Algebraic Topology · Problem 2.2.17

Homology — Allen Hatcher

Mathematics — Topology · cellular-homology

Problem

Show the isomorphism between cellular and singular homology is natural in the following sense: A map

f: X \to Y

that is cellular --- satisfying

f(X^n) \subset Y^n

for all

n

--- induces a chain map

f_*

between the cellular chain complexes of

X

and

Y

, and the map

f_*: H_n^{CW}(X) \to H_n^{CW}(Y)

induced by this chain map corresponds to

f_*: H_n(X) \to H_n(Y)

under the isomorphism

H_n^{CW} \approx H_n