Complex bordism
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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 of
, and
the product vector bundle
over
. A tangential stably complex structure on
is
determined by a choice of an isomorphism
![\displaystyle c_{\mathcal T}\colon {\mathcal T}\!M\oplus \underline{\mathbb R}^k\to \xi](/images/math/2/f/1/2f19edd9c6cb0b41b14d224c85c7312c.png)
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. 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 on
is determined by a choice of a complex bundle
structure in the normal bundle
of an embedding
. A 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 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 on
is
equivalent to a stably complex structure determined by the
isomorphism
![\displaystyle {\mathcal T}(\mathbb{C}P^1)\oplus\underline{\mathbb R}^2\stackrel{\cong}{\longrightarrow} \overline{\eta}\oplus \overline{\eta}](/images/math/4/b/b/4bbdec9b83c90eda22e6035e1b946dc2.png)
where is the Hopf line bundle. On the other hand, the
isomorphism
![\displaystyle {\mathcal T}(\mathbb{C}P^1)\oplus\underline{\mathbb R}^2\stackrel{\cong}{\longrightarrow} \eta\oplus \overline{\eta}\cong \underline{\mathbb C}^2](/images/math/d/e/7/de72e48450e5f811dd4ad211c544427b.png)
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 of
-dimensional complex bordisms and
denoted
. A zero 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
![\displaystyle {\mathcal T}\!M\oplus\underline{\mathbb R}^k\oplus\underline{\mathbb R}^2\stackrel{c_{\mathcal T}\oplus e}{\relbar\joinrel\hspace{-1pt}\relbar\joinrel\hspace{-1pt}\longrightarrow}\xi\oplus\underline{\mathbb C}](/images/math/3/0/0/30016dd60828ef8d72d3e84bf11c43ff.png)
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 space
may also be defined
geometrically, at least for the case when
is a manifold. This can be done along the lines suggested by [Quillen1971] and [Dold1978] by considering special "stably complex" maps
of manifolds
to
. 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:
![\displaystyle \begin{aligned} U_n(X)&=\lim_{k\to\infty}\pi_{2k+n}((X_+)\wedge MU(k)),\\ U^n(X)&=\lim_{k\to\infty}[\Sigma^{2k-n}(X_+),MU(k)] \end{aligned}](/images/math/3/8/8/388203cfec203d491a8ba0f9062a963f.png)
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 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 Complex bordism#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 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 in
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
![\displaystyle X\to M(\nu)\to MU(1)=\mathbb C P^\infty](/images/math/3/6/e/36e602c2b85d008367f6297803ab87d8.png)
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.
If is an oriented manifold, then a choice of complex structure
in the normal bundle of a codimension 2 embedding
is
equivalent to orienting
. The image of the fundamental class of
in the homology of
is 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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