Cobordism
cobordism theory
A generalized cohomology theory determined by spectra of Thom spaces and related to various structures in the stable tangent or normal bundle to a manifold. Cobordism theory is dual (in the sense of -duality) to the theory of bordism.
The simplest example of cobordism is orthogonal or non-oriented cobordism. Let by the group of orthogonal transformations of the Euclidean space
, and
its classifying space. The standard imbedding
defines a mapping
taking the universal fibre bundle
over
into the bundle
, where
is the one-dimensional trivial bundle over
. If
is the Thom space of
, then one obtains a mapping
induced by
, where
is suspension. The sequence
forms a spectrum of spaces and therefore defines a cohomology theory, called the theory of orthogonal cobordism or non-oriented cobordism or
-cobordism; it is denoted by
. The group
of
-dimensional
-cobordism of the pair
is defined as
![]() |
where is the set of homotopy classes of mappings from
into
. Here
,
is the empty set, and by
one means the disjoint union of
and a point. The group
, where
, is called the reduced group of
-dimensional
-cobordism
of
. The generalized homology theory dual to the
-cobordism theory is called
-bordism theory. The groups
of
-dimensional bordism of the pair
are defined as
![]() |
The groups of -dimensional
-bordism of a point are denoted by
and the
-dimensional
-bordism of a point by
; the latter can be described purely geometrically. Furthermore,
,
, so that it can be interpreted both as a cobordism group and a bordism group (see bordism, where it is denoted by
). The total coefficient group of
-cobordism theory, the graded group
, is a ring: multiplication is induced by the Cartesian product of manifolds. Furthermore, for any finite
-complex
the group
is a natural ring with respect to
since the mapping
induced by the imbedding
defines a mapping
, so that
is a multiplicative spectrum of spaces.
The general situation is described as follows. By a structural series one means a sequence of bundles
and mappings
such that
. The mapping
defines a vector bundle
over
, where
. Let
be the Thom space of the bundle
; the above equality defines a mapping
such that the sequence
is a spectrum of spaces, hence defines a cohomology theory. It is called
-cobordism theory and is denoted by
. Thus,
![]() |
The coefficient group of the -cobordism theory is denoted by
. Here,
,
, where
is the coefficient group of the dual
-bordism theory, which admits of a geometric definition using the concept of a so-called
-structure:
-bordancy is defined and the elements of
are interpreted as classes of
-bordant manifolds.
The first examples of cobordism theory arose from series of linear groups. For example, the series of orthogonal groups defines the structural series
, where
,
. The series
defines the structural series
, where
and
is the universal two-sheeted covering corresponding to the inclusion
. The corresponding cobordism theory is called the theory of oriented cobordism; it is denoted by
. The series of unitary groups
defines a theory of unitary or complex cobordism, quasi-complex cobordism, almost-complex cobordism; it is denoted by
. Here the series
is constructed in the following way:
is classifying space of
and the
,
are the mappings of the classifying spaces
and
, respectively, induced by the natural imbeddings
. The series of symplectic groups
defines a theory of symplectic cobordism,
, where
, and the
are constructed in the same way as for the unitary case. There are also cobordism theories corresponding to the series of groups
,
, etc. Finally, the series of identity groups
, where
is a fibre bundle with contractible
, defines a cobordism theory which is the same as the theory of stable cohomotopy groups, and therefore the dual bordism theory is isomorphic to the theory of stable homotopy groups,
,
. An
-manifold is said to be framed (trivialized) since the
-structure is precisely a frame (trivialization) of the stable normal bundle.
-cobordism theory is called trivialized or framed cobordism theory, its
-dimensional coefficient group being denoted by
, so that
. This is the first example of a cobordism; it was due to L.S. Pontryagin, who interpreted the stable homotopy groups of the sphere as (geometrically defined) groups of framed cobordism of a point of
, with the aim of computing the group
.
All these cobordism theories arising from series of linear groups are multiplicative, and therefore for any finite -complex
, the total (graded) cobordism group is a ring. For example, for the series of groups
there is an imbedding
inducing a mapping
![]() |
and therefore a mapping . The spectrum
representing the theory
has the form
,
, hence there exist mappings
so that the spectrum of spaces
is multiplicative.
The development of cobordism theory started from the geometric definition and calculation of the groups ,
,
. An important role was played by Pontryagin's theorem stating that
-bordant manifolds have the same Stiefel number. The study of cobordism theory was advanced by R. Thom. He introduced the spaces
,
and proved the isomorphism
, enabling one to bring into the calculation of the cobordism rings some of the methods of homotopic topology. Thom's constructions stimulated the introduction of
,
, etc., and the corresponding cobordisms. The fundamental problem of the first stage of the development of cobordism theory was the calculation of the cobordism rings of a point.
In the study of the cobordism of a point a big part is played by the characteristic classes: Chern classes for , Stiefel classes for
, Pontryagin and Stiefel classes for
(cf. Characteristic class; Chern class; Pontryagin class). In general, given any structural series
and any multiplicative cohomology theory
in which all bundles
over
are orientable, one can define the characteristic classes as elements of the group
, where
. Furthermore, the corresponding characteristic numbers, which are elements of the rings
are invariant with respect to
-bordancy. Let
be a partition of
and let
be the symmetric function of
variables corresponding to
. The characteristic class
(see Chern class) is denoted by
. The analogous constructions for the Pontryagin and Stiefel classes are denoted by
and
, respectively.
1) Unitary cobordism. The ring is the free graded polynomial algebra in a countable number of homogeneous generators
![]() |
The set ,
, is a system of polynomial generators if and only if
![]() |
where is the partition of
consisting of a single term. One of the systems of polynomial generators of
can be described as follows. Let
be
-dimensional complex projective space. The complex algebraic hypersurface of bidegree
in
is a complex manifold. Its unitary cobordism class is denoted by
,
. It turns out that
![]() |
so that an appropriate linear integer combination of elements of defines a generator of
of degree
.
Since is torsion-free and
, where the
are the Chern classes,
(cf. Chern class), it follows that the Chern numbers (cf. Chern number) completely determine the unitary cobordism class of an almost-complex manifold.
Let be a positive integer and let
,
,
, be a partition of it. There corresponds to each
-dimensional (real dimension) almost-complex manifold
a set
of integers, where the multi-index
runs through all the partitions of
. A set of such integers
is realized as the set of Chern numbers of some almost-complex manifold in the following situation. Let
be the characteristic class given by replacing the Wu generators
in the representation of
with the variables
,
, and let
be the characteristic class given by the product of the functions
. Let
be the value of the characteristic class
on the fundamental class
of the almost-complex manifold
with tangent bundle
.
There exists for a homomorphism a closed almost-complex manifold
such that
for all
if and only if
takes integer values on all the
-dimensional components of each characteristic class
(Stong's theorem, see [1], Chapt. 7). Equivalently, the Hurewicz homomorphism
![]() |
where , is a monomorphism onto a direct summand (Hattori's theorem). Here
denotes reduced
-theory.
2) Non-oriented, or orthogonal, cobordism. Each element of the ring has order
, and
![]() |
that is, is a free polynomial
-algebra. One can choose as generator
any element
with
, for example,
. In this theory there are analogues of the manifolds
, obtained by replacing
by
; a suitable manifold
can serve as a generator of degree
. The Stiefel numbers (cf. Stiefel number) completely define the non-orientable cobordism class of the manifold. The following theorem gives relations among the Stiefel numbers: Given a homomorphism
, there exists a closed
-dimensional manifold
such that
for all
if and only if
for all
, where
. Here
is the full Steenrod operation and
is the full Stiefel class. The ring
is the image of the homomorphism
.
3) Oriented cobordism with ring . All the elements of the torsion subgroup
of this ring have order
. The ring
is the ring of polynomials over
of classes
of degree
, the generators being chosen by the condition
![]() |
The -cobordism class of a manifold is determined by the Pontryagin and Stiefel numbers (cf. Pontryagin number). The signature of the manifold is also an invariant of the cobordism class. The relations among the Stiefel numbers follow from the following fact: The image of the "forgetful" homomorphism
consists precisely of those cobordism classes for which all numbers containing the class
are zero. For any partition
,
![]() |
where is the corresponding Pontryagin number. There do not exist any
-prime relations among the Pontryagin numbers.
Similarly to the introduction of the classes for the unitary cobordism, the classes
are introduced, which are symmetric functions in
. Let
be the characteristic class defining the Hirzebruch
-genus. All relations among the Pontryagin numbers follow from the fact that the Pontryagin numbers are integers and
. The homomorphism
is epimorphic.
4) Special unitary cobordism with ring . A
-manifold
has an
-structure if and only if
. All the elements of the torsion subgroup
have order 2. The kernel of the homomorphism
is precisely
. The group
is finitely generated and
is the ring of polynomials over
of classes
of degree
,
. The torsion subgroup
has the form
when
, while for
,
is a vector space over
the dimension of which is the number of partitions of
. Two
-manifolds are bordant if and only if they have the same characteristic number in integer cohomology and in
-theory.
All relations among the Chern numbers for -dimensional
-manifolds follows from the following:
for all
;
for all
; if
, then
for all
. The image of the homomorphism
consists of the classes
, where
is an oriented manifold all Pontryagin numbers of which containing the class
are even.
The rings and
have also been completely computed. The rings
and
have to date (1986) not been computed. The ring
is the ring of polynomials on
-dimensional generators. All known (1986) elements of
have order 2. (However there is an announcement of an element of order 4 in dimensions
.) With regard to
, the main result here is Serre's theorem on the finiteness of these groups. The ring
of self-adjoint cobordism has also been studied, where the objects are almost-complex manifolds with an operator given in the normal bundle which isomorphically maps the complex structure onto the adjoint. The spectrum of
has been constructed; with regard to the groups
it is known that there is only
-prime torsion, but there are elements of order
for any
, namely
. The image
has also been calculated using the technique of formal groups (cf. Formal group).
A mapping of one cobordism theory into another, for example, , induces a mapping of the spectra
. The cone of this mapping in the category of spectra gives a generalized cohomology theory. The ring of the point of the theory so obtained has the following geometric interpretation. Let
be a
-manifold on the (possibly empty) boundary of which an
-structure is fixed. By introducing the appropriate bordism relation for
-manifolds one obtains the ring
. The groups
,
etc., are introduced in the same way.
So far, smooth manifolds have been considered or, equivalently, linear group representations (the structure series arising from the bundles over ). It is possible to consider various structures on topological manifolds, that is, to start from a group of homeomorphisms (and even proper homotopy equivalences) of
. Here the following examples are known. (Throughout, the letter
denotes passage to the oriented case.)
5) Piecewise-linear cobordism. The objects are piecewise-linear manifolds. The corresponding bordism relation leads to the groups ,
. By defining the group
(or
) as the group of piecewise-linear homeomorphisms of
onto itself that preserve the origin (or the orientation as well), one can introduce the classifying spaces
(or
) and the Thom spaces
(or
) and construct a
(or
) cobordism theory. In this connection,
and
. The groups
have been computed. The cobordism class of a piecewise-linear manifold is completely defined by the characteristic numbers, that is, by the elements of
.
6) Topological cobordism. The objects are topological manifolds for which the groups ,
are defined. By considering the group
of homeomorphisms of
onto itself that preserve the origin, one can define the spaces
and
. The groups
and
have been computed. However, the isomorphism
has been established for all
except
. The absence of a proof of this isomorphism is tied up with the fact that the transitivity theorem on which the isomorphism
is based for topological manifolds, has not been proved in the general case (but it has not been refuted either (1986)).
7) Cobordism of Poincaré complexes ,
. The objects are complexes with Poincaré duality and the bordism is the corresponding equivalence relation. Such complexes have a normal spherical bundle induced from the universal bundle over
(or
). Here
(or
) is an
-space of homotopy equivalences (of degree 1) of the sphere
onto itself. The Thom spectra
and
to which these give rise have finite homotopy groups, whereas the signature defines a non-trivial homomorphism
, so that, a fortiori, the mapping
is not an isomorphism.
Yet another series of examples is given by cobordism of manifolds with singularities of a special type. (This is a very good technique for the construction of various cohomology theories with special properties.) One can construct along these lines a cobordism theory that is the same as ordinary singular cohomology theory and one that is the same as connected -theory.
The second stage in the development of cobordism theory is the study of cobordisms as specific generalized cohomology theories. Let denote one of the fields
or the skew-field of quaternions
, let
be the corresponding series of groups (
,
,
) and let
be the corresponding cobordism theory. A multiplicative generalized cohomology theory
is called
-orientable if any
-vector bundle is
-orientable or, equivalently, if the canonical one-dimensional
-vector bundle
, where
is a projective space, is
-orientable. By an
-orientation of the theory
one means an
-orientation
of the bundle
, and a theory with a chosen orientation is called oriented. The
-cobordism theories have a canonical orientation because of the identification
. The theory
is universal in the class of
-oriented theories, that is, for any
-oriented theory
with
-orientation
there exists a unique multiplicative homomorphism of theories
under which the canonical orientation of the theory
is taken to
. Moreover, when
is one of the fields
, there exist for any
-oriented theory
and any finite
-complex
spectral sequences
and
with
![]() |
![]() |
converging to and natural in
and
, where
is made into an
-module by means of the homomorphism
. If
is the homology theory dual to the
-oriented cohomology theory
, then there is a homomorphism
. In the case when
is the ordinary homology theory, it coincides with Steenrod–Thom realization of cycles (see Steenrod problem). The powerful methods of cobordism theory are connected with formal groups (cf. Formal group, [5]).
The most important and successful applications of cobordism theory are: the proof of the Atiyah–Singer index theorem for an elliptic operator and the general Riemann–Roch theorem; the study of fixed points of group actions; the classification of smooth (or piecewise-smooth) manifolds of given homotopy type; the proof of the theorem on the topological invariance of rational Pontryagin classes, and the solution of the problem of triangulability of topological manifolds.
See also the references in Bordism.
References
[1] | R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) |
[2] | P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) |
[3] | S.P. Novikov, "Methods of algebraic topology from the point of view of cobordism theory" Izv. Akad. Nauk SSSR Ser. Mat. , 31 : 4 (1967) pp. 855–951 (In Russian) |
[4] | T. Bröcker, T. Tom dieck, "Kobordismentheorie" , Springer (1970) |
[5] | V.M. [V.M. Bukhshtaber] Buchstaber, "Cobordisms in problems of algebraic topology" J. Soviet Math. , 7 : 4 (1975) pp. 629–653 Itogi Nauk. i Tekh. Algebra. Geom. Topol. (1975) pp. 231–272 |
Comments
The letter is often used to denote Thom spaces and Thom spectra. Thus, e.g.,
is used to denote the Thom space
and
stands for the spectrum of all the
; similarly one uses
for
, etc. The corresponding generalized cohomology theories are then indicated by the same symbols as is customary for generalized cohomology theories defined by a spectrum; thus,
is the
-th complex cobordism group of
and
is its complex cobordism ring.
A structural series as defined above is often called a
-structure (cf.
-structure, [1]).
The general theorem that the (co)bordism group of -dimensional
-manifolds
is isomorphic to
is known as the Pontryagin–Thom theorem.
A complex structure on a real vector bundle over a manifold
is a vector bundle morphism
such that
. If
is a complex imbedded manifold without boundary
, then multiplication with
on its normal bundle defines a complex structure on that bundle (viewed as a real bundle). A weakly-complex manifold (also called a stably (almost) complex manifold) is a real manifold with a complex structure on its stable normal bundle; i.e. if
denotes the normal bundle of
, then there is a complex structure defined on some
where
stands for the trivial
-dimensional bundle over
,
. The complex bordism groups of a space
, often denoted by
, can now also be interpreted as cobordism classes of mappings
where
is a weakly-complex manifold without boundary. There is a similar interpretation of the complex cobordism groups
, cf. [a3], and for other bordism and cobordism group.
The relation between cobordism theory (and other (generalized) cohomology theories) and formal group theory comes about as follows. A generalized cohomology theory is complex oriented if it has first Chern classes (in a suitable sense; cf. above and [a1], p. 121; [a5], Part II, (2.1)). Let
be the class of the canonical line bundle
over
, the space of lines in
(the fibre of
at
is the line
). Then
and
. Because
is classifying for complex line bundles, there is an
such that
and this induces a ring homomorphism
. The image of
is a power series in two variables, here denoted by
,
, where
stands for
and
for
. Equivalently,
is the power series with coefficients in
such that
. The power series
defines a formal group law.
Conversely, the question arises whether every (one-dimensional commutative) formal group arises as the formal group of a generalized cohomology theory. Here the study of (co)bordism of manifolds with special singularities is important.
It turns out (D. Quillen [a4], cf. also [a12]) that for this formal group law
is a universal formal group law. This universality property shows up in topological terms in the form of the theorem that if
is any complex oriented generalized cohomology theory, then there is a unique transformation of cohomology theories
(linear, degree preserving and multiplicative) such that
, where
means: apply
to the coefficients of
. The logarithm of the formal group law
can be calculated (A.S. Mishchenko, cf. [a12]; cf. Formal group for the notion of logarithm of a formal group law). It is equal to
![]() |
![]() |
where denotes the complex cobordism class of the complex projective space of (complex) dimension
.
On the other hand it is possible to write down explicit formulas for the logarithm of a universal formal group law over , cf. [a2], Chapt. 1. There result explicit formulas for the polynomial generators of
in terms of the
. These formulas take a particularly useful form for the "p-typical" version
of the cohomology theory
. The generalized cohomology theory
, Brown–Peterson cohomology for a prime number
, cf. [a6], is defined by a spectrum
and is such that
with
of degree
. It is such that
is a direct sum of (dimension shifted) copies of
for each space
, functorially in
. Here
stands for the ring of integers localized at
, i.e.
. The theory
can also be defined as the image of an idempotent cohomology operator
(cf., e.g., [a1], Chapt. 4). This operation corresponds to
-typification in formal group theory. The Hazewinkel generators ([a1], pp. 137, 369-370)
of
are defined recursively by
![]() |
![]() |
They arise from the explicit construction of a -typical universal formal group [a8]. A different set of generators
has been given by S. Araki [a7], the Araki generators.
In a certain precise sense, -theory is
-theory for one prime at the time, and currently a great deal of complex cobordism theory is written in terms of
rather than
itself. Combined with the theory of cohomology operations, formal group theory (the rings of operations
and
of
and
also have interpretations in terms of formal groups, cf. [a1], [a9], [a10]), and spectral sequences, notably the Adams–Novikov spectral sequence and the chromatic spectral sequence (cf. [a1], [a11]), complex cobordism and Brown–Peterson cohomology have become a strong calculation tool in algebraic topology, e.g. for the stable homotopy groups of the spheres.
References
[a1] | D.C. Ravenel, "Complex cobordism and stable homotopy groups of spheres" , Acad. Press (1986) |
[a2] | M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) |
[a3] | D. Quillen, "Elementary proofs of some results of cobordism theory using Steenrod operations" Adv. Math. , 7 (1971) pp. 29–56 |
[a4] | D. Quillen, "On the formal group laws of unoriented and complex cobordism theory" Bull. Amer. Math. Soc , 75 (1969) pp. 1293–1298 |
[a5] | J.F. Adams, "Stable homotopy and generalised homology" , Univ. Chicago Press (1974) pp. Part III, Chapt. 12 |
[a6] | E.H. Brown, F.P. Peterson, "A spectrum whose ![]() ![]() |
[a7] | S. Araki, "Typical formal groups in complex cobordism and ![]() |
[a8] | M. Hazewinkel, "Constructing formal groups III: applications to complex cobordism and Brown–Peterson cohomology" J. Pure Appl. Algebra , 10 (1977) pp. 1–18 |
[a9] | P.S. Landweber, "![]() |
[a10] | P.S. Landweber, "Invariant regular ideals in Brown–Peterson cohomology" Duke Math. J. , 42 (1975) pp. 499–505 |
[a11] | H.R. Miller, D.C. Ravenel, W.S. Wilson, "Periodic phenomena of the Adams–Novikov spectral sequence" Ann. of Math. , 106 (1977) pp. 469–516 |
[a12] | V.M. Bukhshtaber, A.S. Mishchenko, S.P. Novikov, "Formal groups and their role in the apparatus of algebraic topology" Russ. Math. Surveys , 26 (1971) pp. 63–90 Uspekhi Mat. Nauk , 26 (1971) pp. 131–154 |
Cobordism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cobordism&oldid=16874