Algebraic Topology · Problem 2.2.36

Homology — Allen Hatcher

Mathematics — Topology · 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.]

Intuitive explanation

The key is to use the long exact sequence of the pair and Mayer–Vietoris to peel away one dimension of the sphere at a time. Start by splitting $S^n$ into two hemispheres; their overlap is $S^{n-1}$ , which lets Mayer–Vietoris relate homology of $X \times S^n$ to that of $X \times S^{n-1}$ . By induction and the relative sequence, you'll see that each sphere dimension contributes a copy of $H_*(X)$ shifted by one degree, building up the direct sum formula.

Intuitive explanation

Related theory

Inner Product Spaces And Gram Schmidt

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