Consider the circle $S^1$ with multiplication given by the complex numbers. Prove that the map $f(x) = x ^n$ , $n$ a positive integer, has degree $n$. What is the degree of the map $g(x) = 1/x$.
This is exercise 25 of Croom's book "basic concepts in algebraic topology". I have not solved it yet.
My approach is that I triangulate the circle as follows: denote $a=(1,0),b=(-1,0),c=(0,1), d=(0,-1)$. The complex contains the 1-simplexes $(ab),(ac),(ad),(bc),(bd)$ and their faces. However, with this triangulation, I computed $H_1(S^1)=\mathbb{Z}^2$, which is wrong.
So could u explain why I was wrong, and how to solve the original problem? Thank you
The first place you went wrong was this:
The 1-simplexes should be $ac$, $cb$, $bd$, and $da$. The edge $ab$ is a diameter of the circle.
With that in hand, perhaps you can complete the problem yourself. One way to solve it is indeed to triangulate, although you'll probably want quite a lot more vertices to make that easier.