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I'm trying to understand cellular homology as a functor from the category (CW-complexes, cellular maps) to the category of abelian groups sequences.

Let $X,Y$ be fixed CW-complexes. My lecturer defined the $n$-th cellular chain group $C_n ^{\text cell} (X)$ as the free abelian group with generators the $n$-cells of $X$, and the $n$-th boundary map $\partial _n ^{\text cell} $ as the morphism s.t., if $A$ is a $n$-cell of $X$ and $\Phi^{(n)}_A :D^n \to X^n $ is its characteristic map in $X$, we have $ \partial_n ^{\text cell} A = \sum_C \epsilon (A,C) C$, where $C$ ranges over the $n-1$-cells of $Y$ and $\epsilon (A,C)$ is the degree of a map $S^{n-1} \to S^{n-1} $ induced by the characteristic maps of $A$ and $B$ (i.e. he uses the cellular boundary formula as a definition).

I know that there is a natural isomorphism between $C_n ^{\text cell}(X) $ and $H_n (X^n ,X^{n-1} )$ (the latter being a singular homology group).

If $f:X \to Y$ is a cellular map, it is clear what is the chain-induced map $f_n :H_n (X^n ,X^{n-1}) \to H_n (Y^n ,Y^{n-1})$, but I'm having troubles understanding what $f_n :C_n ^{\text cell} (X) \to C_n ^{\text cell} (Y)$ is. Given a $n$-cell $A$ of $X$, I would express $f_n (A) $ as a linear combination of the $n$-cells of $Y$ that intersects $f(A)$, but I can't find how to define the coefficients.

Thank you

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    $\begingroup$ I empathize with your question, but I doubt that there will a direct expression, i.e. one that does not pass by the identification of $C_n^{cell}$ with $H_n(X^n,X^{n-1})$, if only for the simple reason that your cellular boundary maps are secretly passing by that interpretation already: they are the maps that correspond to the connecting homomorphism of the triple $(X^n,X^{n-1}, X^{n-2})$ under the natural isomorphism you're quoting. By the way, I think there is a nice exposition of this in Switzer, starting from 10.6. $\endgroup$ Jan 30, 2016 at 9:35
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    $\begingroup$ I think this hinges on what your definition of a CW-complex is. In the definition I know, you want the $n+1$-skeleton to be the $n$-skeleton with some cells attached, but you don't actually pick their "characteristic maps". Then there is no natural isomorphism from your definition of $C_n^{cell}(X)$ to $H_n(X^n,X^{n-1})$ (think of a degree $-1$ map on $S^n$). Once you have picked characteristic maps $c_A$ (and thereby attaching maps $a_A$), you can define the differential explicitly: The coefficient of $A'$ in $\partial_n^{cell}A$ is $deg(c_{A'}^{-1}\circ \pi_{X^n\to X^n/X^{n-1}}\circ a_A)$. $\endgroup$
    – Bertram
    Jan 30, 2016 at 12:23
  • $\begingroup$ Ok, I looked at Switzer again and I think what you want is proposition 10.13, though the formula is a bit convoluted. I also empathize with @Bertram 's point and point to these notes for a nice explanation of the phenomenon. $\endgroup$ Jan 30, 2016 at 15:01

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Let $\alpha$ be an $n$-cell of $X$ and $\beta$ be an $n$-cell of Y. Then $f(\alpha)=\sum_{\beta\in J_n'} y_{\alpha \beta} \beta$. Here $J_n'$ denotes the set of $n$-cells of $Y$. We wish to determine the values of $y_{\alpha \beta}\in \mathbb Z$.

Let $\varphi_\alpha:S^{n-1}\to X^{n-1}$ be the attaching map of $\alpha$. Let $\overline{f}:X^n/X^{n-1}\to Y^n/Y^{n-1}$ be the induced map by $f$ on the quotient.

For every wedge sum $\bigvee_i A_i$ of pointed topological spaces there are canonical retractions $r_j: \bigvee_i A_i \to A_j$. Note that for any CW complex $Z$, $Z^n/Z^{n-1}$ is a wedge of $n$-spheres. Under this identification, we obtain retractions $r_\gamma:Z^n/Z^{n-1}\to S^n$, one for every $n$-cell $\gamma$ of $Z$.

With this notation set up, consider the following commutative diagram:

enter image description here

Proposition: $y_{\alpha \beta}=deg(f_{\alpha \beta})$.

This is theorem 10.13 of Switzer's book "Algebraic topology: Homology and Homotopy", or proposition 3.8 of Lundell and Weingram "The topology of CW complexes". They call $y_{\alpha \beta}$ the "degree with which $\alpha$ is mapped into $\beta$ by $f$", which is a nice name.

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  • $\begingroup$ I should note that Bertram's comment is spot on. You need to be careful. Choose a set of attaching maps for all cells of $X$. This choice defines: an isomorphism as the one you claim, the boundary map on $C_*^{cell}$, and induced maps on $C_*^{cell}$. Using the same set of attaching maps for all three choices gives you a natural isomorphism of chain complexes, where $H_n(X^n,X^{n-1})$ gets the boundary map I alluded to in my first comment above. So you have to be careful, because a CW complex does not come, in the usual definitions, with the attaching maps as part of the data. $\endgroup$ Jan 31, 2016 at 22:30
  • $\begingroup$ So when, in the answer, I said "let $\varphi_\alpha$ be the attaching map of $\alpha$", I was being sloppy. $\endgroup$ Jan 31, 2016 at 22:32
  • $\begingroup$ I found by chance this other question that has the "homological" approach to induced maps. $\endgroup$ Feb 10, 2016 at 14:43

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