Hurewicz map

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 $f \in π_{n} (X, x_{0})$ , consider $f \circ η$ . This gives a $n$ -singular chain in $X$ , and its homology class is precisely the element we are looking for.

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