# Hurewicz theorem

*This fact is related to*: homotopy groups

## Contents

## Statement

### In terms of the Hurewicz homomorphism: absolute version

If is a -connected space with (viz its first homotopy groups vanish) then the Hurewicz map on the homotopy group is an isomorphism:

and moreover, all the reduced homology groups up to are zero. In particular, and for .

In the case , so that is a path-connected space but nothing more is known, the Hurewicz homomorphism from the Fundamental group (?) to the first homology group:

is surjective and has kernel precisely the derived subgroup of , so is isomorphic to the abelianization of .

### In terms of first non-vanishing member: absolute version

Suppose is a Path-connected space (?) that is simply connected. In particular, and are both *trivial* (the one-point set and the trivial group respectively). Then:

- The smallest for which is nontrivial is the same as the smallest for which is nontrivial.
- Both of these groups are isomorphic, and the Hurewicz homomorphism gives an isomorphism.

In the case that we are only given that is a path-connected space, and the Hurewicz homomorphism descends to this natural identification.

### Relative version

*Fill this in later*