# Topology/Deformation Retract

- Definition

A *retraction* is a continuous function to subspace A of X

1.

such that there exists an embedding

2.

satisfying .

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

editThe disc has a deformation retraction to a point, where maps everything to that point and the embedding just fixes that point. Any space that deformation retracts to a point is called *contractable*.

As just mentioned, is a deformation retract of . Note that this is a one way statement, since . 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

edit1. 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.