# Computation of Cellular Homology

I have a few questions about computing cellular homology.

### Setup:

Here is an outline of the process of computing cellular homology as far as I understand it:

1. Put a CW complex structure on your space $X$.
2. Form the cellular chain complex $$\ldots \xrightarrow{d_{n+1}} C_n(X) \xrightarrow{d_n} C_{n-1}(X) \to \ldots \xrightarrow{d_1} C_{0}(X) \to 0$$ where each $C_i(X)$ is the free abelian group generated by the $i$-cells of $X$.
3. As the boundary maps are between free modules it suffices to define the maps on the generators and extend by linearity. In particular, we have the cellular boundary formula $$d_n(e_\alpha^n)=\sum_\beta d_{\alpha \beta} e^{n-1}_\beta$$ where $d_{\alpha \beta}$ is the degree of the composition of the attaching map $S_\alpha^{n-1} \to X^{n-1}$ of $e_\alpha^n$ with the quotient map $q \colon X^{n-1} \to X^{n-1}/(X^{n-1}-e_\beta^{n-1})$.
4. After computing the cellular boundary maps one knows both the images and the kernels of these maps and so may compute the homology by $H_i(X) \cong \ker (d_i) / \text{im}(d_{i+1})$.

### My Question:

My question is how does one in practice compute the degrees $d_{\alpha \beta}$? I will describe my understanding of the process that I have acquired from the examples in Bredon's Topology and Geometry.

The image of the attaching map $S_\alpha^{n-1} \to X^{n-1}$ is a word in $i$-cells for $i \le n-1$. By composing with the quotient map $q$ one is killing all elements in the word except $e_\beta^{n-1}$. This results in a word completely made up of $e_\beta^{n-1}$. Now one simplifies this word into a reduced word. The length of this reduced word is the $d_{\alpha \beta}$ (up to a sign).

Is this understanding correct? In my algebraic topology course this was never really explained explicitly. Thank you very much!

Your description of the attaching map $S^{n-1}_\alpha \to X^{n-1}$ as a "word" is accurate only for the case $n=2$. In higher dimensions, you need something else.
The idea is to perturb the attaching map $\phi_\alpha : S^{n-1}_\alpha \to X^{n-1}$ so that its composition $$F = q_\beta \circ \phi_\alpha : S^{n-1}_\alpha \to (X^{n-1}-e^{n-1}_\beta) \approx S^{n-1}$$ is smooth and has a regular value at a point $p \in e^{n-1}_\beta$ (for instance take $p$ to be the image of the origin under the characteristic map of $e^{n-1}_\beta$). Using local coordinates on $S^{n-1}_\alpha$ at each point of $F^{-1}(p)$, and using the local coordinates on $(X^{n-1}-e^{n-1}_\beta)$ at $p$ given by the characteristic map of $e^{n-1}_\beta$, compute the local degree of $F$ at each point $x \in F^{-1}(p)$: this local degree equals $+1$ if the Jacobian determinant of $F$ at $x$ is positive, and it equals $-1$ if the Jacobian determinant is negative; one or the other is guaranteed by the requirement that $p$ is a regular value. Add up the local degrees over all $x \in F^{-1}(p)$, and you get the number $d_{\alpha\beta}$.