In mathematics, the Tutte homotopy theorem, introduced by Tutte (1958), 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.
The subsets of Q that are unions of circuits are called flats (this is the language used in Tutte's original paper, however in modern usage the flats of a matroid mean something different).
An elementary path is one of the form (X,Y,X), or (X,Y,Z,X) with X,Y,Z all lying in some 2-flat.
A weak form of Tutte's homotopy theorem states that any closed path is homotopic to the trivial path.
A stronger form states a similar result for paths not meeting certain "convex" subsets.