Hurewicz map

Definition

Explicit definition

Let $X$ be a path-connected space. For $n$ a positive integer, the $n^{th}$ Hurewicz map based at $x_0$ of $X$ is a map:

${\pi }_n(X,x_0)\to H_n(X)$

where ${\pi }_n(X,x_0)$ is the $n^{th}$ homotopy group, and $H_n(X)$ is the $n^{th}$ singular homology group.

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

$\eta :{\mathrm{\Delta }}^n\to S^n$

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

Now given any based continuous map $f:(S^n,*)\to (X,x_0)$, consider $f\circ \eta$. 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 ${\pi }_n(X,x_0)$, 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 $f_1\circ \eta$ and $f_2\circ \eta$ 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 $H_n(S^n)=\mathbb{Z}$ 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 ${\mathrm{\Delta }}^n$ 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 $\partial {\mathrm{\Delta }}^n$ 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 $n^{th}$ Hurewicz map is an isomorphism (if $n\ge 2$) and is the map to the abelianization (if $n=1$).
• Freudenthal suspension theorem