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 $f:(X,A)\to (Y,B)$ be a continuous map. Assume that $f:X\to Y$ and $f\vert _A: A\to B$ 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 $X={\mathbb {RP}}^{3}\times L(4,1)$ be the product of the projective space ${\mathbb {RP}}^3=S^3/{\mathbb Z}_2$ with the lens space $L(4,1)=S^3/{\mathbb Z}_4$.
  • 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 $X=G_3({\mathbb R}^6)$ be the Grassmanian of 3-planes in ${\mathbb R}^6$.
  • 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.