Weak homotopy equivalence of topological spaces

This article defines a property of continuous maps between topological spaces

Definition

Definition for path-connected spaces in terms of homotopy groups

Let $A$ and $B$ be path-connected spaces. A weak homotopy equivalence from $A$ to $B$ is a continuous map $f:A\to B$ such that the functorially induced maps $\pi _{n}(f):\pi _{n}(A)\to \pi _{n}(B)$ are group isomorphisms for all $n\geq 1$.

Note that since the maps are homomorphisms anyway, it is enough to require them to be bijective.

Basepoint choice disclaimer for homotopy group isomorphism: To concretely define the map $\pi _{n}(f)$, we need to choose basepoints for $A$ and $B$. Change of basepoint, however, only results in pre-composition and post-composition by isomorphisms and does not affect whether or not the map is an isomorphism.

Equivalent definition for path-connected spaces in terms of homology groups

Let $A$ and $B$ be path-connected spaces. Then a continuous map $f:A\to B$ is a weak homotopy equivalence iff both of these conditions hold:

• The induced map $\pi _{1}(f):\pi _{1}(A)\to \pi _{1}(B)$ is an isomorphism of groups.
• For every bundle of abelian groups ${\mathcal {A}}$ over $B$, the induced map of twisted homology groups $f_{*}:H_{n}(A;f^{*}{\mathcal {A}})\to H_{n}(B;{\mathcal {A}})$ is an isomorphism of groups for all $n\geq 0$.

As above, all of these maps are homomorphisms anyway, so it is enough to require them to be bijective. The above basepoint disclaimer for $\pi _{1}$ also applies here.

If both $A$ and $B$ are simply connected then the criterion is simpler: a continuous map $f:A\to B$ is a weak homotopy equivalence iff the induced map on homology with $\mathbb {Z}$ coefficients $f_{*}:H_{n}(A;\mathbb {Z} )\to H_{n}(B;\mathbb {Z} )$ is an isomorphism of groups for all $n\geq 0$.

Definition for spaces that are not path-connected

Let $A$ and $B$ be topological spaces. A weak homotopy equivalence from $A$ to $B$ is a continuous map $f:A\to B$ such that:

• The functorially induced map $\pi _{0}(f):\pi _{0}(A)\to \pi _{0}(B)$ is a bijection between the set of path components $\pi _{0}(A)$ and the set of path components $\pi _{0}(B)$.
• For every path component of $A$, the restriction of $f$ to a continuous map from that to its image path component of $B$ is a weak homotopy equivalence of path-connected spaces.

Facts

• The existence of a weak homotopy equivalence from $A$ to $B$ does not imply the existence of a weak homotopy equivalence from $B$ to $A$. Thus, to get an equivalence relation on topological spaces, we need to take a symmetric transitive closure. We say that spaces are weak homotopy-equivalent topological spaces if they are in the same equivalence class under the equivalence relation thus obtained.
• The mere fact that $\pi _{n}(A)\cong \pi _{n}(B)$ as abstract groups is not enough to guarantee that $A$ and $B$ are weak homotopy-equivalent, even when $A$ and $B$ are manifolds or CW-spaces (see isomorphic homotopy groups not implies weak homotopy-equivalent). Rather, it is specifically important that the map must induce those isomorphisms.
• The exception to the above is in the case that both $\pi _{n}(A)$ and $\pi _{n}(B)$ are the trivial group/one-point set for all $n$. In this case, any map must induce isomorphisms since that's the only possible map between trivial groups/one-point sets. In this case, the spaces $A$ and $B$ are both weakly contractible spaces.
• Similarly, the mere fact that $\pi _{1}(A)\cong \pi _{1}(B)$ as abstract groups and $H_{n}(A)\cong H_{n}(B)$ as abstract groups does not imply that $A$ and $B$ are weak homotopy-equivalent. See isomorphic homology groups and isomorphic fundamental groups not implies weak homotopy-equivalent. Rather, it is specifically important that the map must induce those isomorphisms.
• The exception to the above is, once again, where the fundamental group and all the homology groups $H_{n},n\geq 1$, are trivial.

Relation with other properties

Stronger properties

• Homotopy equivalence of topological spaces

Weaker properties

• Homology isomorphism of topological spaces