In the mathematical
surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while An ...
the surgery exact sequence is the main technical tool to calculate the
surgery structure set of a compact
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
in dimension
. The
surgery structure set of a compact
-dimensional manifold
is a
pointed set
In mathematics, a pointed set (also based set or rooted set) is an ordered pair (X, x_0) where X is a set and x_0 is an element of X called the base point, also spelled basepoint.
Maps between pointed sets (X, x_0) and (Y, y_0) – called based m ...
which classifies
-dimensional manifolds within the homotopy type of
.
The basic idea is that in order to calculate
it is enough to understand the other terms in the sequence, which are usually easier to determine. These are on one hand the
normal invariants which form
generalized cohomology groups, and hence one can use standard tools of
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...
to calculate them at least in principle. On the other hand, there are the
L-groups which are defined algebraically in terms of
quadratic forms
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
or in terms of
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
es with quadratic structure. A great deal is known about these groups. Another part of the sequence are the
surgery obstruction In mathematics, specifically in surgery theory, the surgery obstructions define a map \theta \colon \mathcal (X) \to L_n (\pi_1 (X)) from the normal invariants to the L-groups which is in the first instance a set-theoretic map (that means not nec ...
maps from normal invariants to the L-groups. For these maps there are certain
characteristic classes
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classe ...
formulas, which enable to calculate them in some cases. Knowledge of these three components, that means: the normal maps, the L-groups and the surgery obstruction maps is enough to determine the structure set (at least up to extension problems).
In practice one has to proceed case by case, for each manifold
it is a unique task to determine the surgery exact sequence, see some examples below. Also note that there are versions of the surgery exact sequence depending on the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
of manifolds we work with: smooth (DIFF), PL, or
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''- dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout m ...
s and whether we take
Whitehead torsion In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau(f) which is an element in the Whitehead group \op ...
into account or not (decorations
or
).
The original 1962 work of
Browder and
Novikov on the existence and uniqueness of manifolds within a
simply-connected homotopy type was reformulated by
Sullivan
Sullivan may refer to:
People
Characters
* Chloe Sullivan, from the television series ''Smallville''
* Colin Sullivan, a character in the film ''The Departed'', played by Matt Damon
* Harry Sullivan (''Doctor Who''), from the British science f ...
in 1966 as a surgery exact sequence.
In 1970
Wall
A wall is a structure and a surface that defines an area; carries a load; provides security, shelter, or soundproofing; or, is decorative. There are many kinds of walls, including:
* Walls in buildings that form a fundamental part of the super ...
developed
non-simply-connected surgery theory and the surgery exact sequence for manifolds with arbitrary
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
.
Definition
The surgery exact sequence is defined as
:
where:
the entries
and
are the
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 of
normal invariants,
the entries
and
are the
L-groups associated to the
group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...