This is in the same vein as my previous question on the representability of the cohomology ring. Why are the homology groups not corepresentable in the homotopy category of spaces? 



Why is homology not (co)representable?
Corepresentable functors preserve products; homology does not.


One replacement is the following. Let X be a CW-complex with basepoint. Then the spaces {K(Z,n)} represent reduced integral homology in the sense that for sufficiently large n, the reduced homology Hk(X) coincides with the homotopy groups of the smash product:



```
pin+k(X ^ K(Z,n)) = [Sn+k, X ^ K(Z,n)]
```

This is some kind of "stabilization", and it factors through taking the n-fold suspension of X. Taking suspensions makes wedges more and more closely related to products. This doesn't make homology representable, but provides some alternative description that's more workable than simply an abstract functor.


