Algebraic Topology · Problem 3.1.8

Cohomology — Allen Hatcher

Mathematics — Topology · cohomology-basic-computations

Problem

Many basic homology arguments work just as well for cohomology even though maps go in the opposite direction. Verify this in the following cases: (a) Compute

H^i(S^n; G)

by induction on

n

in two ways: using the long exact sequence of a pair, and using the Mayer--Vietoris sequence. (b) Show that if

A

is a closed subspace of

X

that is a deformation retract of some neighborhood, then the quotient map

X \to X/A

induces isomorphisms

H^n(X, A; G) \approx \tilde{H}^n(X/A; G)

for all

n

. (c) Show that if

A

is a retract of

X

then

H^n(X; G) \approx H^n(A; G) \oplus H^n(X, A; G)

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