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) MR0248858 Zbl 0181.26604
[2]	P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) MR0176478 Zbl 0125.40103
[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) MR0221509
[4]	T. Bröcker, T. Tom dieck, "Kobordismentheorie" , Springer (1970) MR0275446 Zbl 0211.55501
[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) MR0860042 Zbl 0608.55001
[a2]	M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) MR0506881 MR0463184 Zbl 0454.14020
[a3]	D. Quillen, "Elementary proofs of some results of cobordism theory using Steenrod operations" Adv. Math. , 7 (1971) pp. 29–56 MR0290382 Zbl 0214.50502
[a4]	D. Quillen, "On the formal group laws of unoriented and complex cobordism theory" Bull. Amer. Math. Soc , 75 (1969) pp. 1293–1298 MR0253350 Zbl 0199.26705
[a5]	J.F. Adams, "Stable homotopy and generalised homology" , Univ. Chicago Press (1974) pp. Part III, Chapt. 12 MR0402720 Zbl 0309.55016
[a6]	E.H. Brown, F.P. Peterson, "A spectrum whose cohomology is the algebra of reduced -th powers" Topology , 5 (1966) pp. 149–154
[a7]	S. Araki, "Typical formal groups in complex cobordism and -theory" , Kinokuniya (1973) MR0375354
[a8]	M. Hazewinkel, "Constructing formal groups III: applications to complex cobordism and Brown–Peterson cohomology" J. Pure Appl. Algebra , 10 (1977) pp. 1–18 MR0463182 MR0463183 MR0463184 Zbl 0363.14013
[a9]	P.S. Landweber, " and typical formal groups" Osaka J. Math. , 12 (1975) pp. 357–363 MR377945 Zbl 0311.55003
[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 MR458423
[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 Zbl 0224.57006