What's a Labeling Scheme? I have to learn how to solve problems like the following in the next two weeks:
Let $X$ be the quotient space obtained from an 8-sided polygonal region P by pasting its edges together according to the labelling scheme $aabbcdc^{-1}d^{-1}$.
The problem is, I don't know where to start.  What does this even mean?  I know only some very basic things about algebraic topology, e.g. the fundamental group, covering maps, deformation retracts, and homotopy equivalences.
 A: This section of the Wikipedia article on surfaces gives a good overview of what this means.
Basically, if you see $s$, you label the next edge of the polygon with a clockwise arrow and the label $s$, and where you see $s^{-1}$ you label it $s$ with a counter-clockwise arrow.
Then you identify the points of two edges with the same symbol so the arrows are in the same direction.
A: A precise answer assumes that you have a good understanding of quotient spaces and their relation with equivalence classes.
Assume that each edge of the polygon is assigned a parameterization by the domain $[0,1]$ which goes in the counterclockwise direction around the boundary of the polygon. For instance, if your polygon is literally a Euclidean polygon with straight sides then a side $PQ$ with initial vertex $P$ and terminal vertex $Q$ can be parameterized (using vector operation) as 
$$\gamma(t) = P + t \, (Q-P)
$$
Define the "gluing relation" to be the smallest equivalence relation $\sim$ on the polygon satisfying the following: 


*

*If two sides of the polygon have parameterizations $\gamma_1,\gamma_2$, and if those two sides are labelled with the same letter and the same sign by the given labeling scheme, then $\gamma_1(t)\sim\gamma_2(t)$.

*If two sides of the polygon have parameterizations $\gamma_1,\gamma_2$, and if those two sides are labelled with the same letter and opposite signs by the given labelling scheme, then $\gamma_1(t) \sim \gamma_2(1-t)$.
Then form the quotient space of this equivalence relation. 
One can prove that the quotient space is well-defined up to homeomorphism independent of the choices of parameterizations, depending only on the "labelling scheme".
