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I have a subspace of a unit ball, and I want to prove that this subspace is a retract of the ball.

I know that homology groups and homotopy groups of this subspace vanish, and I strongly believe that this subspace is a CW complex, which would then imply that my subspace is contractible. I also believe that the said subspace has finitely many cells.

I am thus left with the question: Is every contractible subspace of the unit ball a retract of the unit ball? If not, what about every contractible finite CW subcomplex?

Thanks,

Maithreya

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Any open subset will not be a retract since the disk is compact. If you restrict to subcomplexes of the unit disk, then the answer is actually much stronger than yes.

In fact, the space is a deformation retract of the disk. In Hatcher’s chapter 0 he proves that if the inclusion of a CW complex is a homotopy equivalence, the subcomplex is a deformation retract of the complex. In this case, this applies since the inclusion of a contractible space into another is a homotopy equivalence.

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  • $\begingroup$ Thanks Connor for the simple and perfect answer! $\endgroup$ – Maithreya Sitaraman Jul 12 at 6:37

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