In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originate ...
, surgery theory is a collection of techniques used to produce one finite-dimensional
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 n ...
from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while
Andrew Wallace
Andrew Bruce Wallace (born 23 April 1968) is an Australian politician who served as the 31st Speaker of the House of Representatives from November 2021 to April 2022. He is a member of the Liberal Party and has been a member of the House of R ...
called it spherical modification. The "surgery" on a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
''M'' of dimension
, could be described as removing an imbedded
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
of dimension ''p'' from ''M''. Originally developed for differentiable (or,
smooth) manifolds, surgery techniques also apply to
piecewise linear (PL-) and
topological manifolds.
Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with,
handlebody decompositions.
More technically, the idea is to start with a well-understood manifold ''M'' and perform surgery on it to produce a manifold ''M''′ having some desired property, in such a way that the effects on the
homology,
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
s, or other invariants of the manifold are known. A relatively easy argument using
Morse theory shows that a manifold can be obtained from another one by a sequence of spherical modifications if and only if those two belong to a same
cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same di ...
class.
The classification of
exotic sphere
In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of a ...
s by led to the emergence of surgery theory as a major tool in high-dimensional topology.
Surgery on a manifold
A basic observation
If ''X'', ''Y'' are manifolds with boundary, then the boundary of the product manifold is
:
The basic observation which justifies surgery is that the space
can be understood either as the boundary of
or as the boundary of
. In symbols,
:
,
where
is the ''q''-dimensional disk, i.e., the set of points in
that are at distance one-or-less from a given fixed point (the center of the disk); for example, then,
is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
to the unit interval, while
is a circle together with the points in its interior.
Surgery
Now, given a manifold ''M'' of dimension
and an
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
, define another ''n''-dimensional manifold
to be
:
Since
and from the equation from our basic observation before, the gluing is justified then
:
One says that the manifold ''M''′ is produced by a ''surgery'' cutting out
and gluing in
, or by a ''p''-''surgery'' if one wants to specify the number ''p''. Strictly speaking, ''M''′ is a manifold with corners, but there is a canonical way to smooth them out. Notice that the submanifold that was replaced in ''M'' was of the same dimension as ''M'' (it was of
codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals ...
0).
Attaching handles and cobordisms
Surgery is closely related to (but not the same as)
handle attaching. Given an (''n'' + 1)-manifold with boundary (''L'', ∂''L'') and an embedding
: S
''p'' × D
''q'' → ∂''L'', where ''n'' = ''p'' + ''q'', define another (''n'' + 1)-manifold with boundary ''L''′ by
:
The manifold ''L''′ is obtained by "attaching a (''p'' + 1)-handle", with ∂''L''′ obtained from ∂''L'' by a ''p''-surgery
:
A surgery on ''M'' not only produces a new manifold ''M''′, but also a
cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same di ...
''W'' between ''M'' and ''M''′. The ''trace'' of the surgery is the
cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same di ...
(''W''; ''M'', ''M''′), with
:
the (''n'' + 1)-dimensional manifold with boundary ∂''W'' = ''M'' ∪ ''M''′ obtained from the product ''M'' × ''I'' by attaching a (''p'' + 1)-handle D
''p''+1 × D
''q''.
Surgery is symmetric in the sense that the manifold ''M'' can be re-obtained from ''M''′ by a (''q'' − 1)-surgery, the trace of which coincides with the trace of the original surgery, up to orientation.
In most applications, the manifold ''M'' comes with additional geometric structure, such as a map to some reference space, or additional bundle data. One then wants the surgery process to endow ''M''′ with the same kind of additional structure. For instance, a standard tool in surgery theory is surgery on
normal maps
In 3D computer graphics, normal mapping, or Dot3 bump mapping, is a texture mapping technique used for faking the lighting of bumps and dents – an implementation of bump mapping. It is used to add details without using more polygons. A common u ...
: such a process changes a normal map to another normal map within the same bordism class.
Examples
Effects on homotopy groups, and comparison to cell-attachment
Intuitively, the process of surgery is the manifold analog of attaching a cell to a topological space, where the embedding ''φ'' takes the place of the attaching map. A simple attachment of a (''q'' + 1)-cell to an ''n''-manifold would destroy the manifold structure for dimension reasons, so it has to be thickened by crossing with another cell.
Up to homotopy, the process of surgery on an embedding φ: S
''p'' × D
''q'' → ''M'' can be described as the attaching of a (''p'' + 1)-cell, giving the homotopy type of the trace, and the detaching of a ''q''-cell to obtain ''N''. The necessity of the detaching process can be understood as an effect of
Poincaré duality.
In the same way as a cell can be attached to a space to kill an element in some
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
of the space, a ''p''-surgery on a manifold ''M'' can often be used to kill an element
. Two points are important however: Firstly, the element
has to be representable by an embedding φ: S
''p'' × D
''q'' → ''M'' (which means embedding the corresponding sphere with a trivial
normal bundle). For instance, it is not possible to perform surgery on an orientation-reversing loop. Secondly, the effect of the detaching process has to be considered, since it might also have an effect on the homotopy group under consideration. Roughly speaking, this second point is only important when ''p'' is at least of the order of half the dimension of ''M''.
Application to classification of manifolds
The origin and main application of surgery theory lies in the
classification of manifolds
In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain.
Main themes Overview
* Low-dimensional manifolds are classified by geometric struc ...
of dimension greater than four. Loosely, the organizing questions of surgery theory are:
* Is ''X'' a manifold?
* Is ''f'' a diffeomorphism?
More formally, one asks these questions ''up to
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 deform ...
'':
* Does a space ''X'' have the homotopy type of a smooth manifold of a given dimension?
* Is a
homotopy equivalence
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 defo ...
''f'': ''M'' → ''N'' between two smooth manifolds
homotopic
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 deforma ...
to a diffeomorphism?
It turns out that the second ("uniqueness") question is a relative version of a question of the first ("existence") type; thus both questions can be treated with the same methods.
Note that surgery theory does ''not'' give a
complete set of invariants to these questions. Instead, it is
obstruction-theoretic: there is a primary obstruction, and a secondary obstruction called the
surgery obstruction which is only defined if the primary obstruction vanishes, and which depends on the choice made in verifying that the primary obstruction vanishes.
The surgery approach
In the classical approach, as developed 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 ...
and
C. T. C. Wall, surgery is done on
normal maps
In 3D computer graphics, normal mapping, or Dot3 bump mapping, is a texture mapping technique used for faking the lighting of bumps and dents – an implementation of bump mapping. It is used to add details without using more polygons. A common u ...
of degree one. Using surgery, the question "Is the normal map ''f'': ''M'' → ''X'' of degree one cobordant to a homotopy equivalence?" can be translated (in dimensions greater than four) to an algebraic statement about some element in an
L-group of 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 ...