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.
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Let and . Let , given by f(0)=(0,0) and .
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Show that is an isomorphism, for all .
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Show that f is not a homotopy-equivalence.
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Does this contradict Whitehead's theorem? Why, or why not?
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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.
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Show that , for all n > 1.
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Show that X is not homotopy-equivalent to Y.
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Does this contradict Whitehead's theorem? Why, or why not?
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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.
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Show that , for all .
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Show that , for all .
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And yet f is not homotopic to a constant map.
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Let X be a connected CW-complex, with for 1 < i < n, for some n > 1. Let be the Hurewicz homomorphism. Show that .
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Let G be an abelian group.
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Show that Hn+1(K(G,n))=0, for n > 1.
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Show that there is a Moore space M(G,1) if and only if H2(K(G,1))=0.
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For what values of n does there exist a Moore space of type ?
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Let , with attaching map , and .
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Show that can be chosen so that X and Y have the same 3-skeleta.
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Show that (as graded rings).
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Show that (as graded rings).
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Let G be a group, and let be a sequence of -modules.
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Construct a CW-complex X with , and (as -modules).
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If , where , show that is trivial as a -module, for all n>1.
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If , show that is trivial as a -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