Hatcher-Algebraic-Topology · Problem 2.2.36

Homology

Mathematics — Graph Theory · homology-product-spaces

Problem

Show that

H_i(X \times S^n) \approx H_i(X) \oplus H_{i-n}(X)

for all

i

and

n

, where

H_i = 0

for

i < 0

by definition. Namely, show

H_i(X \times S^n) \approx H_i(X) \oplus H_i(X \times S^n, X \times \{x_0\})

and

H_i(X \times S^n, X \times \{x_0\}) \approx H_{i-1}(X \times S^{n-1}, X \times \{x_0\})

. [For the latter isomorphism the relative Mayer--Vietoris sequence yields an easy proof.]