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Canonical static concept record

Fundamental group $\pi_1(X,x_0)$

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Summary

The group of based loops modulo based homotopy, with multiplication by concatenation.

Record metadata

Canonical name: Fundamental group $\pi_1(X,x_0)$
Concept ID: fundamental_group
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebraic topology
Domains: Algebraic topology, Algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

pointed topological space

Data

based-loop homotopy classes

Axioms / constraints

group laws induced by concatenation

Canonically induces

Covering-space classification under standard hypotheses

Concept sources

Wikipedia — Fundamental group — Fundamental group · encyclopedic reference · source ID wp-fundamental-group
Hatcher — Algebraic Topology — Algebraic Topology · textbook · source ID hatcher-algebraic-topology

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Loop space $\Omega X$ Fundamental group $\pi_1(X,x_0)$

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_loop_space_to_fundamental_group_quotient_construction
Relation type: Quotient construction quotient-construction
Direction: source → target
Endpoint roles: source: Quotients to; target: Obtained as quotient of
Authored annotation: take path components

Authored explanation

Take based loops as representatives and identify two loops exactly when they are homotopic through based loops. The quotient set of based-homotopy classes is $\pi_1$ , with multiplication induced by concatenation.

How to interpret this relation type

Identify elements by a stated equivalence relation and equip the quotient with the induced structure. The edge detail must state the representatives and equivalence relation.

Relation sources

Hatcher — Algebraic Topology — Algebraic Topology · textbook · source ID hatcher-algebraic-topology

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