HOMEWORK 6
(1) Compute the homology and cohomology groups with coefficiets in Z, Z/2Z, Z/5Z
for the Klein bottle and space with genus n (n–holes).
(2) What space do we get when we make a hole in torus and glue M¨obius band in
it? Compute its cohomology with coefficients in Z/2Z.
(3) Give an example of two spaces CW–complexes X, Y that have nontrivial homology
groups in dimesion 2 (with coeffs in Z) such that:
(a) H∗(X, Z) ∼= H∗(Y, Z) and H∗(X, Z2) ∼= H∗(Y, Z2).
(b) H∗(X, Z2) ∼= H∗(Y, Z2) and H∗(X, Z) ∼= H∗(Y, Z).
(4) Let us take the following exact sequence C∗ where all maps are 0:
→ Zn → · · · → Z3 → Z2 → 0 → 0
Compute its homology and cohomology groups with coefficients in Z, Zp for
p ∈ N.
Date: March 29, 2013.
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