I'm interested in computing operations on CW complexes, particularly multiple operations successively, such as taking a product and then a quotient, as you might to compute the CW complex structure on the suspension of a CW complex, or on the smash product of CW complexes.

You can accomplish this in part, by naively associating to a CW complex X with $a_k$ k-cells, the polynomial $f_X (t) = \sum_{k=0}^{\infty} a_k t^k$. Then for another CW complex Y with polynomial $f_Y (t)$, the product complex $X\times Y$ has $c_k$ k-cells, $c_k$ the coefficient of $f_X (t) f_Y (t)$. That is, $f_{X\times Y} = f_X f_Y$. You can also compute the $number$ of cells of various dimensions of a quotient of CW complexes in the this way. If A is a subcomplex of X, then $f_{X/A} = f_X - f_A +1$. This (mostly useless) approach is something I made up, but it does feel rather natural, as well.

This does seem at least somewhat effective in giving insight into our problem, namely that we can compute straightforwardly the number of cells in every dimension of the resulting space, so might immediately be able to discern its k-skeleton for certain k, but is clearly insufficient overall, because there are many different structures with the same number of cells in each dimension. I have some hope that a different, likely less lossy, approach of associating something, (a matrix or polynomial?) to a CW complex, would then have 'nice' structure on products and quotients, like $f_{X\times Y} = f_X f_Y$ above.

I have been unable to find this so far, so my question is, is the only approach to finding CW structure on the product or quotient of CW complexes to do all calculations 'by hand' or have a particular general relation already esablished? Example of the latter: $\Sigma (X\times Y) = \Sigma X \vee \Sigma Y \vee \Sigma ( X \wedge Y)$, example of the former: basically any operation in 1-3 dimensions on familiar spaces, e.g. $S^1 \times S^1$, $\Sigma S^1$, $S^1 \wedge S^1$.

I guess I should add that the end goal is to recognize the product/quotient space e.g. $\Sigma X$, $\frac{X\times Y}{Z}$ for some complexes X, Y, Z, as another space. Because just the definition of the operations, $X\times Y$, $X/A$, are very closed form expressions of the CW complex on the resulting space. But it can be rather unclear what that actual space is. For example, being able to see that $\Sigma S^n = S^{n+1}$, not just know the formula for the cells and char maps of the resulting cell space.


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