Deformation retraction/Definition

From Maths
Jump to: navigation, search


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


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