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 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
from spaces to spectra has the following properties: 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
the associated homology theory, it follows that the induced natural transformation of graded abelian groups
is the universal transformation from a homology theory to
factors uniquely through a transformation of homology theories
Assembly maps exist for any homotopy invariant functor, by a simple homotopy-theoretical construction.
As a consequence of the Mayer-Vietoris sequence, the value of an excisive functor on a space
, together with the knowledge how these small subspaces intersect.
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.
Assembly maps are studied in geometric topology mainly for the two functors
, algebraic K-theory of spaces of
In fact, the homotopy fibers of both assembly maps have a direct geometric interpretation when
is a compact topological manifold.
Therefore, knowledge about the geometry of compact topological manifolds may be obtained by studying
, evaluated at a compact topological manifold
, is homotopy equivalent to the space of block structures of
Moreover, the fibration sequence induces a long exact sequence of homotopy groups which may be identified with the surgery exact sequence of
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 on
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
are in 1-to-1 correspondence with elements in the Whitehead group of