I have to answer the next question (for homework):
Find the generators for the fundamental group of the given spaces and write the curves mentioned as combination of the generators:
- The torus: C= the identity
- The cylinder: C= the top circumference
- The Möbius strip: C= the “boundary” of the strip.
Here’s what I understand. First, the fundamental grupo of the torus is isomorphic to $\mathbb{Z}\times \mathbb{Z}$. The generators for this group are (1,0) and (0,1). So the identity should be something like (1,0)*(0,1). I’m not sure what to do.
An example on what I’m asked to do (maybe with another surface or another curve) would be really helpful.