Homotopy

This article is about topology. For chemistry, see Homotopic groups.
The two dashed paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy.

In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from Ancient Greek: ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (/həˈmɒtəpiː/,[1] hə-MO-tə-pee; /ˈhoʊmoʊˌtoʊpiː/,[2] HOH-moh-toh-pee) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.[3]

In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra.

  1. ^ "Homotopy Definition & Meaning". Retrieved 22 April 2022.
  2. ^ "Homotopy Type Theory Discussed - Computerphile". YouTube. Retrieved 22 April 2022.
  3. ^ "Homotopy | mathematics". Encyclopedia Britannica. Retrieved 2019-08-17.