So the degree of z^k between S^1--> S^1 (the circle/1-sphere) is then k , right? The generator is S^1 itself, a single loop and z^k wraps around k times , and k-times is the class k[S^1]. Is that correct?
Yes. The degree of the map S^1\to S^1 given by z\mapsto z^k is k. You're also correct that this would correspond to multiplication by k under the isomorphism with \Z. Pedantic point: you may want to switch up the notation for your generator just to be clearer. Instead of saying S^1, maybe say the 1-simplex \sigma or something like that.
I would love to watch any video lectures on differential geometry or topology by this professor, if any were available.
So the degree of z^k between S^1--> S^1 (the circle/1-sphere) is then k , right? The generator is S^1 itself, a single loop and z^k wraps around k times , and k-times is the class k[S^1]. Is that correct?
Yes. The degree of the map S^1\to S^1 given by z\mapsto z^k is k. You're also correct that this would correspond to multiplication by k under the isomorphism with \Z. Pedantic point: you may want to switch up the notation for your generator just to be clearer. Instead of saying S^1, maybe say the 1-simplex \sigma or something like that.
@@ethanzell4073 Thanks Ethan, nice presentation.