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Canonical static concept record

Higher homotopy group $\pi_n(X,x_0)$

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Summary

For $n\ge 2$ , the group of based homotopy classes of maps $S^n\to X$ ; these groups are abelian.

Record metadata

Canonical name: Higher homotopy group $\pi_n(X,x_0)$
Concept ID: higher_homotopy_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
integer $n\ge2$

Data

based maps from $S^n$ modulo homotopy

Canonically induces

Higher-dimensional homotopy invariants

Concept sources

Wikipedia — Homotopy group — Homotopy group · encyclopedic reference · source ID wp-higher-homotopy-group
Hatcher — Algebraic Topology — Algebraic Topology · textbook · source ID hatcher-algebraic-topology

Incoming relations (arrows to this concept)

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Loop space $\Omega X$ Higher homotopy group $\pi_n(X,x_0)$

Permalink to relation

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

Relation ID: story_loop_space_to_higher_homotopy_group_canonical_construction
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: iterate loops or map spheres

Authored explanation

Higher homotopy groups are obtained from based maps of spheres, equivalently iterated loop spaces.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

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

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Each relation below starts at this concept.