# Hurewicz map

## Contents

## Definition

### Explicit definition

Let be a path-connected space. For a positive integer, the Hurewicz map based at of is a map:

where is the homotopy group, and is the singular homology group.

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

which essentially uses the identification of with the quotient of by the collapse of its boundary to a single point, i.e., a homeomorphism .

Now given any based continuous map , consider . This gives a -singular chain in , and its homology class is precisely the element we are looking for.

To note that this gives a well-defined map on , we need to show that if and are homotopic maps as based continuous maps from to , then and 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 induces a map between and . But and we can thus simply look at the image of the generator of this, to give an element in .

## 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 .

In particular, the Hurewicz map being surjective means that every continuous map from to (and in fact, every formal sum of such continuous maps) is homologous to a continuous map with the property that the entire boundary is mapped to .

### The kernel of the Hurewicz map

The kernel of the Hurewicz map comprises those homotopy classes of maps from to that are nullhomologous. In the case , 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 is -connected, then the Hurewicz map is an isomorphism (if ) and is the map to the abelianization (if ).
- Freudenthal suspension theorem