1 Introduction
The theory of complex bordism is much richer than its unoriented analogue, and at the same time is not as complicated as oriented bordism or other bordism theories with additional structure (B-bordism). Thanks to this, complex cobordism theory has found the most striking and important applications in algebraic topology and beyond. Many of these applications, including the formal group techniques and the Adams-Novikov spectral sequence were outlined in the pioneering work [Novikov1967].
 2 Stably complex structures
A direct attempt to define the bordism relation on complex manifolds fails because the manifold is odd-dimensional and therefore cannot be complex. In order to work with complex manifolds in the bordism theory, one needs to weaken the notion of a complex structure. This leads directly to considering stably complex (also known as weakly almost complex, stably almost complex or quasicomplex) manifolds.
Let denote the tangent bundle of , and the product vector bundle over . A tangential stably complex structure on is determined by a choice of an isomorphism
between the "stable" tangent bundle and a complex vector bundle over . Some of the choices of such isomorphisms are deemed to be equivalent, i.e. determine the same stably complex structures (see details in Chapters II and VII of [Stong1968]). In particular, two stably complex structures are equivalent if they differ by a trivial complex summand. A normal stably complex structure on is determined by a choice of a complex bundle structure on the normal bundle of an embedding . Tangential and normal stably complex structures on determine each other by means of the canonical isomorphism . We therefore may restrict our attention to tangential structures only.
A stably complex manifold is a pair consisting of a manifold and a stably complex structure on it. This is a generalisation of a complex and almost complex manifold (where the latter means a manifold with a choice of a complex structure on , i.e. a stably complex structure with ).
Example 2.1. Let . The standard complex structure on is equivalent to the stably complex structure determined by the isomorphism
where is the Hopf line bundle. On the other hand, the isomorphism
determines a trivial stably complex structure on .
 3 Definition of bordism and cobordism
The bordism relation can be defined between stably complex manifolds. Like the case of unoriented bordism, the set of bordism classes of stably complex manifolds of dimension is an Abelian group with respect to the disjoint union. This group is called the -dimensional complex bordism group and denoted . The zero element is represented by the bordism class of any manifold which bounds and whose stable tangent bundle is trivial (and therefore isomorphic to a product complex vector bundle ). The sphere provides an example of such a manifold. The opposite element to the bordism class in the group may be represented by the same manifold with the stably complex structure determined by the isomorphism
where is given by .
An abbreviated notation for the complex bordism class will be used whenever the stably complex structure is clear from the context.
The complex bordism group and cobordism group of a space may also be defined geometrically, at least for the case when is a manifold. This can be done along the lines suggested by [Quillen1971a] and [Dold1978] by considering special "stably complex" maps of manifolds to . However, nowadays the homotopical approach to bordism has taken over, and the (co)bordism groups are usually defined using the Pontrjagin-Thom construction similarly to the unoriented case:
where is the Thom space of the universal complex -plane bundle , and denotes the set of homotopy classes of pointed maps from to . These groups are -modules and give rise to a multiplicative (co)homology theory. In particular, is a graded ring.
The graded ring with is called the complex cobordism ring; it has nontrivial elements only in nonpositively graded components.
 4 Geometric cobordisms
There is one important case when certain cobordism classes can be represented very explicitly by maps of manifolds.
For any cell complex the cohomology group can be identified with the set of homotopy classes of maps into . Since , every element also determines a cobordism class . The elements of obtained in this way are called geometric cobordisms of . We therefore may view as a subset in , however the group operation in is not obtained by restricting the group operation in (see Formal group laws and genera for the relationship between the two operations).
When is a manifold, geometric cobordisms may be described by submanifolds of codimension 2 with a fixed complex structure on the normal bundle.
Indeed, every corresponds to a homotopy class of maps . The image is contained in some , and we may assume that is transverse to a certain hyperplane . Then is a codimension 2 submanifold in whose normal bundle acquires a complex structure by restriction of the complex structure on the normal bundle of . Changing the map within its homotopy class does not affect the bordism class of the embedding .
Conversely, assume given a submanifold of codimension 2 whose normal bundle is endowed with a complex structure. Then the composition
of the Pontrjagin-Thom collapse map and the map of Thom spaces corresponding to the classifying map of defines an element , and therefore a geometric cobordism.
If is an oriented manifold, then a choice of complex structure on the normal bundle of a codimension 2 embedding is equivalent to orienting . The image of the fundamental class of in the homology of is Poincaré dual to .
 5 Structure results
The complex bordism ring is described as follows.
- is a polynomial ring over generated by the bordism classes of complex projective spaces , .
- Two stably complex manifolds are bordant if and only if they have identical sets of Chern characteristic numbers.
- is a polynomial ring over with one generator in every even dimension , where .
Part 1 can be proved by the methods of [Thom1954]. Part 2 follows from the results of [Milnor1960] and [Novikov1960]. Part 3 is the most difficult one; it was done by [Novikov1960] using the Adams spectral sequence and structure theory of Hopf algebras (see also [Novikov1962] for a more detailed account) and Milnor (unpublished, but see [Thom1995]) in 1960. Another more geometric proof was given by [Stong1965], see also [Stong1968].
 6 Multiplicative generators
 6.1 Preliminaries: characteristic numbers detecting generators
To describe a set of multiplicative generators for the ring we shall need a special characteristic class of complex vector bundles. Let be a complex -plane bundle over a manifold . Write its total Chern class formally as follows:
so that is the th elementary symmetric function in formal indeterminates. These indeterminates acquire a geometric meaning if is a sum of line bundles; then , . Consider the polynomial
and express it via the elementary symmetric functions:
Substituting the Chern classes for the elementary symmetric functions we obtain a certain characteristic class of :
This characteristic class plays an important role in detecting the polynomial generators of the complex bordism ring, because of the following properties (which follow immediately from the definition).
- for .
Given a stably complex manifold of dimension , define its characteristic number by
where is the complex bundle from the definition of the stably complex structure, and the fundamental homology class.
Corollary 6.2. If a bordism class decomposes as where and , then .
It follows that the characteristic number vanishes on decomposable elements of . It also detects indecomposables that may be chosen as polynomial generators. In fact, the following result is a byproduct of the calculation of :
Theorem 6.3. A bordism class may be chosen as a polynomial generator of the ring if and only if
(Ed Floyd was fond of calling the characteristic numbers the "magic numbers" of manifolds.)
 6.2 Milnor hypersurfaces
A universal description of connected manifolds representing the polynomial generators is unknown. Still, there is a particularly nice family of manifolds whose bordism classes generate the whole ring . This family is redundant though, so there are algebraic relations between their bordism classes.
Fix a pair of integers and consider the product . Its algebraic subvariety
is called a Milnor hypersurface. Note that .
The Milnor hypersurface may be identified with the set of pairs , where is a line in and is a hyperplane in containing . The projection describes as the total space of a bundle over with fibre .
Denote by and the projections of onto the first and second factors respectively, and by the Hopf line bundle over a complex projective space; then is the hyperplane section bundle. We have
where , .
Proposition 6.4. The geometric cobordism in corresponding to the element is represented by the submanifold . In particular, the image of the fundamental class in is Poincaré dual to .
Lemma 6.5. We have
Theorem 6.6. The bordism classes multiplicatively generate the complex bordism ring .
Proof. This follows from the fact that
and the previous Lemma.
Example 6.7. We list some bordism groups and generators:
- , generated by a point;
- , generated by , as and ;
- , generated by and , as and ;
- cannot be taken as the polynomial generator , since , while . The bordism class may be taken as .
The previous theorem about the multiplicative generators for has the following important addendum.
Theorem 6.8 (Milnor). Every bordism class with contains a nonsingular algebraic variety (not necessarily connected).
(The Milnor hypersufaces are algebraic, but one also needs to represent by algebraic varieties!) For the proof see Chapter 7 of [Stong1968].
The following question is still open, even in complex dimension 2.
Problem 6.9 (Hirzebruch). Describe the set of bordism classes in containing connected nonsingular algebraic varieties.
Example 6.10. Every class contains a nonsingular algebraic variety, namely, a disjoint union of copies of for and a Riemannian surface of genus for . Connected algebraic varieties are only contained in the bordism classes with .
 6.3 Toric generators and quasitoric representatives in cobordism classes
There is an alternative set of multiplicative generators for the complex bordism ring , consisting of nonsingular projective toric varieties, or toric manifolds. Every therefore supports an effective action of a "big torus" (of dimension half the dimension of the manifold) with isolated fixed points. The construction of is due to [Buchstaber&Ray2001] (see also [Buchstaber&Panov2002] and [Buchstaber&Panov&Ray2007]).
The manifold is constructed as the projectivisation of a sum of line bundles over the bounded flag manifold .
A bounded flag in is a complete flag
for which , contains the coordinate subspace spanned by the first standard basis vectors.
The set of all bounded flags in is a smooth complex algebraic variety of dimension (cf. [Buchstaber&Ray2001]), referred to as the bounded flag manifold. The action of the algebraic torus on given by
where and , induces an action on bounded flags, and therefore endows with a structure of a toric manifold.
is also the total space of a Bott tower, that is, a tower of fibrations with base and fibres in which every stage is the projectivisation of a sum of two line bundles. In particular, is the Hirzebruch surface .
The manifold () consists of pairs , where is a bounded flag in and is a line in . (Here denotes the orthogonal complement to in , so that is the orthogonal complement to in .) Therefore, is the total space of a bundle over with fibre . This bundle is in fact the projectivisation of a sum of line bundles, which implies that is a complex -dimensional toric manifold.
The bundle is the pullback of the bundle along the map taking a bounded flag to its first line . This is described by the diagram
(The bundle , unlike , is not a projectivisation of a sum of line bundles, which prevents the torus action on from lifting to an action on the total space.)
Lemma 6.11. We have .
Proof. We may assume that , as otherwise . We have the equality , the projectivisation of a -plane bundle over . We also have that the map has degree since it is an isomorphism on the affine chart . Furthermore, . The result now follows from Lemma 6.12 below.
Lemma 6.12. Let be a degree map of -dimensional almost complex manifolds, and let be a complex -plane bundle over , . Then
Theorem 6.13 ([Buchstaber&Ray2001]). The bordism classes of toric manifolds multiplicatively generate the complex bordism ring . Therefore, every complex bordism class contains a disjoint union of toric manifolds.
Proof. The first statement follows from the fact that the Milnor hypersurfaces generate the complex bordism ring and the previous Lemma. A product of toric manifolds is toric, but a disjoint union of toric manifolds is not a toric manifold, since toric manifolds are connected by definition.
The manifolds and are not bordant in general, although and by definition.
Connected representatives in cobordism classes cannot be found within toric manifolds because of severe restrictions on their characteristic numbers. (For example, the Todd genus of every toric manifold is 1.) A topological generalisation of toric manifolds was suggested in [Davis&Januszkiewicz1991a] (see also [Buchstaber&Panov2002]). These manifolds have become known as quasitoric. A quasitoric manifold is a smooth manifold of dimension with a locally standard action of an -dimensional torus whose quotient is a simple polytope. Quasitoric manifolds generally fail to be complex or even almost complex, but they always admit stably complex structures [Buchstaber&Ray2001].
Theorem 6.14 ([Buchstaber&Panov&Ray2007]).In dimensions , every complex cobordism class contains a quasitoric manifold, necessarily connected, whose stably complex structure is compatible with the action of the torus.
 7 Adams-Novikov spectral sequence
A principal motivation for [Novikov1967] was to develop a version of the Adams spectral sequence in which mod cohomology (and the Steenrod algebra) are replaced by complex cobordism theory (and its ring of stable cohomology operations), for the purpose of computing stable homotopy groups. The foundations for the Adams-Novikov spectral sequence were laid in this paper, and many applications and computations have followed. An introduction to the work of Novikov on complex cobordism is given in [Adams1974]. The most comprehensive study of the Adams-Novikov spectral sequence is [Ravenel1986], currently available in a second edition from AMS/Chelsea.
 8 References
- [Adams1974] J. F. Adams, Stable homotopy and generalised homology, University of Chicago Press, Chicago, Ill., 1974. MR0402720 (53 #6534) Zbl 0309.55016
- [Buchstaber&Panov&Ray2007] V. M. Buchstaber, T. E. Panov and N. Ray, Spaces of polytopes and cobordism of quasitoric manifolds, Mosc. Math. J. 7 (2007), no.2, 219–242, 350. MR2337880 (2008f:55006) Zbl 1176.55004
- [Buchstaber&Panov2002] V. M. Buchstaber and T. E. Panov, Torus actions and their applications in topology and combinatorics, American Mathematical Society, 2002. MR1897064 (2003e:57039) Zbl 1012.52021
- [Buchstaber&Ray2001] V. M. Buchstaber and N. Ray, Tangential structures on toric manifolds, and connected sums of polytopes, Internat. Math. Res. Notices (2001), no.4, 193–219. MR1813798 (2002b:57043) Zbl 0996.52013
- [Davis&Januszkiewicz1991a] M. W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no.2, 417–451. MR1104531 (92i:52012) Zbl 0733.52006
- [Dold1978] A. Dold, Geometric cobordism and the fixed point transfer, in Algebraic topology (Proc. Conf., Univ. British Columbia, Vancouver, B.C., 1977), Lecture Notes in Math. 673, Springer, Berlin, (1978), 32–87. MR517084 (80g:57052) Zbl 0386.57005
- [Milnor1960] J. Milnor, On the cobordism ring and a complex analogue. I, Amer. J. Math. 82 (1960), 505–521. MR0119209 (22 #9975) Zbl 0095.16702
- [Novikov1960] S. P. Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Soviet Math. Dokl. 1 (1960), 717–720. MR0121815 (22 #12545) Zbl 0094.35902
- [Novikov1962] S. P. Novikov, Homotopy properties of Thom complexes, Mat. Sb. (N.S.) 57 (99) (1962), 407–442. MR0157381 (28 #615) Zbl 0193.51801
- [Novikov1967] S. P. Novikov, Methods of algebraic topology from the point of view of cobordism theory, Math. USSR, Izv. 1, (1967) 827–913. MR0221509 (36 #4561) Zbl 0176.52401
- [Quillen1971a] D. Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 (1971), 29–56 (1971). MR0290382 (44 #7566) Zbl 0214.50502
- [Ravenel1986] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press Inc., Orlando, FL, 1986. MR860042 (87j:55003) Zbl 1073.55001
- [Stong1965] R. E. Stong, Relations among characteristic numbers. I, Topology 4 (1965), 267–281. MR0192515 (33 #740) Zbl 0136.20503
- [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
- [Thom1995] R. Thom, Travaux de Milnor sur le cobordisme, Séminaire Bourbaki, Vol. 5, Exp. No. 180, Soc. Math. France, Paris, (1995), 169–177. MR1603465 Zbl 0116.40402