Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If X is a simply connected Space is it homotopically equivalent to a point in the space, I know this holds in $ \mathbb{R}^n $ but only because of its algebraic properties does it hold for general topologies?

share|improve this question

1 Answer 1

A sphere of dimension 2 is simply connected and not equivalent to a point.

A space homotopically equivalent to a point not only has its fundamental group trivial but also all its higher homotopy groups. There is a famous theorem of Whitehead that says that, provided the space is sufficiently good, having all homotopy groups trivial does imply contractibility; sufficiently good means, for example, being a CW-complex.

share|improve this answer
    
Ah thanks a lot for that, evidently being a bit slow –  user147606 May 3 at 17:23

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.