I am reading Introduction to Algebraic Topology by Rotman and the following is presented as a method to use simplicial homology to compute topological spaces that are homeomorphic to the quotient spaces of polygons.
Now I guess that this is possibly an informal discussion on the authors part. But I find the authors discussion hard to follow, because it seems like we've started with a polygon $P$, added some edges and formed some triangles here and there and then reduced all of it to remove everything we added and ended up with the same polygon $P$. It seems like all the author has done has just considered the polygon $P$ as the simplest possible simplicial complex $K$.
I don't see for example how the triangulation of $P$ induces a triangulation of $X$, since no mention of how this method takes into account the identified edges is made. What exactly is the author trying to convey in these remarks?
How exactly can I use simplicial homology to compute topological spaces that are homeomorphic to the quotient spaces of polygons?
Finally does there exist a better reference that I can look at to learn more about this method of computing homology groups?
Best Answer
I agree, the exposition is a little confusing. It's almost like Rotman is trying to use the machinery of CW complexes without introducing them just yet.
The triangulation of $P$ induces a triangulation of $X$ by considering the induced maps on the chain groups by the quotient map. In practice, we already write the induced chain map on the polygon. For example, in example $7.15$, you see that both the right and left edges are labelled $b$. Thus, you can think of the induced map as just relabelling the edges that get identified. Looking at examples $7.14$ and $7.15$ should really help clarify how this machinery works.
I don't think that's exactly what the author is trying to do. Indeed, (simplicial) homology is not strong enough to "compute" all topological spaces that are homeomorphic to a given quotient space of a polygon. What he does say is that this method can be used to compute homology groups of quotients of polygons, and this together with the remark at the end of page $163$ imply that this method is strong enough to compute the homology groups of any compact connected $2$-manifold i.e. a compact connected surface.
I personally learned the computation of homology through Hatcher's Algebraic Topology. His book is the standard for a graduate course (and is free), however it makes a great second read rather than a first read in my opinion. He uses $\Delta$-complexes instead of simplicial complexes but that's only a minor difference. Other books I have heard good things about are Munkres' Elements of Algebraic Topology. There are also great videos on YouTube. In particular, Harpreet Bedi has three videos on calculating the homology of the Torus, Klein Bottle and $\mathbb{R}P^2$ using $\Delta$-complexes.