Weak homotopy equivalence of topological spaces: Difference between revisions

Latest revision as of 04:38, 19 November 2013

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 $π_{n} (f) : π_{n} (A) \to π_{n} (B)$ are group isomorphisms for all $n \geq 1$ .

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 $π_{n} (f)$ , 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.

Equivalent definition for path-connected spaces in terms of homology groups

Let $A$ and $B$ be path-connected spaces. Then a continuous map $f : A \to B$ is a weak homotopy equivalence iff both of these conditions hold:

The induced map $π_{1} (f) : π_{1} (A) \to π_{1} (B)$ is an isomorphism of groups.
For every bundle of abelian groups $A$ over $B$ , the induced map of twisted homology groups $f_{*} : H_{n} (A; f^{*} A) \to H_{n} (B; A)$ is an isomorphism of groups for all $n \geq 0$ .

As above, all of these maps are homomorphisms anyway, so it is enough to require them to be bijective. The above basepoint disclaimer for $π_{1}$ also applies here.

If both $A$ and $B$ are simply connected then the criterion is simpler: a continuous map $f : A \to B$ is a weak homotopy equivalence iff the induced map on homology with $Z$ coefficients $f_{*} : H_{n} (A; Z) \to H_{n} (B; Z)$ is an isomorphism of groups for all $n \geq 0$ .

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 $π_{0} (f) : π_{0} (A) \to π_{0} (B)$ is a bijection between the set of path components $π_{0} (A)$ and the set of path components $π_{0} (B)$ .
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 $π_{n} (A) ≅ π_{n} (B)$ 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 $π_{n} (A)$ and $π_{n} (B)$ 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 $π_{1} (A) ≅ π_{1} (B)$ as abstract groups and $H_{n} (A) ≅ H_{n} (B)$ 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 $H_{n}, n \geq 1$ , are trivial.

Relation with other properties

Stronger properties

Homotopy equivalence of topological spaces

Weaker properties

Homology isomorphism of topological spaces

@@ Line 11: / Line 11: @@
 ''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.
-===Definition for path-connected spaces in terms of fundamental groups, and homology groups===
+===Equivalent definition for path-connected spaces in terms of homology 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 following two conditions are true:
+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 functorially induced map <math>\pi_1(f): \pi_1(A) \to \pi_1(B)</math> is an isomorphism of [[fundamental group]]s.
+* The induced map <math>\pi_1(f):\pi_1(A) \to \pi_1(B)</math> is an isomorphism of groups.
-* The functorially induced maps <math>H_n(f): H_n(A) \to H_n(B)</math> are group isomorphisms for all <math>n \ge 1</math>.
+* 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>.
-Note that since the maps are homomorphisms anyway, it is enough to require them to be bijective.
+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.
-''Basepoint choice disclaimer for fundamental group isomorphism'': To concretely define the map <math>\pi_1(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.
+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===
@@ Line 28: / Line 28: @@
 * 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==