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 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
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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