In mathematics, assembly maps are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data.

Assembly maps for algebraic K-theory and L-theory play a central role in the topology of high-dimensional manifolds, since their homotopy fibers have a direct geometric interpretation.

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

.

from spaces to spectra has the following properties:

• is homotopy-invariant.

• has Mayer-Vietoris sequences, an equivalent characterization of excision.

• It preserves arbitrary coproducts.

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

is a homotopy-invariant, not necessarily excisive functor. An assembly map is a natural transformation is a homotopy equivalence.

the associated homology theory, it follows that the induced natural transformation of graded abelian groups.

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

Geometric viewpoint

As a consequence of the Mayer-Vietoris sequence, together with the knowledge how these small subspaces intersect. In a cycle representation of the associated homology theory, this means that all cycles must be representable by small cycles. For instance, for singular homology, the excision property is proved by subdivision of simplices, obtaining sums of small simplices representing arbitrary homology classes.

In this spirit, for certain homotopy-invariant functors which are not excisive, the corresponding excisive theory may be constructed by imposing 'control conditions', leading to the field of controlled topology. In this picture, assembly maps are 'forget-control' maps, i.e. they are induced by forgetting the control conditions.

Importance in geometric topology

, algebraic L-theory, algebraic K-theory-theory and their respective assembly maps.

. Moreover, the fibration sequence

induces a long exact sequence of homotopy groups which may be identified with the surgery exact sequence. This may be called the fundamental theorem of surgery theory and was developed subsequently by William Browder, Sergei Novikov, Dennis Sullivan, C. T. C. Wall, Frank Quinn, and Andrew Ranicki.

of the corresponding assembly map is homotopy equivalent to the space of stable h-cobordisms. This fact is called the stable parametrized h-cobordism theorem are in 1-to-1 correspondence with elements in the Whitehead group.

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