What is homotopy in topology?

In topology, a homotopy describes a continuous deformation between two continuous functions from one topological space to another. The term comes from the Ancient Greek words for 'same' and 'place'. It plays a foundational role in algebraic topology through the definition of homotopy groups.

What we know

• Two continuous functions from one topological space to another are homotopic if one can be continuously deformed into the other. [1, 2]
• The word homotopy is derived from the Ancient Greek words homós meaning same or similar, and tópos meaning place. [2]
• A notable use of homotopy is in defining homotopy groups and cohomotopy groups, which are important invariants in algebraic topology. [2]

Where to go next

• the bigger pictureWhat are homology groups in algebraic topology?
• how it workedHow do topological spaces use compactly generated spaces or CW complexes?
• compared withHow are equivalent definitions used for mathematical structures?
• what followedWhat are cohomotopy groups?
• an unexpected connectionWhat is an immersion of differentiable manifolds?

Sources

• [1]homotopy (Wikidata Q746083) · CC0 1.0
• [2]Homotopy · CC BY-SA 4.0
• [3]Homology (mathematics) · CC BY-SA 4.0
• [4]Immersion (mathematics) · CC BY-SA 4.0
• [5]Equivalent definitions of mathematical structures · CC BY-SA 4.0