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So this seems to be a question that it should be easy to find an answer to with just a bit of googling, but alas, I must be unable to type in the right words to search for, because I cannot find anything...

Essentially, my question is this.

Let's say that we're working with CW complexes, cellular homology, and we have two CW complexes $X$ and $Y$. They then have cellular chain complexes $$ \dots \xrightarrow[]{d_{n+3}^{X}} C_{n+2} (X) \xrightarrow[]{d_{n+2}^{X}} C_{n+1} (X) \xrightarrow[]{d_{n+1}^{X}} C_{n} (X) \xrightarrow[]{d_{n}^{X}} C_{n-1} (X) \xrightarrow[]{d_{n-1}^{X}} \dots $$ and $$ \dots \xrightarrow[]{d_{n+3}^{Y}} C_{n+2} (Y) \xrightarrow[]{d_{n+2}^{Y}} C_{n+1} (Y) \xrightarrow[]{d_{n+1}^{Y}} C_{n} (Y) \xrightarrow[]{d_{n}^{Y}} C_{n-1} (Y) \xrightarrow[]{d_{n-1}^{Y}} \dots $$ respectively. Now, it's pretty easy to understand why the $n$-skeleton of the CW complex $X \times Y$ is $$ (X \times Y)^n = \bigcup_{i+j=n} X^{i} \times Y^{j} , $$ but I'm finding it very difficult to find a good text that describes what the chain complex of this thing is, and what the differentials are in terms of the differentials in the chain complexes of $X$ and $Y$. All texts I've found so far just go directly to the Eilenberg-Zilber theorem, and do not devote any time before to discussing what the structure of $C_{\bullet} (X \times Y)$ is on its on terms, rather going directly to explaining that it is isomorphic to $C_{\bullet} (X) \otimes C_{\bullet} (Y)$, and I'd kind of want to see that the former discussion first.

Does anyone have any good reference, or would be able to give an explanation themselves?

Looking forward to your answers.

StormyTeacup
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    You want to understand $C_*(X \times Y)$, so why is giving an isomorphism to a chain complex that we completely understand not a good answer? – Connor Malin Jul 23 '22 at 18:06
  • I suppose my problem is that I'd kind of like to see how one, geometrically, constructs what the differential/boundary operator must be for $C_* (X \times Y)$. So far, I've only seen algebraic treatments of how one defines the tensor product, what the boundary operator is there, and then straight, "and we have the isomorphism $C_* (X) \otimes C_* (Y) \cong C_* (X \times Y)$". I feel like I am sort of missing the geometry. – StormyTeacup Jul 23 '22 at 18:11
  • So the issue is that the texts don't actually provide a proof of the isomorphism? – Connor Malin Jul 23 '22 at 18:12
  • I am sure that they provide a proof that someone can understand. I just cannot understand them. See for instance J. P. May's book A Concise Course in Algebraic Topology, p. 101-3. – StormyTeacup Jul 23 '22 at 18:15
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    Unfortunately, a lot of textbooks immediately apply some magic methods which don't really speak to the geometry. If it is helpful, here is a computation of the homology of a product which avoids the Kunneth theorem. The general case will look similar, but more in depth. – Connor Malin Jul 23 '22 at 18:21
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    Thanks for the link, Connor. As luck would have it, I do believe I've found a book that does provide the kind of elaboration I had in mind. Topology and Geometry by Glen E. Bredon, p. 211 onward. I'll make a full post summarizing his argument if it turns out to be precisely what I was after in case someone in the future has the same problem as I do. – StormyTeacup Jul 23 '22 at 21:49

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