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Math Help - Equivalent Subspaces

  1. #1
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    Equivalent Subspaces

    For each n\in\mathbb{N} give examples of subspaces of \mathbb{R}^n, which are homotopy equivalent but not homeomorphic to each other.

    Give reasons for your answer.

    Any help would be great. Thanks
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  2. #2
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    Quote Originally Posted by ejgmath View Post
    For each n\in\mathbb{N} give examples of subspaces of \mathbb{R}^n, which are homotopy equivalent but not homeomorphic to each other.

    Give reasons for your answer.

    Any help would be great. Thanks

    One open and and one closed intervals (balls if n>1): they're trivially homotopic but not homeomorphic.

    Tonio
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  3. #3
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    Okay, so if I take an open ball B_r({\bf{x}}) in \mathbb{R}^n this is clearly not homeomorphic to a point {\bf{y}} in \mathbb{R}^n as any continous map is not bijective.

    So take a map f:B_r({\bf{x}})\rightarrow {\bf{y}} given by f({\bf{v}})={\bf{y}} for {\bf{v}}\in B_r({\bf{x}}) and then take another map g:{\bf{y}}\rightarrow B_r({\bf{x}}) given by g({\bf{y}})={\bf{x}}

    How do I should that g\circ f and f\circ g are homotopic to the respective identities?
    Last edited by ejgmath; April 26th 2010 at 07:55 AM.
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