TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-03

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Induced fundamental-group maps are well defined, functorial and invariant under based homotopy

Statement

Let $f:(X,x_0)\to(Y,y_0)$ be a continuous map with $f(x_0)=y_0$ . Then

$f_*([\alpha])=[f\circ\alpha]$

is a well-defined group homomorphism. For pointed continuous maps,

$\operatorname{id}_*=\operatorname{id},\qquad (g\circ f)_*=g_*\circ f_*.$

If $f_0,f_1:(X,x_0)\to(Y,y_0)$ are homotopic through a homotopy that keeps $x_0$ at $y_0$ , then $(f_0)_*=(f_1)_*$ .

Facts & Assumptions

Given: Pointed continuous maps between pointed topological spaces and based loops in their domains.

[L1]

The proposed induced map sends $[\alpha]$ to $[f\circ\alpha]$ (The homomorphism on fundamental groups induced by a pointed continuous map).

[L2]

Postcomposition preserves homotopies, and precomposition preserves a homotopy relative to a subspace whose image lies in the fixed subspace (Precomposition and postcomposition by continuous maps preserve homotopies, including their relative form).

[L3]

Loop multiplication traverses the first loop and then the second, using the explicit two-piece concatenation formula (Based loops and the fundamental group).

[L4]

A map between groups is a group homomorphism exactly when it preserves products, and the loop-class operations in question are groups (Monoid homomorphism and group homomorphism, Loop classes form the group $\pi_1(X,x_0)$ under concatenation).

Proof

technique · direct

1.1

If $\alpha_0$ and $\alpha_1$ are endpoint-homotopic, postcomposing their homotopy by $f$ gives an endpoint-homotopy from $f\circ\alpha_0$ to $f\circ\alpha_1$ by [L2]; hence the formula in [L1] is independent of the representative.

L1L2

1.2

The concatenation formulas give the literal equality $f\circ(\alpha*\beta)=(f\circ\alpha)*(f\circ\beta)$ , so $f_*([\alpha][\beta])=f_*([\alpha])f_*([\beta])$ ; thus $f_*$ is a group homomorphism.

L1L3L4

1.3

For every loop $\alpha$ , $\operatorname{id}\circ\alpha=\alpha$ and $(g\circ f)\circ\alpha=g\circ(f\circ\alpha)$ , which proves the identity and composition formulas on every loop class.

1.4

If $H$ is a homotopy from $f_0$ to $f_1$ fixing $x_0$ , precomposition by a based loop $\alpha$ gives an endpoint-fixed path homotopy from $f_0\circ\alpha$ to $f_1\circ\alpha$ by [L2], so $(f_0)_*([\alpha])=(f_1)_*([\alpha])$ .

L1L2

2.1

Steps 1.1--1.4 prove well-definedness, the homomorphism law, functoriality and based-homotopy invariance.

step 1.1step 1.2step 1.3step 1.4∎

Depends on

Used by

The fundamental group is a functor π₁:Top_*toGrp Proposition

Cited to discharge well-definedness by The homomorphism on fundamental groups induced by a pointed continuous map.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 62 results over 16 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, Chapter 1, Induced Homomorphisms (standard reference, not scraped)