Computation of Cellular Homology

Question

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!

Answer 1

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.

One very good way to do these degree calculations is to use ideas of differential topology. This is not usually explained in algebraic topology courses, but nonetheless it is a common and easy method of calculation.

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 $d_{\alpha\beta}$.