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Homotopy Equivalence


Two topological spaces X and Y are homotopy equivalent if there exist continuous maps f:X->Y and g:Y->X, such that the composition f degreesg is homotopic to the identity id_Y on Y, and such that g degreesf is homotopic to id_X. Each of the maps f and g is called a homotopy equivalence, and g is said to be a homotopy inverse to f (and vice versa).

One should think of homotopy equivalent spaces as spaces, which can be deformed continuously into one another.

Certainly any homeomorphism f:X->Y is a homotopy equivalence, with homotopy inverse f^(-1), but the converse does not necessarily hold.

Some spaces, such as any ball B^k, can be deformed continuously into a point. A space with this property is said to be contractible, the precise definition being that X is homotopy equivalent to a point. It is a fact that a space X is contractible, if and only if the identity map id_X is null-homotopic, i.e., homotopic to a constant map.


This entry contributed by Rasmus Hedegaard

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Cite this as:

Hedegaard, Rasmus. "Homotopy Equivalence." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HomotopyEquivalence.html

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