Tutte's homotopy theorem, introduced by W. T. Tutte in 1958, extends the graph-theoretic notion of a path to matroids, showing that any closed path in a matroid can be written as a composition of elementary closed paths, so that in a precise sense every closed path is homotopic to the trivial closed path. The result generalizes an intuition from graph theory, that a cycle can be broken down into simpler cycles, into the more abstract setting of matroid theory, where the underlying structure need not come from a graph at all.
Facts
StatementIn a matroid, any closed path can be written as a composition of elementary closed paths, so that in a precise sense every closed path is homotopic to the trivial closed path. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Proved By
Source Tutte homotopy theorem (Wikipedia)
Sources
1. Tutte's homotopy theorem, Wikipedia
Lede section, opening clause
Tutte's homotopy theorem, introduced by Tutte (1958)
Lede section, closing clause
generalises the concept of "path" from graphs to matroids, and states roughly that closed paths can be written as compositions of elementary closed paths, so that in some sense they are homotopic to the trivial closed path.
View the SourceTutte homotopy theorem (Wikipedia)
Proved By: W. T. Tutte, Lead paragraphQuote, Proved By: W. T. Tutte, Lead paragraph
In mathematics, Tutte's homotopy theorem, introduced by Tutte (1958), generalises the concept of "path" from graphs to matroids, and states roughly
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.