weak homotopy equivalence

A continuous map f:XY between path-connected based topological spacesPlanetmathPlanetmath is said to be a weak homotopy equivalence if for each k1 it induces an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath f*:πk(X)πk(Y) between the kth homotopy groupsMathworldPlanetmath. X and Y are then said to be weakly homotopy equivalent.

Remark 1.

It is not enough for πk(X) to be isomorphic to πk(Y) for all k. The definition requires these isomorphisms to be induced by a space-level map f.

Remark 2.

More generally, two spaces X and Y are defined to be weakly homotopy equivalent if there is a sequence of spaces and maps


in which each map is a weak homotopy equivalence.

Title weak homotopy equivalence
Canonical name WeakHomotopyEquivalence
Date of creation 2013-03-22 13:25:45
Last modified on 2013-03-22 13:25:45
Owner antonio (1116)
Last modified by antonio (1116)
Numerical id 9
Author antonio (1116)
Entry type Definition
Classification msc 55P10
Synonym weak equivalence
Related topic HomotopyEquivalence
Related topic WeakHomotopyAdditionLemma
Related topic ApproximationTheoremForAnArbitrarySpace
Related topic OmegaSpectrum
Related topic WhiteheadTheorem
Defines weakly homotopy equivalent
Defines weakly equivalent