PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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A retract induces an injection on fundamental groups, and a deformation retract induces an isomorphism

Statement

Let $A \subseteq X$ , let $i : A ↪ X$ be the inclusion, and choose $a \in A$ . If $A$ is a retract of $X$ (Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise), then

$i_{*} : π_{1} (A, a) ⟶ π_{1} (X, a)$

is injective. If $A$ is a deformation retract of $X$ , the inclusion and retraction induce mutually inverse fundamental-group isomorphisms at every basepoint of $A$ .

Facts & Assumptions

Given: A subspace $A \subseteq X$ , its inclusion $i : A ↪ X$ , a basepoint $a \in A$ , and a retraction $r : X \to A$ ; in the second clause, a deformation retraction from $X$ onto $A$ .

[F1]

A continuous map $r : X \to A$ is a retraction when $r \circ i = {id}_{A}$ ; for a deformation retract, ${id}_{X}$ is homotopic to $i \circ r$ through a homotopy that fixes every point of $A$ (Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise).

[L1]

For pointed continuous maps, ${id}_{*} = id$ and $(g \circ f)_{*} = g_{*} \circ f_{*}$ ; pointed-homotopic maps induce the same homomorphism on fundamental groups (Induced fundamental-group maps are well defined, functorial and invariant under based homotopy).

Proof

technique · direct

1.1

Since $a \in A$ , both $i : (A, a) \to (X, a)$ and $r : (X, a) \to (A, a)$ are pointed, and $r \circ i = {id}_{A}$ .

givenF1

2.1

Functoriality gives $r_{*} \circ i_{*} = (r \circ i)_{*} = id$ , so $i_{*}$ has a left inverse and is injective.

step 1.1L1algebra

3.1

If $A$ is a deformation retract, the homotopy from ${id}_{X}$ to $i \circ r$ fixes $a$ , so [L1] also gives $i_{*} \circ r_{*} = (i \circ r)_{*} = id$ ; hence $i_{*}$ and $r_{*}$ are mutually inverse isomorphisms.

step 2.1F1L1∎

Depends on

Used by

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Allen Hatcher, Algebraic Topology, Proposition 1.17 (standard reference, not scraped)