What was Poincaré's original idea of the fundamental group? 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.
