Complex bordism
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a cobordism class $u_x\in U^2(X)$. The elements of $U^2(X)$ obtained in this way are called ''geometric cobordisms'' | a cobordism class $u_x\in U^2(X)$. The elements of $U^2(X)$ obtained in this way are called ''geometric cobordisms'' | ||
of $X$. We therefore may view $H^2(X)$ as a subset in $U^2(X)$, however the group operation in $H^2(X)$ is not obtained by | of $X$. We therefore may view $H^2(X)$ as a subset in $U^2(X)$, however the group operation in $H^2(X)$ is not obtained by | ||
− | restricting the group operation in $U^2(X)$ (see [[ | + | restricting the group operation in $U^2(X)$ (see [[Formal group laws and genera|Formal group laws and genera]] for the relationship |
between the two operations). | between the two operations). | ||
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Contents |
1 Introduction
Complex bordisms (also known as unitary bordisms) is the bordism theory of stably complex manifolds. It is one of the most important theory of bordisms with additional structure, or B-bordisms.
The theory of complex bordisms is much richer than its unoriented analogue, and at the same time is not as complicated as oriented bordisms or other bordisms with additional structure (B-bordisms). Thanks to this, the complex cobordism theory found the most stricking and important applications in algebraic topology and beyond. Many of these applications, including the formal group techniques and 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 the complex structures. This leads directly to considering stably complex (also known as weakly almost complex, stably almost complex or quasicomplex) manifolds.
Let denote the tangent bundle ofTex syntax error, and the product vector bundle over
Tex syntax error. A tangential stably complex structure on
Tex syntax erroris
determined by a choice of an isomorphism
between the "stable" tangent bundle and a complex vector
bundle overTex syntax error. Some of the choices of such isomorphisms
are deemed to be equivalent, i.e. determining 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 onTex syntax erroris determined by a choice of a complex bundle
structure in the normal bundle of an embedding . A tangential and normal stably
complex structures onTex syntax errordetermine 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 manifoldTex syntax errorand a stably complex structure
on it. This is a generalisation to 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 onTex syntax erroris
equivalent to a 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 bordisms, the set of bordism classes of stably complex manifolds is an Abelian group with respect to the disjoint union. This group
is called the group ofTex syntax error-dimensional complex bordisms and
denoted . A zero is represented by the bordism
class of any manifoldTex syntax errorwhich 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 manifoldTex syntax errorwith 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 groups of complex bordisms and cobordisms of a spaceTex syntax errormay also be defined geometrically, at least for the case when
Tex syntax erroris a manifold. This can be done along the lines suggested by [Quillen1971] and [Dold1978] by considering special "stably complex" maps of manifolds
Tex syntax errorto
Tex syntax error. However, nowadays the homotopical approach to
bordisms takes 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 . 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 complexTex syntax errorthe 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
ofTex syntax error. 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).
WhenTex syntax erroris a manifold, geometric cobordisms may be described by
submanifolds of codimension 2 with a fixed complex structure in the normal bundle.
Indeed, every corresponds to a homotopy class of maps . The image is contained in some , and we may assume that is transversal to a certain hyperplane .
Then is a codimension 2 submanifold inTex syntax error
whose normal bundle acquires a complex structure by restriction of the complex structure in the normal bundle of . Changing the map within its homotopy class does not affect the bordism class of 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 the the classifying map of defines and element , and therefore a geometric cobordism.
IfTex syntax erroris an oriented manifold, then a choice of complex structure
in the normal bundle of a codimension 2 embedding is
equivalent to orientingTex syntax error. The image of the fundamental class of
Tex syntax errorin the homology of
Tex syntax erroris Poincar\'e dual to .
5 Structure results
6 Multiplicative generators
7 Formal group laws and genera
8 Adams-Novikov spectral sequence
The main references here are [Novikov1967] and [Ravenel1986]
9 References
- [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
- [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
- [Quillen1971] D. Quillen, The Adams conjecture, Topology 10 (1971), 67–80. MR0279804 (43 #5525) Zbl 0219.55013
- [Ravenel1986] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press Inc., Orlando, FL, 1986. MR860042 (87j:55003) Zbl 1073.55001
- [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010
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