Deformation retraction/Definition
Definition
A subspace, [ilmath]A[/ilmath], of a topological space [ilmath](X,\mathcal{ J })[/ilmath] is called a deformation retract of [ilmath]X[/ilmath], if there exists a retraction^{[1]}^{[2]}, [ilmath]r:X\rightarrow A[/ilmath], with the additional property:
- [ilmath]i_A\circ r\simeq\text{Id}_X[/ilmath]^{[1]}^{[2]} (That [ilmath]i_A\circ r[/ilmath] and [ilmath]\text{Id}_X[/ilmath] are homotopic maps)
- Here [ilmath]i_A:A\hookrightarrow X[/ilmath] is the inclusion map and [ilmath]\text{Id}_X[/ilmath] the identity map of [ilmath]X[/ilmath].
Recall that a retraction, [ilmath]r:X\rightarrow A[/ilmath] is simply a continuous map where [ilmath]r\vert_A=\text{Id}_A[/ilmath] (the restriction of [ilmath]r[/ilmath] to [ilmath]A[/ilmath]). This is equivalent to the requirement: [ilmath]r\circ i_A=\text{Id}_A[/ilmath].
- Caution:Be sure to see the warnings on terminology
References
TODO: Mention something about how we must have a homotopy equivalence as a result. If [ilmath]r\circ i_A=\text{Id}_A[/ilmath] then [ilmath]r\circ i_A[/ilmath] and [ilmath]\text{Id}_X[/ilmath] are trivially homotopic. As [ilmath]i_A\circ r\simeq\text{Id}_A[/ilmath] we have the definition of a homotopy equivalence
- ↑ ^{1.0} ^{1.1} An Introduction to Algebraic Topology - Joseph J. Rotman
- ↑ ^{2.0} ^{2.1} Introduction to Topological Manifolds - John M. Lee