Complex bordism

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[edit] 1 Introduction

Complex bordism (also known as unitary bordism) is the bordism theory of stably complex manifolds. It is one of the most important theories of bordism with additional structure, or B-bordism.

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].

[edit] 2 Stably complex structures

A direct attempt to define the bordism relation on complex manifolds fails because the manifold \,W 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 {\mathcal T}\!M denote the tangent bundle of M, and \underline{\mathbb R}^k the product vector bundle M\times\mathbb R^k over M. A tangential stably complex structure on M is determined by a choice of an isomorphism

\displaystyle  c_{\mathcal T}\colon {\mathcal T}\!M\oplus \underline{\mathbb R}^k\to \xi

between the "stable" tangent bundle and a complex vector bundle \xi over M. 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 M is determined by a choice of a complex bundle structure on the normal bundle \nu(M) of an embedding M\hookrightarrow\mathbb R^N. Tangential and normal stably complex structures on M determine each other by means of the canonical isomorphism \mathcal T\!M\oplus\nu(M)\cong\underline{\mathbb R}^N. We therefore may restrict our attention to tangential structures only.

A stably complex manifold is a pair (M,c_{\mathcal T}) consisting of a manifold M and a stably complex structure c_{\mathcal T} 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 {\mathcal T}\!M, i.e. a stably complex structure c_{\mathcal T} with k=0).

Example 2.1. Let M=\CP^1. The standard complex structure on M is equivalent to the stably complex structure determined by the isomorphism

\displaystyle  {\mathcal T}(\CP^1)\oplus\underline{\mathbb R}^2\stackrel{\cong}{\longrightarrow} \overline{\eta}\oplus \overline{\eta}

where \eta is the Hopf line bundle. On the other hand, the isomorphism

\displaystyle  {\mathcal T}(\CP^1)\oplus\underline{\mathbb R}^2\stackrel{\cong}{\longrightarrow} \eta\oplus \overline{\eta}\cong \underline{\mathbb C}^2

determines a trivial stably complex structure on \CP^1.

[edit] 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 [M,c_{\mathcal T}] of stably complex manifolds of dimension n is an Abelian group with respect to the disjoint union. This group is called the n-dimensional complex bordism group and denoted \varOmega^U_n. The zero element is represented by the bordism class of any manifold M which bounds and whose stable tangent bundle is trivial (and therefore isomorphic to a product complex vector bundle M\times\mathbb C^k). The sphere S^n provides an example of such a manifold. The opposite element to the bordism class [M,c_{\mathcal T}] in the group \varOmega^U_n may be represented by the same manifold M 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}{-\hspace{-5pt}-\hspace{-5pt}\longrightarrow}\xi\oplus\underline{\mathbb C}

where e\colon\mathbb R^2\to\mathbb C is given by e(x,y)=x-iy.

An abbreviated notation [M] for the complex bordism class will be used whenever the stably complex structure c_{\mathcal T} is clear from the context.

The complex bordism group U_n(X) and cobordism group U^n(X) of a space X may also be defined geometrically, at least for the case when X is a manifold. This can be done along the lines suggested by [Quillen1971a] and [Dold1978] by considering special "stably complex" maps of manifolds M to X. 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:

\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}

where MU(k) is the Thom space of the universal complex k-plane bundle EU(k)\to BU(k), and [X,Y] denotes the set of homotopy classes of pointed maps from X to Y. These groups are \varOmega_*^U-modules and give rise to a multiplicative (co)homology theory. In particular, U^*(X)=\oplus_n U^n(X) is a graded ring.

The graded ring \varOmega^*_U with \varOmega^{n}_U=\varOmega_{-n}^U is called the complex cobordism ring; it has nontrivial elements only in nonpositively graded components.

[edit] 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 X the cohomology group H^2(X) can be identified with the set [X,\CP^\infty] of homotopy classes of maps into \CP^\infty. Since \CP^\infty=MU(1), every element x\in H^2(X) also determines 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 restricting the group operation in U^2(X) (see Formal group laws and genera for the relationship between the two operations).

When X is a manifold, geometric cobordisms may be described by submanifolds M\subset X of codimension 2 with a fixed complex structure on the normal bundle.

Indeed, every x\in H^2(X) corresponds to a homotopy class of maps f_x\colon X\to\CP^\infty. The image f_x(X) is contained in some \CP^N\subset\CP^\infty, and we may assume that f_x(X) is transverse to a certain hyperplane H\subset\CP^N. Then M_x:=f_x^{-1}(H) is a codimension 2 submanifold in X whose normal bundle acquires a complex structure by restriction of the complex structure on the normal bundle of H\subset\CP^N. Changing the map f_x within its homotopy class does not affect the bordism class of the embedding M_x\to X.

Conversely, assume given a submanifold M\subset X of codimension 2 whose normal bundle is endowed with a complex structure. Then the composition

\displaystyle  X\to M(\nu)\to MU(1)=\CP^\infty

of the Pontrjagin-Thom collapse map X\to M(\nu) and the map of Thom spaces corresponding to the classifying map M\to BU(1) of \nu defines an element x_M\in H^2(X), and therefore a geometric cobordism.

If X is an oriented manifold, then a choice of complex structure on the normal bundle of a codimension 2 embedding M\subset X is equivalent to orienting M. The image of the fundamental class of M in the homology of X is Poincaré dual to x_M\in H^2(X).

[edit] 5 Structure results

The complex bordism ring \varOmega_*^U is described as follows.

Theorem 5.1.

  1. \varOmega_*^U\otimes\mathbb Q is a polynomial ring over \mathbb Q generated by the bordism classes of complex projective spaces \CP^i, i\geqslant1.
  2. Two stably complex manifolds are bordant if and only if they have identical sets of Chern characteristic numbers.
  3. \varOmega_*^U is a polynomial ring over \mathbb Z with one generator a_i in every even dimension 2i, where i\geqslant1.

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].

[edit] 6 Multiplicative generators

[edit] 6.1 Preliminaries: characteristic numbers detecting generators

To describe a set of multiplicative generators for the ring \varOmega_*^U we shall need a special characteristic class of complex vector bundles. Let \xi be a complex k-plane bundle over a manifold M. Write its total Chern class formally as follows:

\displaystyle  c(\xi )=1+c_1(\xi )+\cdots +c_k(\xi )=(1+x_1)\cdots(1+x_k),

so that c_i(\xi )=\sigma_i(x_1,\ldots,x_k) is the ith elementary symmetric function in formal indeterminates. These indeterminates acquire a geometric meaning if \xi is a sum \xi_1\oplus\cdots\oplus\xi_k of line bundles; then x_j=c_1(\xi_j), 1\leqslant j\leqslant k. Consider the polynomial

\displaystyle  P_n(x_1,\ldots x_k)=x_1^n+\cdots +x_k^n

and express it via the elementary symmetric functions:

\displaystyle  P_n(x_1,\ldots ,x_k)=s_n(\sigma_1,\ldots,\sigma_k).

Substituting the Chern classes for the elementary symmetric functions we obtain a certain characteristic class of \xi:

\displaystyle  s_n(\xi)=s_n(c_1(\xi),\ldots,c_k(\xi))\in H^{2n}(M).

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.

  1. s_n(\xi)=0 for 2n>\dim M.
  2. s_n(\xi\oplus\eta)=s_n(\xi)+s_n(\eta).

Given a stably complex manifold (M,c_{\mathcal T}) of dimension 2n, define its characteristic number by

\displaystyle  s_n[M] := s_n(\xi)\langle M\rangle\in\mathbb Z

where \xi is the complex bundle from the definition of the stably complex structure, and \langle M\rangle\in H_{2n}(M) the fundamental homology class.

Corollary 6.2. If a bordism class [M]\in\varOmega_{2n}^U decomposes as [M_1]\times[M_2] where \dim M_1>0 and \dim M_2>0, then s_n[M]=0.

It follows that the characteristic number s_n vanishes on decomposable elements of \varOmega^U_{2n}. It also detects indecomposables that may be chosen as polynomial generators. In fact, the following result is a byproduct of the calculation of \varOmega^U_*:

Theorem 6.3. A bordism class [M]\in\varOmega_{2n}^U may be chosen as a polynomial generator a_n of the ring \varOmega_*^U 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}

(Ed Floyd was fond of calling the characteristic numbers s_n[M] the "magic numbers" of manifolds.)

[edit] 6.2 Milnor hypersurfaces

A universal description of connected manifolds representing the polynomial generators a_n\in\varOmega_*^U is unknown. Still, there is a particularly nice family of manifolds whose bordism classes generate the whole ring \varOmega_*^U. This family is redundant though, so there are algebraic relations between their bordism classes.

Fix a pair of integers j\geqslant i\geqslant0 and consider the product \CP^i\times\CP^j. Its algebraic subvariety

\displaystyle  H_{ij}=\{ (z_0:\ldots :z_i)\times (w_0:\ldots :w_j)\in \CP^i\times \CP^j\colon z_0w_0+\cdots +z_iw_i=0\}

is called a Milnor hypersurface. Note that H_{0j}\cong\mathbb C P^{j-1}.

The Milnor hypersurface H_{ij} may be identified with the set of pairs (l,\alpha), where l is a line in \mathbb C^{i+1} and \alpha is a hyperplane in \mathbb C^{j+1} containing l. The projection (l,\alpha)\mapsto l describes H_{ij} as the total space of a bundle over \mathbb CP^i with fibre \mathbb CP^{j-1}.

Denote by p_1 and p_2 the projections of \CP^i\times\CP^j onto the first and second factors respectively, and by \eta the Hopf line bundle over a complex projective space; then \bar\eta is the hyperplane section bundle. We have

\displaystyle  H^*(\CP^i\times\CP^j)=\mathbb Z[x,y]/(x^{i+1}=0,\;y^{j+1}=0)

where x=p_1^*c_1(\bar\eta), y=p_2^*c_1(\bar\eta).

Proposition 6.4. The geometric cobordism in \CP^i\times\CP^j corresponding to the element x+y\in H^2(\CP^i\times\CP^j) is represented by the submanifold H_{ij}. In particular, the image of the fundamental class \langle H_{ij}\rangle in H_{2(i+j-1)}(\CP^i\times\mathbb C P^j) is Poincaré dual to x+y.

Proof. Click here - opens a separate pdf file.


Lemma 6.5. We have

\displaystyle  s_{i+j-1}[H_{ij}]=\begin{cases} j,&\text{if \ $i=0$, i.e. $H_{ij}=\CP^{j-1}$};\\ 2,&\text{if \ $i=j=1$};\\ 0,&\text{if \ $i=1$, $j>1$};\\ -\binom{i+j}i,&\text{if \ $i>1$}. \end{cases}

Proof. Click here - opens a separate pdf file.


Theorem 6.6. The bordism classes \{[H_{ij}],\;0\leqslant i\leqslant j\} multiplicatively generate the complex bordism ring \varOmega_*^U.

Proof. This follows from the fact that

\displaystyle  \mathop{\text{g.c.d.}}\Bigl({\textstyle\binom{n+1}i},\;1\leqslant i\leqslant n\Bigr)= \begin{cases} p, & \text{if \ $n=p^k-1$,}\\ 1, & \text{otherwise,} \end{cases}

and the previous Lemma.


Example 6.7. We list some bordism groups and generators:

  • \varOmega_{2i+1}^U=0;
  • \varOmega_0^U=\mathbb Z, generated by a point;
  • \varOmega_2^U=\mathbb Z, generated by [\mathbb CP^1], as 1=2^1-1 and s_1[\CP^1]=2;
  • \varOmega_4^U=\mathbb Z\oplus\mathbb Z, generated by [\mathbb CP^1\times\mathbb CP^1] and [\mathbb CP^2], as 2=3^1-1 and s_2[\CP^2]=3;
  • [\CP^3] cannot be taken as the polynomial generator a_3\in\varOmega_6^U, since s_3[\CP^3]=4, while s_3(a_3)=\pm2. The bordism class [H_{22}]+[\CP^3] may be taken as a_3.

The previous theorem about the multiplicative generators for \varOmega_*^U has the following important addendum.

Theorem 6.8 (Milnor). Every bordism class x\in\varOmega_n^U with n>0 contains a nonsingular algebraic variety (not necessarily connected).

(The Milnor hypersufaces are algebraic, but one also needs to represent -[H_{ij}] 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 \varOmega_*^U containing connected nonsingular algebraic varieties.

Example 6.10. Every class k[\CP^1]\in\varOmega^U_2 contains a nonsingular algebraic variety, namely, a disjoint union of k copies of \CP^1 for k>0 and a Riemannian surface of genus (1-k) for k\leqslant0. Connected algebraic varieties are only contained in the bordism classes k[\CP^1] with k\leqslant1.

[edit] 6.3 Toric generators and quasitoric representatives in cobordism classes

There is an alternative set of multiplicative generators \{[B_{ij}],0\leqslant i\leqslant j\} for the complex bordism ring \varOmega_*^U, consisting of nonsingular projective toric varieties, or toric manifolds. Every B_{ij} therefore supports an effective action of a "big torus" (of dimension half the dimension of the manifold) with isolated fixed points. The construction of B_{ij} is due to [Buchstaber&Ray2001] (see also [Buchstaber&Panov2002] and [Buchstaber&Panov&Ray2007]).

Milnor hypersurfaces H_{ij} are not toric manifolds for i>1, because of a simple cohomological obstruction (see Proposition 5.43 in [Buchstaber&Panov2002]).

The manifold B_{ij} is constructed as the projectivisation of a sum of j line bundles over the bounded flag manifold B_i.

A bounded flag in \mathbb C^{n+1} is a complete flag

\displaystyle  \mathcal U=\{U_1\subset U_2\subset\ldots\subset U_{n+1}=\mathbb C^{n+1},\;\; \dim U_i=i\}

for which U_k,\;\;{}2\leqslant k\leqslant n, contains the coordinate subspace \mathbb C^{k-1} spanned by the first k-1 standard basis vectors.

The set B_n of all bounded flags in \mathbb C^{n+1} is a smooth complex algebraic variety of dimension n (cf. [Buchstaber&Ray2001]), referred to as the bounded flag manifold. The action of the algebraic torus (\mathbb C^\times)^n on \mathbb C^{n+1} given by

\displaystyle  (t_1,\ldots,t_n)\cdot(w_1,\ldots,w_n,w_{n+1})=(t_1w_1,\ldots,t_nw_n,w_{n+1}),

where (t_1,\ldots,t_n)\in(\mathbb C^\times)^n and (w_1,\ldots,w_n,w_{n+1})\in\mathbb C^{n+1}, induces an action on bounded flags, and therefore endows B_n with a structure of a toric manifold.

B_n is also the total space of a Bott tower, that is, a tower of fibrations with base \CP^1 and fibres \CP^1 in which every stage is the projectivisation of a sum of two line bundles. In particular, B_2 is the Hirzebruch surface H_1.

The manifold B_{ij} (0\leqslant i\leqslant j) consists of pairs (\mathcal U,W), where \mathcal U is a bounded flag in \mathbb C^{i+1} and W is a line in U_1^\bot\oplus\mathbb C^{j-i}. (Here U_1^\bot denotes the orthogonal complement to U_1 in \mathbb C^{i+1}, so that U_1^\bot\oplus\mathbb C^{j-i} is the orthogonal complement to U_1 in \mathbb C^{j+1}.) Therefore, B_{ij} is the total space of a bundle over B_i with fibre \CP^{j-1}. This bundle is in fact the projectivisation of a sum of j line bundles, which implies that B_{ij} is a complex (i+j-1)-dimensional toric manifold.

The bundle B_{ij}\to B_i is the pullback of the bundle H_{ij}\to\mathbb CP^i along the map f\colon B_i\to\CP^i taking a bounded flag \mathcal U to its first line U_1\subset\mathbb C^{i+1}. This is described by the diagram

\displaystyle  \begin{array}{ccc} B_{ij}\; & \longrightarrow &\; H_{ij}\\ \downarrow & & \downarrow\\ B_i\; & \stackrel f\longrightarrow &\; \CP^i \end{array}.

(The bundle H_{ij}\to\mathbb CP^i, unlike B_{ij}\to B_i, is not a projectivisation of a sum of line bundles, which prevents the torus action on \CP^i from lifting to an action on the total space.)

Lemma 6.11. We have s_{i+j-1}[B_{ij}]=s_{i+j-1}[H_{ij}].

Proof. We may assume that j>1, as otherwise B_{ij}=H_{ij}=\CP^1. We have the equality H_{ij}=\CP(\xi), the projectivisation of a j-plane bundle \xi over \CP^i. We also have that the map f \colon B_i\to \CP^i has degree 1 since it is an isomorphism on the affine chart \{\mathcal U\in B_i : U_1\not\subset \Cc^i\}. Furthermore, B_{ij}=\CP(f^*\xi). The result now follows from Lemma 6.12 below.


Lemma 6.12. Let f\colon M \to N be a degree d map of 2i-dimensional almost complex manifolds, and let \xi be a complex j-plane bundle over N, j>1. Then

\displaystyle  s_{i+j-1}[\CP(f^*\xi)] = d \cdot s_{i+j-1}[\CP(\xi)].

Proof. Click here - opens a separate pdf file.


Theorem 6.13 ([Buchstaber&Ray2001]). The bordism classes of toric manifolds \{[B_{ij}],\;0\leqslant i\leqslant j\} multiplicatively generate the complex bordism ring \varOmega_*^U. 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 H_{ij} and B_{ij} are not bordant in general, although H_{0j}=B_{0j}=\CP^{j-1} and H_{1j}=B_{1j} 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 2n with a locally standard action of an n-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 >2, every complex cobordism class contains a quasitoric manifold, necessarily connected, whose stably complex structure is compatible with the action of the torus.

[edit] 7 Adams-Novikov spectral sequence

A principal motivation for [Novikov1967] was to develop a version of the Adams spectral sequence in which mod p 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.

[edit] 8 References

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