Thread: Deformation retracts?

Let .

Is X a retract of X ?

Is it a deformation retract of X ?


I would think one would use the fact that if X were a deformation retract of X , then their fundamental groups would be isomorphic.

But the fundamental group of X is ZXZ, while I think the fundamental group of X is ZX{0}. Would this be enough to justify it's not a deformation retract? If yes, I have a feeling that X is a retract of X . But how do you prove this?

You are quite right.

A deformation retract is a special case of a homotopy equivalence. Hence, if a subspace was a deformation retract of the space , the two spaces would indeed have the same fundamental group (which they don't in your case).

To show that is however a retract of , you just use the first map that comes to mind, i.e. use the map given by , so you map all of the second copy of to the point

Thanks, this makes a lot of sense.