Weak homotopy equivalence of topological spaces
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.
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.
- 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.