# homotopy equivalence

Definition Suppose that $X$ and $Y$ are topological spaces and $f:X\to Y$ is a continuous map. If there exists a continuous map $g:Y\to X$ such that $f\circ g\simeq id_{Y}$ (i.e. $f\circ g$ is http://planetmath.org/node/1584homotopic to the identity mapping on $Y$), and $g\circ f\simeq id_{X}$, then $f$ is a homotopy equivalence. 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 $X\simeq Y$.

## 0.0.1 Properties

1. 1.

Any homeomorphism $f:X\to Y$ is obviously a homotopy equivalence with $g=f^{-1}$.

2. 2.

For topological spaces, homotopy equivalence is an equivalence relation.

3. 3.

A topological space $X$ is (by definition) contractible, if $X$ is homotopy equivalent to a point, i.e., $X\simeq\{x_{0}\}$.

## References

• 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