Weak homotopy equivalence of topological spaces
From Topospaces
This article defines a property of continuous maps between topological spaces
Contents
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.
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.