DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-03

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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The homomorphism on fundamental groups induced by a pointed continuous map

Definition

Let $f:X\to Y$ be continuous and let $x_0\in X$ . Composition sends a loop $\alpha$ at $x_0$ to the loop $f\circ\alpha$ at $f(x_0)$ . Using the loop classes and fundamental group of Based loops and the fundamental group, the proposed induced homomorphism is

$f_*:\pi_1(X,x_0)\longrightarrow\pi_1(Y,f(x_0)),\qquad f_*([\alpha]):=[f\circ\alpha].$

The next theorem proves that this value is independent of the representative, that it is a group homomorphism in the sense of Monoid homomorphism and group homomorphism, and that induced maps respect identities, composition and homotopies that fix the basepoint.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 59 results over 15 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)