Weak homotopy equivalence of topological spaces: Difference between revisions

Revision as of 15:48, 29 July 2011

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.

Definition for path-connected spaces in terms of fundamental groups, and homology 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 following two conditions are true:

The functorially induced map $π_{1} (f) : π_{1} (A) \to π_{1} (B)$ is an isomorphism of fundamental groups.
The functorially induced maps $H_{n} (f) : H_{n} (A) \to H_{n} (B)$ 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 fundamental group isomorphism: To concretely define the map $π_{1} (f)$ , 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 $π_{0} (f) : π_{0} (A) \to π_{0} (B)$ 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.

Relation with other properties

Stronger properties

Homotopy equivalence of topological spaces

Weaker properties

Homology isomorphism of topological spaces

@@ Line 3: / Line 3: @@
 ==Definition==
-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 maps of <math>\pi_n(f):\pi_n(A) \to \pi_n(B)</math> are isomorphisms for all <math>n \ge 0</math>. Note that:
+===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.
+===Definition for path-connected spaces in terms of fundamental groups, and 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:
+* The functorially induced map <math>\pi_1(f): \pi_1(A) \to \pi_1(B)</math> is an isomorphism of [[fundamental group]]s.
+* 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>.
+Note that since the maps are homomorphisms anyway, it is enough to require them to be bijective.
+''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.
+===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.
-# For <math>n = 0</math>, we have a set map <math>\pi_0(f):\pi_0(A) \to \pi_0(B)</math> from the [[defining ingredient::space of path components]] of <math>A</math> to the [[space of path components]] of <math>B</math>. We simply require this set map to be a bijection.
-# For <math>n \ge 1</math>, we have a group homomorphism <math>\pi_n(f):\pi_n(A) \to \pi_n(B)</math> of [[defining ingredient::homotopy group]]s. We require this map to be a group isomorphism. Note that a group homomorphism is an isomorphism iff it is a bijection, so this is equivalent to requiring the map to be a set bijection.
 ==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 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.
+* 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.
 ==Relation with other properties==