Assembly Map - Homotopy-theoretical Viewpoint

Homotopy-theoretical Viewpoint

It is a classical result that for any generalized homology theory on the category of topological spaces (assumed to be homotopy equivalent to CW-complexes), there is a spectrum such that

where .

The functor from spaces to spectra has the following properties:

• It is homotopy-invariant (preserves homotopy equivalences). This reflects the fact that is homotopy-invariant.
• It preserves homotopy co-cartesian squares. This reflects that fact that has Mayer-Vietoris sequences, an equivalent characterization of excision.
• It preserves arbitrary coproducts. This reflects the disjoint-union axiom of .

A functor from spaces to spectra fulfilling these properties is called excisive.

Now suppose that is a homotopy-invariant, not necessarily excisive functor. An assembly map is a natural transformation from some excisive functor to such that is a homotopy equivalence.

If we denote by the associated homology theory, it follows that the induced natural transformation of graded abelian groups is the universal transformation from a homology theory to, i.e. any other transformation from some homology theory factors uniquely through a transformation of homology theories .

Assembly maps exist for any homotopy invariant functor, by a simple homotopy-theoretical construction.