show that a quotient space is a CW complex.

Question

Here is what I am thinking about:

I want to show that the quotient space obtained from a polygon $P$ by identifying some of its edges together in pairs is a CW complex.

I can assume without proof that $P$ is homeomorphic to $D^2$ along a homeomorphism that identifies the union of edges with $\partial D^2,$ but still I do not know how to write a formula to this quotient space.

I am guessing that I can say the following:

That the sides are identified in pairs means the following. There is an enumeration $a_1, b_1, \dots , a_n, b_n$ of the edges of $P$ (not necessarily in cyclic order but without repetitions) and for each $k = 1, 2,\dots , n$ a homeomorphism $\phi_k : a_k \to b_k$ so that desired identification space $S$ is obtained from $P$ by identifying $x \in a_k \subset \partial P$ with $\phi_k(x) \in b_k \subset \partial P$.

Is this explanation for the quotient space correct? If not, please help me to modify it.

But then how can I prove that it is a CW complex using the following theorem:

(Prop A.2 on p.521 of Hatcher) Given a Hassdorff space $X$ and a family of maps $\Phi_{\alpha}: D_{\alpha}^n \to X,$ then these maps are the characteristic maps of a CW complex structure on $X$ iff:

$(i)$ Each $\Phi_{\alpha}$ is injective on interior of $D_{\alpha}^n.$

$(ii)$ The open cells $\Phi_{\alpha}(e_{\alpha})$ partition $X.$

$(iii)$ For each cell $e_{\alpha}^n, \Phi_{\alpha}(\partial D_{\alpha}^n)$ is contained in the union of a finite number of cells of dimension less than n.

$(iv)$ A subset of $X$ is closed iff it meets the closure of each cell of $X$ in a closed set.

Any help Will be greatly appreciated!

Answer 1

Hints:

1. Construct an obvious family of maps $\{\Phi_{\alpha}\}$ consisting of homeomorphisms from the 0-disk to each vertex of $P$, from the 1-disk to each edge of $P$, and from the 2-disk to the face of $P.$ You can use the theorem you stated to demonstrate that these are the characteristic maps of a CW complex structure on $P.$ To be explicit, assume $P$ is equipped with the subspace topology inherited from the usual topology on $\mathbb{R}^2.$

2. The identification you described induces an equivalence relation on $P$, yielding the quotient (identification) space $P / \sim,$ equipped with the final topology; i.e. the finest topology such that the map $q:P\to P/\sim$ given by $q(x) = [x]$ is continuous.

3. Consider the family of maps $\{\tilde{\Phi}_{\alpha}\},$ where $\tilde{\Phi}_{\alpha} = q \circ \Phi_{\alpha}.$ These are now maps from the 0/1/2-disk into the identification space $P/\sim.$ Some of these maps will be redundant - for example, if two vertices or edges of $P$ are identified in the quotient, the images in $P/\sim$ of their associated maps from the family $\{\tilde{\Phi}_{\alpha}\}$ will be the same. Once these redundant maps have been removed from the family, you should be able to show that the remaining maps satisfy the criteria of the stated theorem.