weak homotopy equivalence

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

Remark 1.

It is not enough for ${\pi }_k\mathit{}\mathrm{(}X\mathrm{)}$ to be isomorphic to ${\pi }_k\mathit{}\mathrm{(}Y\mathrm{)}$ for all $k\mathrm{.}$ The definition requires these isomorphisms to be induced by a space-level map $f\mathrm{.}$

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

$$X\to X_1\leftarrow X_2\to X_3\leftarrow \mathrm{\cdots }\to X_n\leftarrow Y$$

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