The question ''How did Poincaré discover the fundamental group?'' has been asked before here. But I don't understand the idea of the fundamental group from Poincaré point of view in his paper. I read the Hatcher's book concerning the idea of fundamental group. But I think that this is different from original idea from Poincaré. Is there any reference which explains Poincaré idea perfectly and originally? I want to understand what are the Betti numbers and the fundamental group suppose to do from Poincare point of view and what is difference between them?
Thanks in advance.
I think the Analysis situs website is exactly what you're looking after.
Small catch, though: it's in French.