In
mathematics, assembly maps are an important concept in
geometric topology. From the
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
-theoretical viewpoint, an assembly map is a
universal
Universal is the adjective for universe.
Universal may also refer to:
Companies
* NBCUniversal, a media and entertainment company
** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal
** Universal TV, a ...
approximation of a homotopy invariant
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
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
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
and
L-theory
In mathematics, algebraic ''L''-theory is the ''K''-theory of quadratic forms; the term was coined by C. T. C. Wall,
with ''L'' being used as the letter after ''K''. Algebraic ''L''-theory, also known as "Hermitian ''K''-theory",
is important in ...
play a central role in the topology of high-dimensional
manifolds, since their
homotopy fiber In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag ''(See Chapter 11 for construction.)'' is part of a constructio ...
s have a direct geometric
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-complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This clas ...
es), there is a
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
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 the fact that
has
Mayer-Vietoris sequence Mayer-Vietoris may refer to:
* Mayer–Vietoris axiom
* Mayer–Vietoris sequence
In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of top ...
s, an equivalent characterization of excision.
* It preserves arbitrary
coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduc ...
s. 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
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
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 group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s
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.
Geometric viewpoint
As a consequence of the
Mayer-Vietoris sequence Mayer-Vietoris may refer to:
* Mayer–Vietoris axiom
* Mayer–Vietoris sequence
In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of top ...
, the value of an excisive functor on a space
only depends on its value on 'small' subspaces of
, 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
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
, the excision property is proved by subdivision of
simplices
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
, 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
, algebraic
L-theory
In mathematics, algebraic ''L''-theory is the ''K''-theory of quadratic forms; the term was coined by C. T. C. Wall,
with ''L'' being used as the letter after ''K''. Algebraic ''L''-theory, also known as "Hermitian ''K''-theory",
is important in ...
of
, and
,
algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
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
- and
-theory and their respective assembly maps.
In the case of
-theory, the homotopy fiber
of the corresponding assembly map
, 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
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
of homotopy groups which may be identified with the
surgery exact sequence
In the mathematical surgery theory the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in dimension >4. The surgery structure set \mathcal (X) of a compact n-dimensional manifold X is ...
of
. This may be called the fundamental theorem of surgery theory and was developed subsequently by
William Browder,
Sergei Novikov,
Dennis Sullivan
Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate ...
,
C. T. C. Wall
Charles Terence Clegg "Terry" Wall (born 14 December 1936) is a British mathematician, educated at Marlborough College, Marlborough and Trinity College, Cambridge. He is an :wikt:emeritus, emeritus professor of the University of Liverpool, where ...
,
Frank Quinn, and
Andrew Ranicki
Andrew Alexander Ranicki (born Andrzej Aleksander Ranicki; 30 December 1948 – 21 February 2018) was a British mathematician who worked on algebraic topology. He was a professor of mathematics at the University of Edinburgh.
Life
Ranicki was ...
.
For
-theory, the homotopy fiber
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
.
Surgery theory
K-theory