Hurewicz map

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:

$$\eta:\Delta^n \to S^n$$

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

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.

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 $$H_n(X)$$.

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 $$\Delta^n$$ 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 $$\partial \Delta^n$$ 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