Subspaces of R^n Homotopy Equivalent But Not Homeomorphic

[jaymath]

For each n in N give examples of subspaces of R^n, which are homotopy equivalent but NOT homeomorphic to each other.

Give reasons for your answer.

I'm working along the lines of open and closed intervals in R and balls in R^n with n>1. Although I'm struggling with the reasoning.

Any help would be great.

 

[owlpride]

That should work. What are you struggling with?

 

[jaymath]

Im struggling with the argument. As I said I am working along the lines of balls homotopy equivalent to points. So for example in R^2 the closed unit disk is clearly not homemorphic to a single point, it fails as a bijection. But they are homotopy equivalent. But I can't see how there are continuous functions mapping from one to the other?

I can see to go from the disk to a point you just make every point in the disk equal to a single point, I think that's continuous? But how would you go the other way.

Thanks

 

[owlpride]

How you get a continuous map from a point to a disk? Just map that point anywhere you want! The center of the disk is probably the most convenient. Let's call this map f, and the map that takes the entire disk to the point g.

Then you have to show that f g: disk -> disk and g f: pt -> pt are homotopic to the identity map on their respective receiving spaces. Can you do that?

 

[jaymath]

Ah, okay, I was missing the fact that it need not be bijective, so as you say it can go anywhere.