Consider 2-simplex (see Hatcher's example p105). Its boundary map is the sum of 1-simplices with signs. For instance, if you follow the counterclockwise route, , where .
"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?
A boundary map of 1-simplex is . In Hatcher's example, has a simplicial structure that consists of one edge (1-simplex) and one vertex (0-simplex). In this case, .
We know that .
Now consider . A boundary of a 0-simplex (vertex) is an empty simplex. Thus . Now we see that has a single generator (0-simplex v) and its group is . This generator is not an image of a boundary map , we have .
Similary, as seen above. Thus has a single generator (1-simplex "e") and its group is . Similarly, this generator is not an image of a boundary map (actually we have no 2-simplex here), we have .