Attaching a 2-cell by a word In Example 2.36 on Pg 141 of Hatcher's Algebraic Topology, he writes: ... one 2-cell attached by the product of commutators $[a_1,b_1] \ldots$
Can someone please explain to me what is meant by attaching a 2-cell by a word. I am assuming it means that the attaching map can be given by a word in some free group. But, I cannot make sense of this. 
Thank you for your time.
 A: You start with a wedge of $2g$ oriented circles, labeled by $a_i,b_i$. Then each letter in the word corresponds to gluing part of the boundary of the 2-cell to that $1$-cell, in the orientation prescribed by whether you have the letter or its inverse. You get the complete word by travelling around the boundary of the $2$-cell.
Also, look at the picture on page 5 of Hatcher. This explains the construction in detail. For example, the genus 3 surface pictured there has word $[a,b][c,d][e,f]$ as you travel around the boundary of the $2$-cell.
Edit: Here is an actual formua in the case of the torus. We have two circles identified at a point. Let them be parameterized by $\theta_1$ and $\theta_2$ respectively where $\theta_1,\theta_2\colon[0,2\pi]\to S^1$, with basepoint $\theta_1(0)=\theta_2(0)$. Okay, now think of the $2$-cell as a unit square $[0,1]\times[0,1]$. The function from the boundary of the square to the wedge of two circles is given by
$$(0,y)\mapsto  \theta_2(2\pi y),\,\,\,\, (1,y)\mapsto \theta_2(2\pi y)$$
$$ (x,0)\mapsto\theta_1(2\pi x),\,\,\,\,\, (x,1)\mapsto\theta_1(2\pi x)$$
A: I think you just need more familiarity with the general notion of adjunction spaces of the form $B \cup _fX$ where $f: A \to B$ and $A$ is a closed subspace of $B$, with inclusion $i: A \to B$. The most important property of this is that the square of maps 
$$\begin{matrix}
A & \to & B \\
\downarrow&& \downarrow\\
X &\to& B \cup _fX
\end{matrix}$$
is a pushout, i.e. you can construct in a unique way  continuous functions $B \cup _fX \to Y$ by giving a pair of functions $g: X \to Y, h: B \to Y$ such that $gi= hf$. In this way you do not need to visualise $B \cup_fX$, though a few simple examples are useful, since this is the way we use them. See my book "Topology and groupoids". 
In the case $A=S^1$, $X= E^2$ then a map $f: A \to B$ represents an element of $\pi_1(B)$, and if this is a free group then its elements can be written as words in the generators. 
October 5: More can be explained through an example. Suppose $K$ is the Klein Bottle obtained in the usual way as an identification of a square $\sigma$ with sides in order
$a,b,-a,b$.
We would like to write $$\partial \sigma = a+b-a+b.$$
The RHS is clearly an element of the free group on two generators $a,b$ which correspond to the two circles of the $1$-skeleton $K^1$ of the Klein Bottle. But what is the LHS, and
 what is $\partial$? 
It turns out that this is  $\partial : \pi_2(K,K^1,v) \to \pi_1(K^1,v)$ , that this has the structure of crossed module, and the second relative homotopy group 
 is the free crossed module on one generator $\sigma$. 
This result is not so easy to prove, but it is the start of lifting the nonabelian ideas of the fundamental group and groupoid to dimension $2$. See the book (with free pdf) advertised here. 
