Are two topological spaces homotopically equivalent if they are homeomorphic? If one is a deform retract of the other? Is there a way to use quotient spaces here or is that something else?
2026-05-14 08:42:27.1778748147
How do I show homotopy equivalence between two topological spaces?
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I think that the point made by the question is: how to construct homotopy equivalences which are not just homeomorphisms or deformation retracts?
A useful way is a glueing theorem for homotopy equivalences, published in 1968, now available in Chapter 7, "Cofibrations", of the book Topology and Groupoids. See also further details and references in this mathoverflow discussion.
I found this result in the 1960s through generalising the result that a homotopy equivalence of spaces $f:Y \to Z $ induces an isomorphism of homotopy groups $\pi_n(Y,y) \to \pi_n(Z,f(y))$. The proof of that involves operation of the fundamental groupoid on homotopy groups at various base points. So I looked at generalising from $(S^n,1)$ to $(X,A)$, in order to see what was going on.