Assembly map

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. Equivariant assembly maps are used to formulate the Farrell–Jones conjectures in K- and L-theory.
Homotopy-theoretical viewpoint.
It is a classical result that for any generalized homology theory formula_1 on the category of topological spaces (assumed to be homotopy equivalent to CW-complexes), there is a spectrum formula_2 such that
where formula_4.
The functor formula_5 from spaces to spectra has the following properties:
A functor from spaces to spectra fulfilling these properties is called excisive.
Now suppose that formula_9 is a homotopy-invariant, not necessarily excisive functor. An assembly map is a natural transformation formula_10 from some excisive functor formula_11 to formula_9 such that formula_13 is a homotopy equivalence.
If we denote by formula_14 the associated homology theory, it follows that the induced natural transformation of graded abelian groups formula_15 is the universal transformation from a homology theory to formula_16, i.e. any other transformation formula_17 from some homology theory formula_18 factors uniquely through a transformation of homology theories formula_19.
Assembly maps exist for any homotopy invariant functor, by a simple homotopy-theoretical construction.
Geometric viewpoint.
As a consequence of the Mayer-Vietoris sequence, the value of an excisive functor on a space formula_20 only depends on its value on 'small' subspaces of formula_20, 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.
Assembly maps are studied in geometric topology mainly for the two functors formula_22, algebraic L-theory of formula_20, and formula_24, algebraic K-theory of spaces of formula_20. In fact, the homotopy fibers of both assembly maps have a direct geometric interpretation when formula_20 is a compact topological manifold. Therefore, knowledge about the geometry of compact topological manifolds may be obtained by studying formula_27- and formula_28-theory and their respective assembly maps.
In the case of formula_28-theory, the homotopy fiber formula_30 of the corresponding assembly map formula_31, evaluated at a compact topological manifold formula_32, is homotopy equivalent to the space of block structures of formula_32. Moreover, the fibration sequence
induces a long exact sequence of homotopy groups which may be identified with the surgery exact sequence of formula_32. 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.
For formula_36-theory, the homotopy fiber formula_37 of the corresponding assembly map is homotopy equivalent to the space of stable h-cobordisms on formula_32. This fact is called the stable parametrized h-cobordism theorem, proven by Waldhausen-Jahren-Rognes. It may be viewed as a parametrized version of the classical theorem which states that equivalence classes of h-cobordisms on formula_32 are in 1-to-1 correspondence with elements in the Whitehead group of formula_40.