Well, I've been to over two weeks' worth of lectures and reading the relevant chapters in Hatcher, but I still have no idea how homology works.
Here's what I understand
you have n-simplices like points, lines, triangles, tetrahedrons, etc.
then you have boundary maps, which take the simplex and their rotation and give the faces. For example if you had a triangle with vertices x,y,z then the boundary map of this simplex gives [x,y]-[y,z]+[z,x]...
Ok. So [x,y] would be the 1-simplex between 0 and 1. What does the sum [x,y]-[y,z]+[z,x]
The first example in the chapter is computing the homology groups of the circle. Okay, so a circle is one vertex v and one edge e.
The explanation given in Hatcher includes:
"Then and are both Z and the boundary map is zero since ."
I'm lost. This is supposed to be the easiest example, and I have no clue what he's rambling on about. Could someone help?