Weak homotopy equivalence of topological spaces

2013-11-19T04:38:05Z

Cmalk: inserted twisted homology criterion

{{continuous map property}}

==Definition==

===Definition for path-connected spaces in terms of homotopy groups===

Let <math>A</math> and <math>B</math> be [[path-connected space]]s. A '''weak homotopy equivalence''' from <math>A</math> to <math>B</math> is a continuous map <math>f:A \to B</math> such that the functorially induced maps <math>\pi_n(f):\pi_n(A) \to \pi_n(B)</math> are group isomorphisms for all <math>n \ge 1</math>.

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 <math>\pi_n(f)</math>, we need to choose basepoints for <math>A</math> and <math>B</math>. 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 <math>A</math> and <math>B</math> be [[path-connected space]]s. Then a continuous map <math>f:A \to B</math> is a weak homotopy equivalence iff both of these conditions hold:

* The induced map <math>\pi_1(f):\pi_1(A) \to \pi_1(B)</math> is an isomorphism of groups.
* For every bundle of abelian groups <math>\mathcal A</math> over <math>B</math>, the induced map of twisted homology groups <math>f_*:H_n(A;f^* \mathcal A) \to H_n(B;\mathcal A)</math> is an isomorphism of groups for all <math>n \ge 0</math>.

As above, all of these maps are homomorphisms anyway, so it is enough to require them to be bijective. The above basepoint disclaimer for <math>\pi_1</math> also applies here.

If both <math>A</math> and <math>B</math> are [[Simply connected space|simply connected]] then the criterion is simpler: a continuous map <math>f:A \to B</math> is a weak homotopy equivalence iff the induced map on homology with <math>\mathbb Z</math> coefficients <math>f_*:H_n(A; \mathbb Z) \to H_n(B; \mathbb Z)</math> is an isomorphism of groups for all <math>n \ge 0</math>.

===Definition for spaces that are not path-connected===

Let <math>A</math> and <math>B</math> be [[topological space]]s. A '''weak homotopy equivalence''' from <math>A</math> to <math>B</math> is a continuous map <math>f:A \to B</math> such that:

* The functorially induced map <math>\pi_0(f): \pi_0(A) \to \pi_0(B)</math> is a bijection between the [[set of path components]] <math>\pi_0(A)</math> and the set of path components <math>\pi_0(B)</math>.
* For every path component of <math>A</math>, the restriction of <math>f</math> to a continuous map from that to its image path component of <math>B</math> is a weak homotopy equivalence of path-connected spaces.

==Facts==

* The existence of a weak homotopy equivalence from <math>A</math> to <math>B</math> does not imply the existence of a weak homotopy equivalence from <math>B</math> to <math>A</math>. 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 <math>\pi_n(A) \cong \pi_n(B)</math> as abstract groups is not enough to guarantee that <math>A</math> and <math>B</math> are weak homotopy-equivalent, even when <math>A</math> and <math>B</math> are [[manifold]]s or [[CW-space]]s (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 <math>\pi_n(A)</math> and <math>\pi_n(B)</math> are the trivial group/one-point set for all <math>n</math>. 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 <math>A</math> and <math>B</math> are both [[weakly contractible space]]s.
* Similarly, the mere fact that <math>\pi_1(A) \cong \pi_1(B)</math> as abstract groups and <math>H_n(A) \cong H_n(B)</math> as abstract groups does ''not'' imply that <math>A</math> and <math>B</math> 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 <math>H_n, n \ge 1</math>, are trivial.

==Relation with other properties==

===Stronger properties===

* [[Homotopy equivalence of topological spaces]]

===Weaker properties===

* [[Homology isomorphism of topological spaces]]

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Weak homotopy equivalence of topological spaces