Topological retraction

Deformation retraction

Let π‘Œ βŠ†π‘‹ be a subspace and πœ„ :π‘Œ →𝑋 the inclusion. A deformation retraction π‘Ÿ :𝑋 β†’π‘Œ is a is a retraction, i.e. π‘Ÿπœ„ =idπ‘Œ, such that πœ„π‘Ÿ ≃id𝑋 homotopy

π‘Ÿπœ„=idπ‘Œ[πœ„π‘Ÿ]=[id𝑋]

Hence in Category of topological spaces, π‘Ÿ is a left inverse of πœ„, but in NaΓ―ve homotopy category [π‘Ÿ] is a proper inverse of [πœ„].

Properties

  • Clearly if π‘Œ is a deformation retract of 𝑋, π‘Œ ≃𝑋. Thus a deformation retraction is a stronger kind of Homotopy equivalence.

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