Weak homotopy equivalence

Definition

Let $X$ and $Y$ be topological spaces, and let $f \colon X \to Y$ be a continuous map. The map $f$ is a weak homotopy equivalence if both of the following hold:

$f$ induces a bijection $\pi_0(X) \to \pi_0(Y)$ between the sets of path components;
for every $x \in X$ and every $n \ge 1$ , the induced map $f_* \colon \pi_n(X, x) \to \pi_n(Y, f(x))$ of homotopy groups is an isomorphism.

Remarks

Example: every homotopy equivalence $f$ with homotopy inverse $g$ is a weak homotopy equivalence, since $g_*$ inverts $f_*$ up to the change-of-base-point isomorphisms along the homotopies.

Example: all homotopy groups of the Warsaw circle are trivial, so the map from it to a point is a weak homotopy equivalence. This map is not a homotopy equivalence.

Non-example: the inclusion $S^1 \hookrightarrow D^2$ of the boundary circle into the disk is a bijection on $\pi_0$ . It fails clause 2, because $\pi_1(S^1) \cong \Z$ while $\pi_1(D^2)$ is trivial.

Weak homotopy equivalences induce isomorphisms on singular homology and cohomology. A weak homotopy equivalence is also called a weak equivalence.

Reading path

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Tag	`SREW4HZRM4`
URI	`m2://SREW4HZRM4`
Kind	definition
Writing	W0 stub
MSC	55P10, 55Q05
Defines	Weak homotopy equivalence
Uses	Homotopy group, Continuous map
Changed	2 November 2026

Weak homotopy equivalence

Definition

Remarks

Reading path

Further reading

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