# Hurewicz map is well-defined

From Topospaces

## Statement

### Loose statement

Let be a path-connected space. For a positive integer, we want to show that the Hurewicz map based at of is a well-defined map:

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

### The strict map

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.

### What we need to show

To note that this is indeed well-defined, 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.

## Proof

*Fill this in later*