Let $M_g$ be the closed orientable surface of genus $g$ with its usual $CW$ structure consisting of one $0$-cell, $2g$ $1$-cells, and one $2$-cell attached by the product of the commutators $[a_1,b_1] \dots [a_g, b_g]$. The associated cellular chain complex is $$ 0 \to \mathbb{Z} \overset{d_2}{\to} \mathbb{Z}^{2g} \overset{d_1}{\to} \mathbb{Z} \to 0$$
In the example it is simply stated that $d_2$ is $0$ because each $a_i$ or $b_i$ appears with its inverse in the product of the commutators and so the $\Delta_{\alpha \beta}$ are homotopic to constant maps.
Ok. This seems like a very hand-wavy explanation and not applicable to more general situations.
How do I compute the degree of $\Delta_{\alpha \beta}: \partial D_n^{\alpha} \to S_{\beta}^{n-1}$? Is there a computation to this. I have seen solution that this can be done using matrices but I cannot actually find an explanation calculating degrees using matrices.
Anyway, I am very lost on how to compute degree and Hatcher's explanation are a bit too heuristic for me at the moment.
The matrix computation I am mentioning looks something like this.
http://www.math.ku.dk/~moller/f03/algtop/opg/S2.2.pdf
I would really appreciate an explanation :)
(No need to go through the example in the link, a very simple situation as that for the Torus would suffice)