Cw complex $\Sigma_g$

Question

Consider the oriented connected compact surface $\Sigma_g$ of genus $g$ with its standard CW structure. How do I write down the attaching map for the single $2$-cell and how can it be proven that it is continuous? I am interested in a formal (algebra standards) proof not a dirty topologists proof (write down a picture and claim map, which is not even defined formally, is continuous).

Answer 1

As @StefanHamcke indicates in his comment, what topologists learn to do in their training is to take their knowledge of CW complexes as learned from books like Hatcher's or Spanier's, and to instantly translate intuitive descriptions of CW complexes into rigorous descriptions. This process is mechanical and uninteresting once you learn how to do it. However, until you have reached the mechanical and uninteresting stage, you should study how to carry out this process.

Here is what you get if you carry out out this translation process starting with the standard picture of pairing up the sides of an octagon to form $\Sigma_2$ by using the pairing formula $AB\bar A \bar B C D \bar C \bar D$.

Start with the $0$-skeleton $\Sigma^0_2$: a single point $p$.

Define the $1$-skeleton $\Sigma^1_2$ by attaching four edges to $p$. More specifically, starting with a disjoint union $X = \{p\} \cup \alpha \cup \beta \cup \gamma \cup \delta$ where $\alpha,\beta,\gamma,\delta$ are each homeomorphic to $[0,1]$, form the quotient space $\Sigma^1_2$ of $X$ by attaching each of the points of $\partial \alpha, \partial \beta, \partial \gamma, \partial\delta$ to the point $p$. Let $\pi : X \to \Sigma^1_2$ be the quotient map. (Here I assume that one knows how to translate the "attaching" language into a specific description of the partition which defines the quotient map: the "attaching" language describes a relation on a set, and one takes the equivalence relation generated by that relation).

Define four paths $A,B,C,D :[0,1] \to \Sigma^1_2$: take the homeomorphisms from $[0,1]$ to $\alpha,\beta,\gamma,\delta$ resp.; compose with the inclusion maps into $X$; and compose with the quotient map $\pi : X \to \Sigma^1_2$. Each of these four paths is a composition of continuous maps therefore continuous. Each of these paths has both of its endpoints at $p$.

Define a path $f : [0,1] \to \Sigma^1_2$ by converting the pairing formula $AB\bar A \bar B C D \bar C \bar D$ into a concatenation formula and picking a way to associate, for example
$$f = A* (B* (\bar A* (\bar B* (C* (D* (\bar C* \bar D)))))) $$ (Here I assume one knows the formula for concatenation of two paths and for the reverse of a path, and the lemma regarding the continuity of these operations. Also, feel free to associate in any other fashion.)

Define $\Sigma_2$ to be the quotient space of the disjoint union $\Sigma^1_2 \cup D^2$ by identifying $e^{2 \pi i t} \in \partial D^2$ with $f(t) \in \Sigma^1_2$.