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 ...
, cobordism is a fundamental
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
on the class of
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
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 ...
s of the same dimension, set up using the concept of the
boundary (French ''
bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dimension are ''cobordant'' if their
disjoint union is the ''boundary'' of a compact manifold one dimension higher.
The boundary of an (''n'' + 1)-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 ...
''W'' is an ''n''-dimensional manifold ∂''W'' that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by
René Thom
René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958.
He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he becam ...
for
smooth manifolds (i.e., differentiable), but there are now also versions for
piecewise linear and
topological manifolds.
A ''cobordism'' between manifolds ''M'' and ''N'' is a compact manifold ''W'' whose boundary is the disjoint union of ''M'' and ''N'',
.
Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right. Cobordism is a much coarser equivalence relation than
diffeomorphism or
homeomorphism
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 isom ...
of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to
diffeomorphism or
homeomorphism
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 isom ...
in dimensions ≥ 4 – because the
word problem for groups cannot be solved – but it is possible to classify manifolds up to cobordism. Cobordisms are central objects of study 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 ...
and
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 ...
. In geometric topology, cobordisms are
intimately connected with
Morse theory, and
''h''-cobordisms are fundamental in the study of high-dimensional manifolds, namely
surgery theory. In algebraic topology, cobordism theories are fundamental
extraordinary cohomology theories, and
categories of cobordisms are the domains of
topological quantum field theories.
Definition
Manifolds
Roughly speaking, an ''n''-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 ...
''M'' is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
locally (i.e., near each point)
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 an open subset of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
A
manifold with boundary is similar, except that a point of ''M'' is allowed to have a neighborhood that is homeomorphic to an open subset of the
half-space
:
Those points without a neighborhood homeomorphic to an open subset of Euclidean space are the boundary points of
; the boundary of
is denoted by
. Finally, a
closed manifold
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example i ...
is, by definition, a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
manifold without boundary (
.)
Cobordisms
An
-dimensional ''cobordism'' is a
quintuple consisting of an
-dimensional compact differentiable manifold with boundary,
; closed
-manifolds
,
; and
embeddings
,
with disjoint images such that
:
The terminology is usually abbreviated to
. ''M'' and ''N'' are called ''cobordant'' if such a cobordism exists. All manifolds cobordant to a fixed given manifold ''M'' form the ''cobordism class'' of ''M''.
Every closed manifold ''M'' is the boundary of the non-compact manifold ''M'' × [0, 1); for this reason we require ''W'' to be compact in the definition of cobordism. Note however that ''W'' is ''not'' required to be connected; as a consequence, if ''M'' = ∂''W''
1 and ''N'' = ∂''W''
2, then ''M'' and ''N'' are cobordant.
Examples
The simplest example of a cobordism is the unit interval . It is a 1-dimensional cobordism between the 0-dimensional manifolds , . More generally, for any closed manifold ''M'', (; , ) is a cobordism from ''M'' × to ''M'' × .
If ''M'' consists of a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
, and ''N'' of two circles, ''M'' and ''N'' together make up the boundary of a
pair of pants ''W'' (see the figure at right). Thus the pair of pants is a cobordism between ''M'' and ''N''. A simpler cobordism between ''M'' and ''N'' is given by the disjoint union of three disks.
The pair of pants is an example of a more general cobordism: for any two ''n''-dimensional manifolds ''M'', ''M''′, the disjoint union
is cobordant to the
connected sum The previous example is a particular case, since the connected sum
is isomorphic to
The connected sum
is obtained from the disjoint union
by surgery on an embedding of
in
, and the cobordism is the trace of the surgery.
Terminology
An ''n''-manifold ''M'' is called ''null-cobordant'' if there is a cobordism between ''M'' and the empty manifold; in other words, if ''M'' is the entire boundary of some (''n'' + 1)-manifold. For example, the circle is null-cobordant since it bounds a disk. More generally, a ''n''-sphere is null-cobordant since it bounds a (''n'' + 1)-disk. Also, every orientable surface is null-cobordant, because it is the boundary of a
handlebody. On the other hand, the 2''n''-dimensional
real projective space is a (compact) closed manifold that is not the boundary of a manifold, as is explained below.
The general ''bordism problem'' is to calculate the cobordism classes of manifolds subject to various conditions.
Null-cobordisms with additional structure are called
fillings. ''Bordism'' and ''cobordism'' are used by some authors interchangeably; others distinguish them. When one wishes to distinguish the study of cobordism classes from the study of cobordisms as objects in their own right, one calls the equivalence question ''bordism of manifolds'', and the study of cobordisms as objects ''cobordisms of manifolds''.
The term ''bordism'' comes from French , meaning boundary. Hence bordism is the study of boundaries. ''Cobordism'' means "jointly bound", so ''M'' and ''N'' are cobordant if they jointly bound a manifold; i.e., if their disjoint union is a boundary. Further, cobordism groups form an extraordinary ''cohomology theory'', hence the co-.
Variants
The above is the most basic form of the definition. It is also referred to as unoriented bordism. In many situations, the manifolds in question are
oriented
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
, or carry some other additional structure referred to as
G-structure. This gives rise to
"oriented cobordism" and "cobordism with G-structure", respectively. Under favourable technical conditions these form a
graded ring called the cobordism ring
, with grading by dimension, addition by disjoint union and multiplication by
cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\t ...
. The cobordism groups
are the coefficient groups of a
generalised homology theory.
When there is additional structure, the notion of cobordism must be formulated more precisely: a ''G''-structure on ''W'' restricts to a ''G''-structure on ''M'' and ''N''. The basic examples are ''G'' = O for unoriented cobordism, ''G'' = SO for oriented cobordism, and ''G'' = U for
complex cobordism using ''stably''
complex manifolds. Many more are detailed by
Robert E. Stong.
In a similar vein, 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
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.
Instead of considering additional structure, it is also possible to take into account various notions of manifold, especially
piecewise linear (PL) and
topological manifolds. This gives rise to bordism groups
, which are harder to compute than the differentiable variants.
Surgery construction
Recall that in general, if ''X'', ''Y'' are manifolds with boundary, then the boundary of the product manifold is .
Now, given a manifold ''M'' of dimension ''n'' = ''p'' + ''q'' and an
embedding define the ''n''-manifold
:
obtained by
surgery
Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical specialty that uses operative manual and instrumental techniques on a person to investigate or treat a pa ...
, via cutting out the interior of
and gluing in
along their boundary
:
The trace of the surgery
:
defines an elementary cobordism (''W''; ''M'', ''N''). Note that ''M'' is obtained from ''N'' by surgery on
This is called reversing the surgery.
Every cobordism is a union of elementary cobordisms, by the work of
Marston Morse,
René Thom
René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958.
He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he becam ...
and
John Milnor.
Examples
As per the above definition, a surgery on the circle consists of cutting out a copy of
and gluing in
The pictures in Fig. 1 show that the result of doing this is either (i)
again, or (ii) two copies of
For surgery on the 2-sphere, there are more possibilities, since we can start by cutting out either
or
Morse functions
Suppose that ''f'' is a
Morse function on an (''n'' + 1)-dimensional manifold, and suppose that ''c'' is a critical value with exactly one critical point in its preimage. If the index of this critical point is ''p'' + 1, then the level-set ''N'' := ''f''
−1(''c'' + ε) is obtained from ''M'' := ''f''
−1(''c'' − ε) by a ''p''-surgery. The inverse image ''W'' := ''f''
−1(
'c'' − ε, ''c'' + ε defines a cobordism (''W''; ''M'', ''N'') that can be identified with the trace of this surgery.
Geometry, and the connection with Morse theory and handlebodies
Given a cobordism (''W''; ''M'', ''N'') there exists a smooth function ''f'' : ''W'' →
, 1such that ''f''
−1(0) = ''M'', ''f''
−1(1) = ''N''. By general position, one can assume ''f'' is Morse and such that all critical points occur in the interior of ''W''. In this setting ''f'' is called a Morse function on a cobordism. The cobordism (''W''; ''M'', ''N'') is a union of the traces of a sequence of surgeries on ''M'', one for each critical point of ''f''. The manifold ''W'' is obtained from ''M'' ×
, 1by attaching one
handle for each critical point of ''f''.
The Morse/Smale theorem states that for a Morse function on a cobordism, the flowlines of ''f''′ give rise to a
handle presentation of the triple (''W''; ''M'', ''N''). Conversely, given a handle decomposition of a cobordism, it comes from a suitable Morse function. In a suitably normalized setting this process gives a correspondence between handle decompositions and Morse functions on a cobordism.
History
Cobordism had its roots in the (failed) attempt by
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...
in 1895 to define
homology purely in terms of manifolds . Poincaré simultaneously defined both homology and cobordism, which are not the same, in general. See
Cobordism as an extraordinary cohomology theory for the relationship between bordism and homology.
Bordism was explicitly introduced by
Lev Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely d ...
in geometric work on manifolds. It came to prominence when
René Thom
René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958.
He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he becam ...
showed that cobordism groups could be computed by means of
homotopy theory, via the
Thom complex construction. Cobordism theory became part of the apparatus of
extraordinary cohomology theory
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
, alongside
K-theory. It performed an important role, historically speaking, in developments in topology in the 1950s and early 1960s, in particular in the
Hirzebruch–Riemann–Roch theorem, and in the first proofs of the
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the sp ...
.
In the 1980s the
category with compact manifolds as
objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ai ...
and cobordisms between these as
morphisms played a basic role in the Atiyah–Segal axioms for
topological quantum field theory
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.
Although TQFTs were invented by physicists, they are also of mathe ...
, which is an important part of
quantum topology.
Categorical aspects
Cobordisms are objects of study in their own right, apart from cobordism classes. Cobordisms form a
category whose objects are closed manifolds and whose morphisms are cobordisms. Roughly speaking, composition is given by gluing together cobordisms end-to-end: the composition of (''W''; ''M'', ''N'') and (''W'' ′; ''N'', ''P'') is defined by gluing the right end of the first to the left end of the second, yielding (''W'' ′ ∪
''N'' ''W''; ''M'', ''P''). A cobordism is a kind of
cospan: ''M'' → ''W'' ← ''N''. The category is a
dagger compact category.
A
topological quantum field theory
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.
Although TQFTs were invented by physicists, they are also of mathe ...
is a
monoidal functor
In category theory, monoidal functors are functors between monoidal category, monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categor ...
from a category of cobordisms to a category of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s. That is, it is a
functor whose value on a disjoint union of manifolds is equivalent to the tensor product of its values on each of the constituent manifolds.
In low dimensions, the bordism question is relatively trivial, but the category of cobordism is not. For instance, the disk bounding the circle corresponds to a nullary (0-ary) operation, while the cylinder corresponds to a 1-ary operation and the pair of pants to a binary operation.
Unoriented cobordism
The set of cobordism classes of closed unoriented ''n''-dimensional manifolds is usually denoted by
(rather than the more systematic
); it is an
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 ...
with the disjoint union as operation. More specifically, if
'M''and
'N''denote the cobordism classes of the manifolds ''M'' and ''N'' respectively, we define