Topology/Deformation Retract

Definition

A retraction is a continuous function to subspace A of X

1. $r:X\to A$

such that there exists an embedding

2. $\iota :A\hookrightarrow X$

satisfying $r\circ \iota =id_{A}$.

The purpose of this construction is to shrink down some of the topologically irrelevant wiggles of a space, or otherwise simplify to help find basic properties.

A deformation retraction is a stronger property where a homotopy exists that takes the identity to a retraction.

For example there is a deformation retraction of an open ended cylinder to a circle, despite the fact that they are not homeomorphic. There are some topological properties preserved in this way and they are of interest in algebraic topology.

Examples

The disc has a deformation retraction to a point, where $r$ maps everything to that point and the embedding $\iota$ just fixes that point. Any space that deformation retracts to a point is called contractable.

As just mentioned, $S^{1}$ is a deformation retract of $[0,1]\times S^{1}$. Note that this is a one way statement, since $[0,1]\times S^{1}\not \subset S^{1}$. More generally we can say that we can `divide out' by terms that are contractable in a Cartesian product. So the unit n-cube is always contractable.

Exercises

1. Find, explicitly, a deformation retraction from the unit n-cube to the point.

2. Although deformation retractions are not reflexive, show that they are transitive.