I'm self studying Munkres' Elements of Algebraic Topology.
The Exercise 59.1 of the book is following.
Exercise 59.1: $X$ and $Y$ are topological space. prove that $H_m(X\times Y)$ is isomorphic to $\bigoplus_{p+q=m} H_p(X;H_q(Y))$ for any $m$.
I know by Eilenberg-Zilber Theorem singular chain complex of the product space $X\times Y$ is chain equivalent to the tensor product of singular chain of $X$ and singular chain of $Y$. But how can we use that to prove above exercise? Also, the proof is purely algebraic? or is there some topological intuition behind above isomorphism?
Thank you for any solutions or hints in advance!
I think you need results of the following type for free chain complexes $C,D$ of abelian groups.
IF $H(D)$ denotes the homology of $D$ considered as a graded group, or as a chain complex with zero differential, then there is a morphism of chain complexes $g: D \to H(D)$ inducing an isomorphism of homology.
Let $g : D \to E$ be a morphism of chain complexes. If $g$ induces an isomorphism in homology, then so also does $1 \otimes g: C \otimes D \to C \otimes E$.
These should be in your text somwhere!
(I am familiar with these in cohomology, since they were relevant to my 1961 thesis on function spaces, part of which was published here, with notation and terminology now antiquated. It contains some examples of non naturality.)