Deformation retraction

Let $𝑌 \subseteq 𝑋$ be a subspace and $𝜄 : 𝑌 \to 𝑋$ the inclusion. A deformation retraction $𝑟 : 𝑋 \to 𝑌$ is a is a retraction, i.e. $𝑟 𝜄 = i d 𝑌$ , such that $𝜄 𝑟 ≃ i d 𝑋$ #m/def/homotopy

𝑟 𝜄 = i d 𝑌 [𝜄 𝑟] = [i d 𝑋]

Hence in $𝖳 𝗈 𝗉$ , $𝑟$ is a left inverse of $𝜄$ , but in $𝖳 𝗈 𝗉 / ≃$ $[𝑟]$ is a proper inverse of $[𝜄]$ .

Properties

Clearly if $𝑌$ is a deformation retract of $𝑋$ , $𝑌 ≃ 𝑋$ . Thus a deformation retraction is a stronger kind of Homotopy equivalence.

#state/tidy | #lang/en | #SemBr