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 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 \ge 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.

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 \ge 1, are trivial.

Relation with other properties

Stronger properties

Weaker properties