TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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The inclusion of a deformation retract is a homotopy equivalence with the retraction as homotopy inverse

Statement

If $A$ is a deformation retract of $X$ , with inclusion $i:A\hookrightarrow X$ and retraction $r:X\to A$ , then $i$ is a homotopy equivalence and $r$ is a homotopy inverse of $i$ .

Facts & Assumptions

Given: A deformation retraction $(r,H)$ of $X$ onto $A$ , with inclusion $i:A\hookrightarrow X$ .

[A1]

Retraction gives $r\circ i=\operatorname{id}_A$ , and deformation retraction gives $i\circ r\simeq_A\operatorname{id}_X$ (Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise).

[A2]

A continuous map is a homotopy equivalence when it has a continuous map whose composites with it are homotopic to the identity maps (Homotopy equivalences, homotopy inverses and spaces of the same homotopy type).

Proof

technique · direct

1.1

The equality $r\circ i=\operatorname{id}_A$ is in particular a homotopy $r\circ i\simeq\operatorname{id}_A$ , while $i\circ r\simeq_A\operatorname{id}_X$ is in particular an ordinary homotopy.

2.1

Hence $r$ satisfies both homotopy-inverse conditions for $i$ , so $i$ is a homotopy equivalence with homotopy inverse $r$ .

step 1.1A2∎

Depends on

Used by

For n≥1, the punctured Euclidean space ℝⁿ∖{0} is homotopy equivalent to Sⁿ⁻¹ Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 13 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

A. Hatcher, Algebraic Topology, Section 0 (standard reference, not scraped)