What is homotopy in mathematics?

Answer 1

This article is about topology. For chemistry, see Homotopic groups.

The two bold paths shown above are homotopic relative to their endpoints. Thin lines mark isocontours of one possible homotopy.

In topology, two continuous functions from one topological space to another are called homotopic (Greek homos = identical and topos = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. An outstanding use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain pathological spaces. Consequently most algebraic topologists work with compactly generated spaces, CW complexes, or spectra.