
Let be a continuous map. Assume that and are homotopy equivalences.
- Show that $f_*:H_*(X,A)\to H_*(Y,B)$ is an isomorphism.
- Give an example where $H_*(X,A)\not\cong H_*(Y,B)$ , although X is homotopy equivalent to Y, and A is homotopy equivalent to B.
Let be the product of the projective space with the lens space .
- Find a CW-decomposition of X.
- Determine the chain complex $(C_{\bullet}(X),d)$ associated to that cell decomposition.
- Compute the homology groups H_*(X).
Let be the Grassmanian of 3-planes in .
- Find a CW-decomposition of X.
- Determine the chain complex $(C_{\bullet}(X),d)$ associated to that cell decomposition.
- Compute the homology groups H_*(X).
Let T_g be the orientable surface of genus g.
- Describe a 2-fold cover $p: T_3 \to T_2$ .
- Compute $p_*:H_1(T_3) \to H_1(T_2)$ .
- Compute $p_*:H_2(T_3) \to H_2(T_2)$ .
Let $X=({\mathbb C}^2\setminus \{0\})/(z_1,z_2)\thicksim (2z_1,2z_2)$ . Compute H_*(X).
Show that ${\mathbb {RP}}^2$ is not a retract of ${\mathbb {RP}}^3$ .
Problem 1, in Bredon's book, p. 259.
Problem 5, in Bredon's book, p. 259.

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Created by Alexandru I. Suciu, Sat Mar 21, 1998

alexsuciu@neu.edu

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