Hurewicz map: Difference between revisions

Latest revision as of 22:01, 9 January 2011

Definition

Explicit definition

Let $X$ be a path-connected space. For $n$ a positive integer, the $n^{t h}$ Hurewicz map based at $x_{0}$ of $X$ is a map:

$π_{n} (X, x_{0}) \to H_{n} (X)$

where $π_{n} (X, x_{0})$ is the $n^{t h}$ homotopy group, and $H_{n} (X)$ is the $n^{t h}$ singular homology group.

The map is defined as follows. First define a map:

$η : Δ^{n} \to S^{n}$

which essentially uses the identification of $S^{n}$ with the quotient of $Δ^{n}$ by the collapse of its boundary to a single point, i.e., a homeomorphism $Δ^{n} / \partial Δ^{n} \to S^{n}$ .

Now given any based continuous map $f : (S^{n}, *) \to (X, x_{0})$ , consider $f \circ η$ . This gives a $n$ -singular chain in $X$ , and its homology class is precisely the element we are looking for.

To note that this gives a well-defined map on $π_{n} (X, x_{0})$ , we need to show that if $f_{1}$ and $f_{2}$ are homotopic maps as based continuous maps from $(S^{n}, *)$ to $(X, x_{0})$ , then $f_{1} \circ η$ and $f_{2} \circ η$ are both in the same homology class. Further information: Hurewicz map is well-defined

Hands-off definition

Here is an alternative description of the map. We use the fact that $f : S^{n} \to X$ induces a map between $H_{n} (S^{n})$ and $H_{n} (X)$ . But $H_{n} (S^{n}) = Z$ and we can thus simply look at the image of the generator of this, to give an element in $H_{n} (X)$ .

Facts

The image of the Hurewicz map

The image of the Hurewicz map is a subgroup comprising those singular homology classes that are represented by a singular simplex with the property that all points of the boundary get mapped to the basepoint $x_{0}$ .

In particular, the Hurewicz map being surjective means that every continuous map from $Δ^{n}$ to $X$ (and in fact, every formal sum of such continuous maps) is homologous to a continuous map with the property that the entire boundary $\partial Δ^{n}$ is mapped to $x_{0}$ .

The kernel of the Hurewicz map

The kernel of the Hurewicz map comprises those homotopy classes of maps from $S^{n}$ to $(X, x_{0})$ that are nullhomologous. In the case $n = 1$ , the explanation lies in non-commutativity, i.e., by cutting and rearranging the pieces of the map, we can get a nullhomotopic map.

Related facts

Hurewicz theorem: This states that if $X$ is $(n - 1)$ -connected, then the $n^{t h}$ Hurewicz map is an isomorphism (if $n \geq 2$ ) and is the map to the abelianization (if $n = 1$ ).
Freudenthal suspension theorem

@@ Line 1: / Line 1: @@
 ==Definition==
+===Explicit definition===
 Let <math>X</math> be a [[path-connected space]]. For <math>n</math> a positive integer, the <math>n^{th}</math> [[Hurewicz map]] based at <math>x_0</math> of <math>X</math> is a map:
@@ Line 11: / Line 13: @@
 <math>\eta:\Delta^n \to S^n</math>
-which essentially uses the identification of <math>S^n</math> with the boundary of <math>\Delta^n</math>.
+which essentially uses the identification of <math>S^n</math> with the quotient of <math>\Delta^n</math> by the collapse of its boundary to a single point, i.e., a homeomorphism <math>\Delta^n/\partial \Delta^n \to S^n</math>.
+Now given any based continuous map <math>f: (S^n,*) \to (X,x_0)</math>, consider <math>f \circ \eta</math>. This gives a <math>n</math>-[[singular chain]] in <math>X</math>, and its homology class is precisely the element we are looking for.
+To note that this gives a well-defined map on <math>\pi_n(X,x_0)</math>, we need to show that if <math>f_1</math> and <math>f_2</math> are [[homotopic map]]s as based continuous maps from <math>(S^n,*)</math> to <math>(X,x_0)</math>, then <math>f_1 \circ \eta</math> and <math>f_2 \circ \eta</math> are both in the same homology class. {{further|[[Hurewicz map is well-defined]]}}
-Now given any <math>f \in \pi_n(X,x_0)</math>, consider <math>f \circ \eta</math>. This gives a <math>n</math>-singular chain in <math>X</math>, and its homology class is precisely the element we are looking for.
+===Hands-off definition===
 Here is an alternative description of the map. We use the fact that <math>f: S^n \to X</math> induces a map between <math>H_n(S^n)</math> and <math>H_n(X)</math>. But <math>H_n(S^n) = \mathbb{Z}</math> and we can thus simply look at the image of the generator of this, to give an element in <math>H_n(X)</math>.
@@ Line 19: / Line 25: @@
 ==Facts==
-* [[Hurewicz theorem]]
+===The image of the Hurewicz map===
+The image of the Hurewicz map is a subgroup comprising those singular homology classes that are represented by a [[singular simplex]] with the property that all points of the boundary get mapped to the basepoint <math>x_0</math>.
+In particular, the Hurewicz map being surjective means that every continuous map from <math>\Delta^n</math> to <math>X</math> (and in fact, every formal sum of such continuous maps) is homologous to a continuous map with the property that the entire boundary <math>\partial \Delta^n</math> is mapped to <math>x_0</math>.
+===The kernel of the Hurewicz map===
+The kernel of the Hurewicz map comprises those homotopy classes of maps from <math>S^n</math> to <math>(X,x_0)</math> that are nullhomologous. In the case <math>n = 1</math>, the explanation lies in non-commutativity, i.e., by cutting and rearranging the pieces of the map, we can get a nullhomotopic map.
+===Related facts===
+* [[Hurewicz theorem]]: This states that if <math>X</math> is <math>(n - 1)</math>-connected, then the <math>n^{th}</math> Hurewicz map is an isomorphism (if <math>n \ge 2</math>) and is the map to the abelianization (if <math>n = 1</math>).
 * [[Freudenthal suspension theorem]]