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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 relation ...
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 British ...
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 Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
(French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dimension are ''cobordant'' if their
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (th ...
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 became w ...
for
smooth 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 ...
s (i.e., differentiable), but there are now also versions for piecewise linear and
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 mathe ...
s. A ''cobordism'' between manifolds ''M'' and ''N'' is a compact manifold ''W'' whose boundary is the disjoint union of ''M'' and ''N'', \partial W=M \sqcup 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 In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...
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 isomorphi ...
of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...
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 isomorphi ...
in dimensions ≥ 4 – because the
word problem for groups In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group ''G'' is the algorithmic problem of deciding whether two words in the generators represent the same elem ...
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 originated i ...
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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
. In geometric topology, cobordisms are intimately connected with
Morse theory In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiabl ...
, and ''h''-cobordisms are fundamental in the study of high-dimensional manifolds, namely
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 ...
. In algebraic topology, cobordism theories are fundamental
extraordinary cohomology theories 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 ...
, and categories of cobordisms are the domains of
topological quantum field theories 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 mathem ...
.


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 points ...
locally In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). P ...
(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 isomorphi ...
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, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
\R^n. A
manifold with boundary 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 ne ...
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 M; the boundary of M is denoted by \partial M. 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 ...
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 British ...
manifold without boundary (\partial M=\emptyset.)


Cobordisms

An (n+1)-dimensional ''cobordism'' is a
quintuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is def ...
(W; M, N, i, j) consisting of an (n+1)-dimensional compact differentiable manifold with boundary, W; closed n-manifolds M, N; and
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 gi ...
s i\colon M \hookrightarrow \partial W, j\colon N \hookrightarrow\partial W with disjoint images such that :\partial W = i(M) \sqcup j(N)~. The terminology is usually abbreviated to (W; M, N). ''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 const ...
, and ''N'' of two circles, ''M'' and ''N'' together make up the boundary of a
pair of pants Trousers (British English), slacks, or pants are an item of clothing worn from the waist to anywhere between the knees and the ankles, covering both legs separately (rather than with cloth extending across both legs as in robes, skirts, and dr ...
''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 M \sqcup M' is cobordant to the
connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
M\mathbinM'. The previous example is a particular case, since the connected sum \mathbb^1\mathbin\mathbb^1 is isomorphic to \mathbb^1. The connected sum M\mathbinM' is obtained from the disjoint union M \sqcup M' by surgery on an embedding of \mathbb^0 \times \mathbb^n in M \sqcup M', 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 In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds. Hand ...
. On the other hand, the 2''n''-dimensional
real projective space In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properties Construction A ...
\mathbb^(\R) 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, or carry some other additional structure referred to as
G-structure In differential geometry, a ''G''-structure on an ''n''- manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''. The notion of ''G''-structures includes var ...
. This gives rise to "oriented cobordism" and "cobordism with G-structure", respectively. Under favourable technical conditions these form a
graded ring In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
called the cobordism ring \Omega^G_*, 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\ti ...
. The cobordism groups \Omega^G_* 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 In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it ...
using ''stably''
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
s. Many more are detailed by Robert E. Stong. In a similar vein, a standard tool in
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 ...
is surgery on normal maps: such a process changes a normal map to another normal map within the same bordism class. Instead of considering additional structure, it is also possible to take into account various notions of manifold, especially piecewise linear (PL) and
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 mathe ...
s. This gives rise to bordism groups \Omega_*^(X), \Omega_*^(X), 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 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 gi ...
\varphi : \mathbb^p \times \mathbb^q \subset M, define the ''n''-manifold :N := (M - \operatorname\varphi) \cup_ \left(\mathbb^\times \mathbb^\right) 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 pat ...
, via cutting out the interior of \mathbb^p \times \mathbb^q and gluing in \mathbb^ \times \mathbb^ along their boundary :\partial \left (\mathbb^p \times \mathbb^q \right) = \mathbb^p \times \mathbb^ = \partial \left( \mathbb^ \times \mathbb^ \right). The trace of the surgery :W := (M \times I) \cup_ \left(\mathbb^ \times \mathbb^q\right) defines an elementary cobordism (''W''; ''M'', ''N''). Note that ''M'' is obtained from ''N'' by surgery on \mathbb^\times \mathbb^ \subset N. This is called reversing the surgery. Every cobordism is a union of elementary cobordisms, by the work of
Marston Morse Harold Calvin Marston Morse (March 24, 1892 – June 22, 1977) was an American mathematician best known for his work on the ''calculus of variations in the large'', a subject where he introduced the technique of differential topology now known a ...
,
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 became w ...
and
John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
.


Examples

As per the above definition, a surgery on the circle consists of cutting out a copy of \mathbb^0 \times \mathbb^1 and gluing in \mathbb^1 \times \mathbb^0. The pictures in Fig. 1 show that the result of doing this is either (i) \mathbb^1 again, or (ii) two copies of \mathbb^1 For surgery on the 2-sphere, there are more possibilities, since we can start by cutting out either \mathbb^0 \times \mathbb^2 or \mathbb^1 \times \mathbb^1.


Morse functions

Suppose that ''f'' is a
Morse function In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
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 A handle is a part of, or attachment to, an object that allows it to be grasped and manipulated by hand. The design of each type of handle involves substantial ergonomic issues, even where these are dealt with intuitively or by following tra ...
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 "The ...
in 1895 to define
homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...
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 due ...
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 became w ...
showed that cobordism groups could be computed by means of
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...
, via the
Thom complex In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact sp ...
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 In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...
. 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 In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex alge ...
, 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 space ...
. In the 1980s 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) * ...
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 ...
and cobordisms between these as
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s 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 mathem ...
, which is an important part of
quantum topology Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology. Dirac notation provides a viewpoint of quantum mechanics which becomes amplified into a framework that can embrace the amplitudes associa ...
.


Categorical aspects

Cobordisms are objects of study in their own right, apart from cobordism classes. Cobordisms form a
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) * ...
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 In category theory, a span, roof or correspondence is a generalization of the notion of relation between two objects of a category. When the category has all pullbacks (and satisfies a small number of other conditions), spans can be considered as ...
: ''M'' → ''W'' ← ''N''. The category is a
dagger compact category In category theory, a branch of mathematics, dagger compact categories (or dagger compact closed categories) first appeared in 1989 in the work of Sergio Doplicher and John E. Roberts on the reconstruction of compact topological groups from thei ...
. 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 mathem ...
is a
monoidal functor In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two ...
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 In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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 \mathfrak_n (rather than the more systematic \Omega_n^); 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 commut ...
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 = \sqcup N/math>; this is a well-defined operation which turns \mathfrak_n into an abelian group. The identity element of this group is the class
emptyset Emptyset is a Bristol-based production project, formed in 2005 by James Ginzburg and Paul Purgas. Ginzburg and Purgas say that by working across performance, installation and the moving image they are examining the physical properties of sound, ...
/math> consisting of all closed ''n''-manifolds which are boundaries. Further we have + =
emptyset Emptyset is a Bristol-based production project, formed in 2005 by James Ginzburg and Paul Purgas. Ginzburg and Purgas say that by working across performance, installation and the moving image they are examining the physical properties of sound, ...
/math> for every ''M'' since M \sqcup M = \partial (M \times ,1. Therefore, \mathfrak_n is a vector space over \mathbb_2, the field with two elements. The cartesian product of manifolds defines a multiplication N]= \times N so :\mathfrak_* = \bigoplus_\mathfrak_n is a
graded algebra In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the se ...
, with the grading given by the dimension. The cobordism class \in \mathfrak_n of a closed unoriented ''n''-dimensional manifold ''M'' is determined by the Stiefel–Whitney
characteristic number 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 ...
s of ''M'', which depend on the stable isomorphism class of the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
. Thus if ''M'' has a stably trivial tangent bundle then 0 \in \mathfrak_n. In 1954
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 became w ...
proved :\mathfrak_* = \mathbb_2 \left i \geqslant 1, i \neq 2^j - 1 \right/math> the polynomial algebra with one generator x_i in each dimension i \neq 2^j - 1. Thus two unoriented closed ''n''-dimensional manifolds ''M'', ''N'' are cobordant, = \in \mathfrak_n, if and only if for each collection \left(i_1, \cdots, i_k\right) of ''k''-tuples of integers i \geqslant 1, i \neq 2^j - 1 such that i_1 + \cdots + i_k = n the Stiefel-Whitney numbers are equal :\left\langle w_(M) \cdots w_(M), \right\rangle = \left\langle w_(N) \cdots w_(N), \right\rangle \in \mathbb_2 with w_i(M) \in H^i\left(M; \mathbb_2\right) the ''i''th Stiefel-Whitney class and \in H_n\left(M; \mathbb_2\right) the \mathbb_2-coefficient
fundamental class In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fundamen ...
. For even ''i'' it is possible to choose x_i = \left mathbb^i(\R)\right/math>, the cobordism class of the ''i''-dimensional
real projective space In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properties Construction A ...
. The low-dimensional unoriented cobordism groups are :\begin \mathfrak_0 &= \Z/2, \\ \mathfrak_1 &= 0, \\ \mathfrak_2 &= \Z/2, \\ \mathfrak_3 &= 0, \\ \mathfrak_4 &= \Z/2 \oplus \Z/2, \\ \mathfrak_5 &= \Z/2. \end This shows, for example, that every 3-dimensional closed manifold is the boundary of a 4-manifold (with boundary). The
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
\chi(M) \in \Z modulo 2 of an unoriented manifold ''M'' is an unoriented cobordism invariant. This is implied by the equation :\chi_ = \left(1 - (-1)^ \right)\chi_W for any compact manifold with boundary W. Therefore, \chi: \mathfrak_i \to \Z/2 is a well-defined group homomorphism. For example, for any i_1, \cdots, i_k \in\mathbb :\chi \left( \mathbb^ (\R) \times \cdots \times \mathbb^(\R) \right) = 1. In particular such a product of real projective spaces is not null-cobordant. The mod 2 Euler characteristic map \chi: \mathfrak_ \to \Z/2 is onto for all i \in \mathbb, and a group isomorphism for i = 1. Moreover, because of \chi(M \times N) = \chi(M)\chi(N), these group homomorphism assemble into a homomorphism of graded algebras: :\begin \mathfrak \to \mathbb_2 \\[] \mapsto \chi(M) x^ \end


Cobordism of manifolds with additional structure

Cobordism can also be defined for manifolds that have additional structure, notably an orientation. This is made formal in a general way using the notion of ''X''-structure (or
G-structure In differential geometry, a ''G''-structure on an ''n''- manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''. The notion of ''G''-structures includes var ...
). Very briefly, the
normal bundle In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion). Definition Riemannian manifold Let (M,g) be a Riemannian m ...
ν of an immersion of ''M'' into a sufficiently high-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
\R^ gives rise to a map from ''M'' to the
Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
, which in turn is a subspace of the
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free acti ...
of the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
: ν: ''M'' → Gr(''n'', ''n'' + ''k'') → ''BO''(''k''). Given a collection of spaces and maps ''Xk'' → ''Xk''+1 with maps ''Xk'' → ''BO''(''k'') (compatible with the inclusions ''BO''(''k'') → ''BO''(''k''+1), an ''X''-structure is a lift of ν to a map \tilde \nu: M \to X_k. Considering only manifolds and cobordisms with ''X''-structure gives rise to a more general notion of cobordism. In particular, ''Xk'' may be given by ''BG''(''k''), where ''G''(''k'') → ''O''(''k'') is some group homomorphism. This is referred to as a
G-structure In differential geometry, a ''G''-structure on an ''n''- manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''. The notion of ''G''-structures includes var ...
. Examples include ''G'' = ''O'', the orthogonal group, giving back the unoriented cobordism, but also the subgroup SO(''k''), giving rise to
oriented 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 dim ...
, the
spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a L ...
, the unitary group ''U''(''k''), and the trivial group, giving rise to
framed cobordism Framed may refer to: Common meanings *A painting or photograph that has been placed within a picture frame *Someone falsely shown to be guilty of a crime as part of a frameup Film and television * ''Framed'' (1930 film), a pre-code crime action ...
. The resulting cobordism groups are then defined analogously to the unoriented case. They are denoted by \Omega^G_*.


Oriented cobordism

Oriented cobordism is the one of manifolds with an SO-structure. Equivalently, all manifolds need to be oriented and cobordisms (''W'', ''M'', ''N'') (also referred to as ''oriented cobordisms'' for clarity) are such that the boundary (with the induced orientations) is M \sqcup (-N), where −''N'' denotes ''N'' with the reversed orientation. For example, boundary of the cylinder ''M'' × ''I'' is M \sqcup (-M): both ends have opposite orientations. It is also the correct definition in the sense 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 ...
. Unlike in the unoriented cobordism group, where every element is two-torsion, 2''M'' is not in general an oriented boundary, that is, 2 'M''≠ 0 when considered in \Omega_*^. The oriented cobordism groups are given modulo torsion by :\Omega_*^\otimes \Q =\Q \left _\mid i \geqslant 1 \right the polynomial algebra generated by the oriented cobordism classes :y_=\left mathbb^(\Complex) \right \in \Omega_^ of the
complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
s (Thom, 1952). The oriented cobordism group \Omega_*^ is determined by the Stiefel–Whitney and Pontrjagin
characteristic number 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 ...
s (Wall, 1960). Two oriented manifolds are oriented cobordant if and only if their Stiefel–Whitney and Pontrjagin numbers are the same. The low-dimensional oriented cobordism groups are : :\begin \Omega_0^ &= \Z, \\ \Omega_1^ &= 0, \\ \Omega_2^ &= 0, \\ \Omega_3^ &= 0, \\ \Omega_4^ &= \Z, \\ \Omega_5^ &= \Z_2. \end The
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
of an oriented 4''i''-dimensional manifold ''M'' is defined as the signature of the intersection form on H^(M) \in \Z and is denoted by \sigma(M). It is an oriented cobordism invariant, which is expressed in terms of the Pontrjagin numbers by the
Hirzebruch signature theorem In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem) is Friedrich Hirzebruch's 1954 result expressing the signature of a smooth closed oriented manifold by a linear combi ...
. For example, for any ''i''1, ..., ''ik'' ≥ 1 :\sigma \left (\mathbb^(\Complex) \times \cdots \times \mathbb^(\Complex) \right) = 1. The signature map \sigma:\Omega_^ \to \Z is onto for all ''i'' ≥ 1, and an isomorphism for ''i'' = 1.


Cobordism as an extraordinary cohomology theory

Every
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
theory (real, complex etc.) has an
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 ...
called
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...
. Similarly, every cobordism theory Ω''G'' has an
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 ...
, with homology ("bordism") groups \Omega^G_n(X) and cohomology ("cobordism") groups \Omega^n_G(X) for any space ''X''. The generalized homology groups \Omega_*^G(X) are covariant in ''X'', and the generalized cohomology groups \Omega^*_G(X) are contravariant in ''X''. The cobordism groups defined above are, from this point of view, the homology groups of a point: \Omega_n^G = \Omega_n^G(\text). Then \Omega^G_n(X) is the group of ''bordism'' classes of pairs (''M'', ''f'') with ''M'' a closed ''n''-dimensional manifold ''M'' (with G-structure) and ''f'' : ''M'' → ''X'' a map. Such pairs (''M'', ''f''), (''N'', ''g'') are ''bordant'' if there exists a G-cobordism (''W''; ''M'', ''N'') with a map ''h'' : ''W'' → ''X'', which restricts to ''f'' on ''M'', and to ''g'' on ''N''. An ''n''-dimensional manifold ''M'' has a fundamental homology class 'M''∈ ''Hn''(''M'') (with coefficients in \Z/2 in general, and in \Z in the oriented case), defining a natural transformation :\begin \Omega^G_n(X) \to H_n(X) \\ (M,f) \mapsto f_* \end which is far from being an isomorphism in general. The bordism and cobordism theories of a space satisfy the
Eilenberg–Steenrod axioms In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homo ...
apart from the dimension axiom. This does not mean that the groups \Omega^n_G(X) can be effectively computed once one knows the cobordism theory of a point and the homology of the space ''X'', though the
Atiyah–Hirzebruch spectral sequence In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet, i ...
gives a starting point for calculations. The computation is only easy if the particular cobordism theory reduces to a product of ordinary homology theories, in which case the bordism groups are the ordinary homology groups :\Omega^G_n(X)=\sum_H_p(X;\Omega^G_q(\text)). This is true for unoriented cobordism. Other cobordism theories do not reduce to ordinary homology in this way, notably
framed cobordism Framed may refer to: Common meanings *A painting or photograph that has been placed within a picture frame *Someone falsely shown to be guilty of a crime as part of a frameup Film and television * ''Framed'' (1930 film), a pre-code crime action ...
, oriented cobordism and
complex cobordism In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it ...
. The last-named theory in particular is much used by algebraic topologists as a computational tool (e.g., for the
homotopy groups of spheres In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure ...
). Cobordism theories are represented by Thom spectra ''MG'': given a group ''G'', the Thom spectrum is composed from the
Thom space In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact sp ...
s ''MGn'' of the standard vector bundles over the
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free acti ...
s ''BGn''. Note that even for similar groups, Thom spectra can be very different: ''MSO'' and ''MO'' are very different, reflecting the difference between oriented and unoriented cobordism. From the point of view of spectra, unoriented cobordism is a product of Eilenberg–MacLane spectra – ''MO'' = ''H''((''MO'')) – while oriented cobordism is a product of Eilenberg–MacLane spectra rationally, and at 2, but not at odd primes: the oriented cobordism spectrum ''MSO'' is rather more complicated than ''MO''.


See also

* ''h''-cobordism *
Link concordance In mathematics, two links L_0 \subset S^n and L_1 \subset S^n are concordant if there exists an embedding f : L_0 \times ,1\to S^n \times ,1/math> such that f(L_0 \times \) = L_0 \times \ and f(L_0 \times \) = L_1 \times \. By its nature, link ...
*
List of cohomology theories This is a list of some of the ordinary and generalized (or extraordinary) homology and cohomology theories in algebraic topology that are defined on the categories of CW complexes or spectra. For other sorts of homology theories see the links at ...
*
Symplectic filling In mathematics, a filling of a manifold ''X'' is a cobordism ''W'' between ''X'' and the empty set. More to the point, the ''n''-dimensional topological manifold ''X'' is the boundary of an (''n'' + 1)-dimensional manifold ''W''. Perhaps ...
*
Cobordism hypothesis In mathematics, the cobordism hypothesis, due to John C. Baez and James Dolan, concerns the classification of extended topological quantum field theories (TQFTs). In 2008, Jacob Lurie outlined a proof of the cobordism hypothesis, though the detail ...
* Cobordism ring *
Timeline of bordism This is a timeline of bordism, a topological theory based on the concept of the boundary of a manifold. For context see timeline of manifolds. Jean Dieudonné Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a Frenc ...


Notes


References

*
John Frank Adams John Frank Adams (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory. Life He was born in Woolwich, a suburb in south-east London, and attended Bedford School. He began research ...
, ''Stable homotopy and generalised homology'', Univ. Chicago Press (1974). * *
Michael F. Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded th ...
, ''Bordism and cobordism'' Proc. Camb. Phil. Soc. 57, pp. 200–208 (1961). * * * * * Sergei Novikov, ''Methods of algebraic topology from the point of view of cobordism theory'', Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 855–951. *
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 due ...
, ''Smooth manifolds and their applications in homotopy theory'' American Mathematical Society Translations, Ser. 2, Vol. 11, pp. 1–114 (1959). *
Daniel Quillen Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 197 ...
, ''On the formal group laws of unoriented and complex cobordism theory'' Bull. Amer. Math. Soc., 75 (1969) pp. 1293–1298. * Douglas Ravenel, ''Complex cobordism and stable homotopy groups of spheres'', Acad. Press (1986). * * Yuli B. Rudyak, ''On Thom spectra, orientability, and (co)bordism'', Springer (2008). * Robert E. Stong, ''Notes on cobordism theory'', Princeton Univ. Press (1968). * *
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 became w ...
, ''Quelques propriétés globales des variétés différentiables'',
Commentarii Mathematici Helvetici The ''Commentarii Mathematici Helvetici'' is a quarterly peer-reviewed scientific journal in mathematics. The Swiss Mathematical Society started the journal in 1929 after a meeting in May of the previous year. The Swiss Mathematical Society sti ...
28, 17-86 (1954). *


External links


Bordism
on the Manifold Atlas.
B-Bordism
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