- #1

- 431

- 0

But I've just got onto a question which should be fairly easy but its suddenly made me realise how badly I understand singular homology.

Basically, it says to work out the degree of the map z^n when considered on the Riemann sphere.

So it would be a good idea to understand the generating element for the 2nd homology group of the 2-sphere. But this is where I got stuck.

I can't picture how the simplex "wraps" round the sphere in some sense. I mean you could use 2 of them, one for each hemisphere which have the opposite orientation for their boundary so that the boundary cancels out. But then you are left with something that has one orientation on the top and another at the bottom.

Another question (that I really should have found out) is are homotopic simplexes represented by the same element in homology. And if not, how do we (intuitively) see that two simplexes are the same thing in homology? I feel this is quite essential in understanding the elements of homology. Like when we have 2 simplexes like this, are we to imagine that we have "glued" them together so to speak?

My inkling is that the identity for the 2nd homology group is mapped to the element n (so z^n has degree n) since you can imagine each circle of points with modulus r being mapped to a circle which winds around n times with modulus r^n. The only thing is then however, that the points 1 and infinity are only mapped to once (by 0 and infinity). So surely if the identity of the homology group was mapped to n, each point should have been covered n times.

I'm sorry for the extremely basic question and how badly I explained it, but its difficult to explain why you don't understand something! I just need to get this sorted because its so basic and is doing my head in!!! Any help will be very much appreciated :)