Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Suppose X is a path-connected, locally compact, Hausdorff space and Y is its one-point compactification. Let G be the fundamental group of X and H be the fundamental group of Y. Is it true that the embedding of X into Y always induces a monomorphism G->H? More generally, what is the relationship between G and H?

share|improve this question

2 Answers 2

More generally, if A and X0 are any finite CW complexes, and f : A → X0 is any map, let Y be the mapping cone of f, and let X be Y with the cone point removed; then Y is the one-point compactification of X, and the inclusion X0 → X is a homotopy equivalence. (David's example is the case A = X0 = S1, f = id.) So any map X → Y which is the mapping cone of something is homotopy equivalent to a one-point compactification.

I don't think you can realize any map of groups as the induced map on π1 of a mapping cone (0 → Z/2Z?) but you can realize (G → 0, 0 → a free group, ...)

share|improve this answer

It is not always a monomorphism. Let X be R^2 \setminus { (x,y) : x^2 + y^2 < 1 }. Then X has fundamental group the integers, but Y is contractible.

share|improve this answer
3  
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). –  Chris Schommer-Pries Oct 27 '09 at 17:09
    
You are right. I have edited to fix the error. –  David Speyer Oct 27 '09 at 17:16
    
I think the good old inversion z |-> 1/z makes this easier to see: i.e., just take the closed unit disk in the plane with the origin removed. –  Pete L. Clark Dec 27 '09 at 10:53

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.