Hurewicz map

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.

Now given any $f\in {\pi }_n(X,x_0)$, consider $f\circ \eta$. This gives a $n$-singular chain in $X$, and its homology class is precisely the element we are looking for.

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

• Hurewicz theorem
• Freudenthal suspension theorem