Suppose $X$ and $Y$ are two topological spaces and $f:X\rightarrow Y$ is a homotopy equivalence. My problem is about the restriction $f|_A$ for some $A \subset X$. Is $f|_A:A \rightarrow f(A)$ a homotopy equivalence too? If No, Under what condition $f|_A$ is homotopy equivalence?
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3No, consider $f:D^n \rightarrow $ where $$ is the one point space and restricting to $S^{n-1}$. – Noel Lundström Jun 13 '20 at 07:58
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@NoelLundström Thanks a lot. Is there any condition on $A$ or $f$ implies homotopy equivalence ? – Vahid Shams Jun 13 '20 at 08:12
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A condition assuring that $f|_A:A \rightarrow f(A)$ is a homotopy equivalence is that $A \hookrightarrow X$ and $f(A) \hookrightarrow Y$ are homotopy equivalences. But that is very restrictive.
In general you cannot hope that $f|_A$ is a homotopy equivalence. Let $\phi : A \to B$ be any surjective map. The cones $CA = (A \times I)/(A \times\{1\}), CB = (B \times I)/(B \times\{1\})$ are contractible and contain $A, B$ as subspaces (identified with $A \times\{0\}, B \times\{0\}$). The cone map $f = C\phi : CA \to CB, C\phi([a,t]) = [\phi(a),t]$, is a homotopy equivalence, but $f|_A$ is not unless we started with a homotopy equivalence $\phi$.
Paul Frost
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