Understanding cellular homology: degree of attaching map of a two cell

Question

I am working towards an understanding of cellular homology as explained here on Wikipedia.

To help me I am calculating a simple example:

I have two problems: good mathematical notation and actual correctness of what I'm doing.

First, let me show what work I did:

For notation I use $\partial$ to denote the boundary of a cell. Also, if $e_1$ is any $1$-cell, I let $\partial e_1 = \{a,b\}$ where $a$ is the label for the positive end and $b$ the label for the boundary point with negative orientation.

Then I calculated the attaching maps as follows:

$$\begin{array}{cc} f_2: \partial e_2 \to e_1^2 \sqcup e_1^1 & e^{i\theta}\mapsto e^{i\theta} \\ f_{11}: \partial e_1^1 \to e_0^1 \sqcup e_0^2 & a \mapsto e_0^2, b\mapsto e_0^1 \\ f_{12}: \partial e_1^2 \to e_0^1 \sqcup e_0^2 & a \mapsto e_0^2, b\mapsto e_0^1 \end{array}$$

Using these attaching maps and the definition of degree for attaching maps in dimension $0$ I calculated the following degrees for these maps:

$$ \deg (f_{11} \text{ at } e_0^1) = 1, \deg (f_{11} \text{ at } e_0^2) = -1$$

$$ \deg (f_{12} \text{ at } e_0^1) = 1, \deg (f_{12} \text{ at } e_0^2) = -1$$

For $f_2$ I have:

$$ \deg (f_2 \text{ at } X_1/(X_1 - e_1^1)) = -1$$ $$ \deg (f_2 \text{ at } X_1/(X_1 - e_1^2)) = 1$$

My justification for the degrees of $f_2$ is that $X_1/(X_1 - e_1^1)$ looks like this:

And this maps $S^1$ one time CCW around the origin. The space $X_1/(X_1 - e_1^2)$ looks like this:

And this maps $S^1$ one time around the origin clockwise so the degree is $-1$ by definition.

My questions are:

(1) Is the notation $\partial e_1^1 \to e_0^1 \sqcup e_0^2$ for the domain and codomain of the attaching maps correct? I could not find a worked out example anywhere and this is my own notation.

(2) Similarly I am unsure whether it is good to write $a \mapsto > e_0^2, b\mapsto e_0^1$ for the attaching maps. Is this how it is usually written down?

(3) Is my argument for the degrees of the map $f_2$ correct? I did calculate the homology groups and they turned out correct but of course this doesn't mean that everything I did here was correct.

Basically, this question boils down to checking whether I completely understand the definition of cellular homology.

Answer 1

First, a remark: it's common to denote the dimension of a cell (or anything, really) as an exponent, not a subscript. So for example $e^2$ is usually used for a cell of dimension $2$, not $e_2$. That's not a huge problem though.

As for your questions:

#1: This notation is correct, but only for the map attaching a 1-cell to the 0-skeleton. In general the n-skeleton is not the disjoint union $\sqcup$ of the cells. In particular you wrote $\partial e_2 \to e_1^2 \sqcup e_1^1$: this looks like the $1$-cells are not attached, which is not the case.

If you want to use a more common notation, I suggest calling your CW complex something like $X$, and the $k$-skeletons $X^{(k)}$. Then the attaching map is $\partial e_{k+1} \to X^{(k)}$, for any dimension $k$.

#2: This one is OK, in my opinion. I'm a bit more unsure about $e^{i \theta} \mapsto e^{i \theta}$, but it's clear from the context what that means. But really, in practice you don't write down the attaching map explicitly like that, a picture is generally enough (though props to you for wanting to understand what's going on behind the picture).

#3: This looks OK to me.