And maybe for us novices, what does the conjecture say?

Also the long line has zero homotopy groups, but is not contractible. It's too long! (but it is also a manifold, sort of...)

right. That was embarrassingly easy. Thanks.

I think you mean { (x,y) : x^2 + y^2 < 1}. i.e. you want to leave the boundary circle there. Then Y is contractible. Otherwise Y is a pinched torus and also has fundamental group the integers (although the induced map is zero).

Okay great! I see that Thm 4.2 also has the assumption that A is a faithful R-module. Can that be removed? or is it automatic?