TOPICS
Search

Homotopy


A continuous transformation from one function to another. A homotopy between two functions f and g from a space X to a space Y is a continuous map G from X×[0,1]|->Y such that G(x,0)=f(x) and G(x,1)=g(x), where × denotes set pairing. Another way of saying this is that a homotopy is a path in the mapping space Map(X,Y) from the first function to the second.

Two mathematical objects are said to be homotopic if one can be continuously deformed into the other. The concept of homotopy was first formulated by Poincaré around 1900 (Collins 2004).


See also

Cohomotopy Group, h-Cobordism, Homotopic, Homotopy Axiom, Homotopy Class, Homotopy Theory, Homotopy Type Explore this topic in the MathWorld classroom

Explore with Wolfram|Alpha

References

Aubry, M. Homotopy Theory and Models. Boston, MA: Birkhäuser, 1995.Collins, G. P. "The Shapes of Space." Sci. Amer. 291, 94-103, July 2004.Krantz, S. G. "The Concept of Homotopy" §10.3.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 132-133, 1999.

Referenced on Wolfram|Alpha

Homotopy

Cite this as:

Weisstein, Eric W. "Homotopy." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Homotopy.html

Subject classifications