Difference between revisions of "Deformation retraction"

Latest revision as of 20:13, 11 May 2016

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].

Warnings on terminology

Some authors define a deformation retract to be what we would call a strong deformation retraction.

TODO: Make a table or something

References

1. ↑ ^{1.0 1.1} An Introduction to Algebraic Topology - Joseph J. Rotman
2. ↑ ^{2.0 2.1} Introduction to Topological Manifolds - John M. Lee

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