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?