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I have some questions about homotopy. Before starting here a definition:

  • A topological space $X$ is called contractible if $X$ is homotopy equivalent with a one-point-space
  • Suppose $X$ a toplogical space, $Y\subset X$. A retraction $r:X\rightarrow Y$ is a continu function such that $r_{|Y}=id_Y$, thus r(y)=y $\forall y\in Y$ and $f(x)\in Y\ \forall x\in X$

Now the questions:

  • Suppose $X\equiv X'$ and $Y\equiv Y'$ homotopy equivalences, prove that $X\times Y\equiv X'\times Y'$
  • Prove that every star-shaped subspace $X\subset\Bbb{R^n}$ is contractible
  • $id_{S^{n-1}}$ not homotopic to constant map (thus $S^{n-1}$ not contractible) $\Longrightarrow$ non-existence of retractions $D^n\rightarrow S^{n-1}$
  • Brouwers Fix Point Theorem $\Longrightarrow$ Poincare-Miranda Theorem

I have no idea how to solve the questions and implications ... Can someone maybe help me? Thank you :)

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Hint: what can you say (in terms of homotopic functions) if $X$ is homotopy equivalent to a point? – Sigur Jan 12 '13 at 18:29
What is Poincare-Miranda Theorem? – lee Jan 13 '13 at 1:56
Please, post only one question in one post. Posting several questions in the same post is discouraged and such questions may be closed, see meta. – Cameron Buie Oct 11 '13 at 3:32

(1)If X,Y is homotopy equivalent to X',Y', then there exists $f_1(f_2)$ from X to X' (X' to X), so that the compositions of them are equivalent to identity. And there exists $g_1(g_2)$ from Y to Y' (Y' to Y), and the compositions of these two maps are equivalent to identity. You just check that $f_1×g_1$:X×Y→X ′ ×Y ′ , $f_1×g_1$(x,y)=($f_1(x)$,$g_1(y)$) is a homotopy inverse of $f_2×g_2$. (2)X is star-shaped, thus there is a point x of X, such for any other point x' in X, tx+(1-t)x' is also in X for any $0<t<1$. And the map I×X→X,F(t,x')=tx+(1-t)x' is a homotopy from the identity map on X and the constant map on X. (3)If there is a retraction $D^n →Sn−1$ , then because $D^n$ is contractible, you would have a contration of S^n-1(first inclusion into D^n, then composite to the contraction homotopy of D^n, finally retract to S^n-1).

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