Weak homotopy equivalence of topological spaces

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This article defines a property of continuous maps between topological spaces

Definition

Definition for path-connected spaces in terms of homotopy groups

Let and be path-connected spaces. A weak homotopy equivalence from to is a continuous map such that the functorially induced maps are group isomorphisms for all .

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 , we need to choose basepoints for and . 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 and be path-connected spaces. Then a continuous map is a weak homotopy equivalence iff both of these conditions hold:

  • The induced map is an isomorphism of groups.
  • For every bundle of abelian groups over , the induced map of twisted homology groups is an isomorphism of groups for all .

As above, all of these maps are homomorphisms anyway, so it is enough to require them to be bijective. The above basepoint disclaimer for also applies here.

If both and are simply connected then the criterion is simpler: a continuous map is a weak homotopy equivalence iff the induced map on homology with coefficients is an isomorphism of groups for all .

Definition for spaces that are not path-connected

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

  • The functorially induced map is a bijection between the set of path components and the set of path components .
  • For every path component of , the restriction of to a continuous map from that to its image path component of is a weak homotopy equivalence of path-connected spaces.

Facts

  • The existence of a weak homotopy equivalence from to does not imply the existence of a weak homotopy equivalence from to . 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 as abstract groups is not enough to guarantee that and are weak homotopy-equivalent, even when and 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 and are the trivial group/one-point set for all . 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 and are both weakly contractible spaces.
  • Similarly, the mere fact that as abstract groups and as abstract groups does not imply that and 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 , are trivial.

Relation with other properties

Stronger properties

Weaker properties