Types of topological retractions
Contents
Definitions
Retraction
Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be considered a s subspace of [ilmath]X[/ilmath]. A continuous map, [ilmath]r:X\rightarrow A[/ilmath] is called a retraction if^{[1]}:
- The restriction of [ilmath]r[/ilmath] to [ilmath]A[/ilmath] (the map [ilmath]r\vert_A:A\rightarrow A[/ilmath] given by [ilmath]r\vert_A:a\mapsto r(a)[/ilmath]) is the identity map, [ilmath]\text{Id}_A:A\rightarrow A[/ilmath] given by [ilmath]\text{Id}_A:a\mapsto a[/ilmath]
If there is such a retraction, we say that: [ilmath]A[/ilmath] is a retract^{[1]} of [ilmath]X[/ilmath].
Deformation retraction
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^{[2]}^{[1]}, [ilmath]r:X\rightarrow A[/ilmath], with the additional property:
- [ilmath]i_A\circ r\simeq\text{Id}_X[/ilmath]^{[2]}^{[1]} (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
Strong deformation retraction
Strong deformation retraction/Definition
References
- ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} Introduction to Topological Manifolds - John M. Lee
- ↑ ^{2.0} ^{2.1} An Introduction to Algebraic Topology - Joseph J. Rotman