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 .
Is H1(X,A) isomorphic to
? Explain why, or why not.
-
- Let X and Y be two finite CW complexes. Show that
.
- Let A and B
be two subcomplexes of X such that .
Show that .
- Let be a continuous map. Assume that
and are homotopy equivalences.
- Show that is an isomorphism.
- Give an example where
, 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 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 associated to
that cell decomposition.
- Compute the homology groups H*(X).
- Let Tg be the orientable surface of genus g.
- Describe a 2-fold cover .
- Compute .
- Compute
.
- Let .
Compute H*(X).
- Show that is not a retract of
.
- 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
http://www.math.neu.edu/~suciu/mth3107/top2final/index.html