Complex bordism
An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 09:39, 1 April 2011 and the changes since publication. |
The user responsible for this page is Taras Panov. No other user may edit this page at present. |
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 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 Poincare dual to
.
5 Structure results
Complex bordism ring is described as follows.
Theorem 5.1.
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 Adams spectral sequence and structure theory of Hopf algebras (see also [Novikov1962] for a more detailed account) and Milnor (unpublished) in 1960. Another more geometric proof was given by [Stong1965], see also [Stong1968].
6 Multiplicative generators
6.1 Preliminaries: characteristic number sn
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 formally its total Chern class as
follows:
![\displaystyle c(\xi )=1+c_1(\xi )+\ldots +c_k(\xi )=(1+x_1)\dots(1+x_k),](/images/math/a/3/c/a3c8725885d1bbf430dc8a594c139256.png)
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
![\displaystyle P_n(x_1,\ldots x_k)=x_1^n+\ldots +x_k^n](/images/math/4/9/8/498103a80c4ccec7fdd609ffa60d0241.png)
and express it via the elementary symmetric functions:
![\displaystyle P_n(x_1,\ldots ,x_k)=s_n(\sigma_1,\dots ,\sigma_k).](/images/math/5/e/3/5e3851321a40d2a6e550ee86c473365d.png)
Substituting the Chern classes for the elementary symmetric
functions we obtain a certain characteristic class of :
![\displaystyle s_n(\xi)=s_n(c_1(\xi),\ldots,c_k(\xi))\in H^{2n}(M).](/images/math/2/3/6/2360f310e594c7a63f73cd45d9bbc38d.png)
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).
Proposition 6.1.
-
for
.
-
.
Given a stably complex manifold of
dimension
, define its characteristic number
![\displaystyle s_n[M]=s_n(\xi)\langle M\rangle\in\mathbb Z](/images/math/8/9/c/89c6094854b72cf3c246560dbf4a5426.png)
where is the complex bundle from the definition of 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
![\displaystyle s_n[M]=\begin{cases} \pm1, &\text{if $n\ne p^k-1$ for any prime $p$;}\\ \pm p, &\text{if $n=p^k-1$ for some prime $p$.} \end{cases}](/images/math/c/8/5/c85430ad17dca6960d6989d8d71d9c5d.png)
6.2 Milnor hypersurfaces Hij
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 superfluous though, so there are algebraic relations between
their bordism classes.
Fix a pair of integers and consider the product
. Its algebraic subvariety
![\displaystyle H_{ij}=\{ (z_0:\ldots :z_i)\times (w_0:\ldots :w_j)\in \mathbb{C}P^i\times \mathbb{C}P^j\colon z_0w_0+\ldots +z_iw_i=0\}](/images/math/6/a/9/6a9946cd67f0fb82c02ae794552a2f71.png)
is called the Milnor hypersurface. Note that .
Denote by and
the projections
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
![\displaystyle H^*(\mathbb C P^i\times\mathbb C P^j)=\mathbb Z[x,y]/(x^{i+1}=0,\;y^{j+1}=0)](/images/math/5/a/6/5a67f0af059ef2437ca32f0cc4ed98bf.png)
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 Poincare dual to
.
See the proof.
Lemma 6.5. We have
![\displaystyle s_{i+j-1}[H_{ij}]=\begin{cases} j,&\text{if \ $i=0$, i.e. $H_{ij}=\mathbb C P^{j-1}$};\\ 2,&\text{if \ $i=j=1$};\\ 0,&\text{if \ $i=1$, $j>1$};\\ -\binom{i+j}i,&\text{if \ $i>1$}. \end{cases}](/images/math/3/d/2/3d25a19b1a59aff9251fd5cc03455c4e.png)
See the proof.
Theorem 6.6.
The bordism classes multiplicatively
generate the bordism ring
.
{{endthm}
Proof.
This follows from the fact that
Tex syntax error
and the previous Lemma.
Example 6.7.
-
=0;
-
=\mathbb Z, generated by a point;
-
=\mathbb Z, generated by
, as
and
;
-
=\mathbb Z\oplus\mathbb \Z, 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 specification.
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.
Theorem 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 Bij and quasitoric representatives in cobordism classes
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
- [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
- [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
- [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
This page has not been refereed. The information given here might be incomplete or provisional. |