homotopy equivalence

Definition Suppose that X and Y are topological spacesMathworldPlanetmath and f:XY is a continuous map. If there exists a continuous map g:YX such that fgidY (i.e. fg is http://planetmath.org/node/1584homotopicMathworldPlanetmath to the identity mapping on Y), and gfidX, then f is a homotopy equivalenceMathworldPlanetmathPlanetmath. This homotopy equivalence is sometimes called strong homotopy equivalence to distinguish it from weak homotopy equivalence.

If there exist a homotopy equivalence between the topological spaces X and Y, we say that X and Y are homotopy equivalent, or that X and Y are of the same homotopy type. We then write XY.

0.0.1 Properties

  1. 1.

    Any homeomorphism f:XY is obviously a homotopy equivalence with g=f-1.

  2. 2.

    For topological spaces, homotopy equivalence is an equivalence relationMathworldPlanetmath.

  3. 3.

    A topological space X is (by definition) contractibleMathworldPlanetmath, if X is homotopy equivalent to a point, i.e., X{x0}.


  • 1 A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. Also available http://www.math.cornell.edu/ hatcher/AT/ATpage.htmlonline.
Title homotopy equivalence
Canonical name HomotopyEquivalence
Date of creation 2013-03-22 12:13:22
Last modified on 2013-03-22 12:13:22
Owner matte (1858)
Last modified by matte (1858)
Numerical id 14
Author matte (1858)
Entry type Definition
Classification msc 55P10
Related topic HomotopyOfMaps
Related topic WeakHomotopyEquivalence
Related topic Contractible
Related topic HomotopyInvariance
Related topic ChainHomotopyEquivalence
Related topic PathConnectnessAsAHomotopyInvariant
Related topic TheoremOnCWComplexApproximationOfQuantumStateSpacesInQAT
Defines homotopy equivalent
Defines homotopically equivalent
Defines homotopy type
Defines strong homotopy equivalence