(1) General case: Let $X$ be a topological space and $A$ be its subspace. Suppose $X$ and $A$ are homotopically equivalent to each other, then is $A$ necessarily a deformation retract of $X$?

(2) Special case: I think even if the inclusion $i: A \hookrightarrow X$ is a homotopic equivalence, we still don't know whether its homotopic inverse is a retraction $r:X \to A$. In this special case, is $A$ necessarily a deformation retract of $X$?

I am very confused about both (1) and (2) and find it hard to prove them or come up with counterexamples. Thanks in advance!


Corollary 0.20 of Hatcher's book: If $(X, A)$ satisfies the homotopy extension property(CW pair, for instance) and the inclusion $A֓ \hookrightarrow X$ is a homotopy equivalence, then $A$ is a deformation retract of $X$ .

  • 2
    $\begingroup$ I think that this is answered in Hatcher, Alg. Top., Page 18, ex. 6. (b), or also (7). The answer seems to be "no". $\endgroup$ – Peter Franek Mar 5 '17 at 20:41
  • $\begingroup$ (1) is false even if $A$ is a single point, if you work with strong retractions, as Hatcher does, see the exercises in chapter 0 $\endgroup$ – Alessandro Codenotti Mar 5 '17 at 20:48
  • $\begingroup$ @AlessandroCodenotti If $A$ is a point, then there is no difference between strong and weak d.r., and/or between (1) and (2). $\endgroup$ – Peter Franek Mar 5 '17 at 21:14
  • $\begingroup$ you're right, I didn't quite remember what a weak deformation retraction is $\endgroup$ – Alessandro Codenotti Mar 5 '17 at 21:39

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