This is a take-home exam, due Monday, June 15, at 9AM. Do 6 of the following 7 problems. Give complete proofs or justifications for each statement you make. Show all your work.

1. Let Y = {(x,y) ÎR² |  x > 0, y = sin(1/x) or x = 0, -1 £ y £ 1} and X = {0,1}. Let f:X® Y, given by f(0) = (0,0) and f(1) = (1/p,0).
   1. Show that f_*:p_n(X)® p_n(Y) is an isomorphism, for all n ³ 0.
   2. Show that f is not a homotopy-equivalence.
   3. Does this contradict Whitehead's theorem? Why, or why not?

2. Let X be a connected, finite CW-complex, with p₁(X) having a non-trivial element of finite order. Let Y = X^~ x K(p₁(X),1), where X^~ is the universal cover of X.
   1. Show that p_n(X) @ p_n(Y), for all n ³ 0.
   2. Show that X is not homotopy-equivalent to Y.
   3. Does this contradict Whitehead's theorem? Why, or why not?

3. Let f: T³ ® S² be the composite of the Hopf bundle map p:S³® S² and the quotient map q:T³® S³, which collapses the 2-skeleton of the 3-torus to a point.
   1. Show that f_* = 0:p_n(T³)® p_n(S²), for all n ³ 0.
   2. Show that f_* = 0:H_n(T³)® H_n(S²), for all n > 0.
   3. And yet f is not homotopic to a constant map.

4. Let X be a connected CW-complex, with p_i(X) = 0 for 1 < i < n, for some n ³ 2. Let h:p_n(X)® H_n(X) be the Hurewicz homomorphism. Show that H_n(X)/h(p_n(X)) @ H_n(K(p₁(X),1).

5. Let G be an abelian group.
   1. Show that H_n+1(K(G,n)) = 0, for n > 1.
   2. Show that there is a Moore space M(G,1) if and only if H₂(K(G,1)) = 0.
   3. For what values of n does there exist a Moore space of type M(Zⁿ,1)?

6. Let X = CP² U e³, with attaching map S²- ^p ® S² = CP¹ Ì CP², and Y = M(Z_p,2) V S⁴.
   1. Show that M(Z_p,2) can be chosen so that X and Y have the same 3-skeleta.
   2. Show that H^*(X;Z) @ H^*(Y;Z) (as graded rings).
   3. Show that H^*(X;Z_p) is not isomorphic to H^*(Y;Z_p) (as graded rings).

7. Let G be a group, and let {M_n}_n be a sequence of ZG-modules.
   1. Construct a CW-complex X with p₁(X) = G, and p_n(X) = M_n (as ZG-modules).
   2. If X = K(G,1)xY, where p₁(Y) = 0, show that p_n(X) is trivial as a ZG-module, for all n > 1.
   3. If X = RPⁿ, show that p_n(X) = Z is trivial as a ZZ₂-module if and only if n is odd.

Created by Alexandru I. Suciu, Th Jun 11, 1998
alexsuciu@neu.edu