it is my first time asking a question on this site. I am stuck on trying to prove a thing which seems very simple and almost trivial, but I still can't manage to get formal. The problem is to find an explicit homotopy equivalence of pairs $(R^n, R^n \setminus \lbrace 0 \rbrace)$ and $(I^n, \delta I^n)$ where I denotes the unit interval. The thing is, I know how to build one homotopy equivalence between $R^n$ and $I^n$ and one between $R^n \setminus \lbrace 0 \rbrace$ and $\delta I^n$ but I can't find any between the pairs. The problem is that a homotopy $$F:(R^n, R^n \setminus \lbrace 0 \rbrace) \times I \rightarrow (R^n, R^n \setminus \lbrace 0 \rbrace)$$ for example, must respect the pair, so $F(x,t)\in R^n \setminus \lbrace 0 \rbrace$ for each $x \in R^n \setminus \lbrace 0 \rbrace$. Thank you in advance for any suggestion.
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Are you sure that the pairs in you question are homotopy equivalent? After a little bit of thought I feel they shouldn't be. Am I right in thinking that $\delta I^n$ means the boundary of $I^n$? I will give it a bit more thought in a while and see how one could prove they are different. – mcwiggler May 25 '18 at 13:50
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No, I finally got an answer... This exercise was given by my professor, and he just told us it was wrong. The pairs are not equivalent actually! (Btw yes, $\delta I^n$ is the boundary of $I^n$) – Damiano Vinciarelli May 26 '18 at 17:04
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As you now know, it is not true. If you are in need of an explicit proof see Is $(I^n, \partial I^n)$ homotopy equivalent to $(R^n, R^n\setminus \left\{0\right\})$. The proof uses some non-elementary facts.
Paul Frost
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