# weak homotopy equivalence

A continuous map $f:X\rightarrow Y$ between path-connected based topological spaces is said to be a weak homotopy equivalence if for each $k\geq 1$ it induces an isomorphism $f_{*}:\pi_{k}(X)\rightarrow\pi_{k}(Y)$ between the $k$th homotopy groups. $X$ and $Y$ are then said to be weakly homotopy equivalent.

###### Remark 1.

It is not enough for $\pi_{k}(X)$ to be isomorphic to $\pi_{k}(Y)$ 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

 $X\to X_{1}\leftarrow X_{2}\to X_{3}\leftarrow\cdots\to X_{n}\leftarrow Y$

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