Complex bordism

Revision as of 16:40, 10 March 2010

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 ${{Authors|Taras Panov}} == Introduction == <wikitex>; <!--A unitary struture $\bar \nu$ on a manifold $M$ is a choice of weak complex structure on the stable normal bundle of $M$. By the [[B-Bordism#Pontrjagin-Thom isomorphism|Pontrjagin-Thom isomorphism]] the bordism groups of closed unitary manifolds $(M, \bar \nu)$ are isomorphic to the homotopy groups of the Thom spectrum $MU$, $\Omega_*^{U} \cong \pi_n(MU)$.--> ''Complex bordisms'' (also known as ''unitary bordisms'') is the [[Bordism and cobordism|bordism theory]] of [[#Stably complex structures|stably complex manifolds]]. It is one of the most important theory of bordisms with additional structure, or [[B-Bordism|B-bordisms]]. The theory of complex bordisms is much richer than its [[Bordism and cobordism#Unoriented bordism|unoriented]] analogue, and at the same time is not as complicated as [[Oriented bordism|oriented bordisms]] or other bordisms with additional structure ([[B-Bordism|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 laws and genera|formal group techniques]] and [[#Adams-Novikov spectral sequence|Adams-Novikov spectral sequence]] were outlined in the pioneering work \cite{Novikov1968}. </wikitex> == Stably complex structures == <wikitex>; A direct attempt to define the [[Bordism and cobordism#Bordism relation|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 the complex structures. 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 $$ c_{\mathcal T}\colon {\mathcal T}\!M\oplus \underline{\mathbb R}^k\to \xi $$ 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. determining the same stably complex structures (see details in Chapters II and VII of \cite{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 in the normal bundle $\nu(M)$ of an embedding $M\hookrightarrow\mathbb R^N$. A 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 to 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 $k=0$). {{beginthm|Example}} Let $M=\mathbb{C}P^1$. The standard complex structure on $M$ is equivalent to a stably complex structure determined by the isomorphism $$ {\mathcal T}(\mathbb{C}P^1)\oplus\underline{\mathbb R}^2\stackrel{\cong}{\longrightarrow} \overline{\eta}\oplus \overline{\eta} $$ where $\eta$ is the Hopf line bundle. On the other hand, the isomorphism $$ {\mathcal T}(\mathbb{C}P^1)\oplus\underline{\mathbb R}^2\stackrel{\cong}{\longrightarrow} \eta\oplus \overline{\eta}\cong \underline{\mathbb C}^2 $$ determines a trivial stably complex structure on $\mathbb C P^1$. {{endthm}} </wikitex> == Definition of bordism and cobordism == == Geometric cobordisms == == Structure results == == Multiplicative generators == == Formal group laws and genera == == Adams-Novikov spectral sequence == == References == {{#RefList:}} [[Category:Manifolds]] {{Stub}}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 the complex structures. 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. 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 $M$ is determined by a choice of a complex bundle structure in the normal bundle $\nu(M)$ of an embedding $M\hookrightarrow\mathbb R^N$. A 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 to 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 $M=\mathbb{C}P^1$. The standard complex structure on $M$ is equivalent to a stably complex structure determined by the isomorphism

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

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

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

determines a trivial stably complex structure on $\mathbb C P^1$.

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 $[M,c_{\mathcal T}]$ of stably complex manifolds is an Abelian group with respect to the disjoint union. This group is called the group of $n$-dimensional complex bordisms and denoted $\varOmega^U_n$. A zero 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

$$\displaystyle {\mathcal T}\!M\oplus\underline{\mathbb R}^k\oplus\underline{\mathbb R}^2\stackrel{c_{\mathcal T}\oplus e}{\relbar\joinrel\hspace{-1pt}\relbar\joinrel\hspace{-1pt}\longrightarrow}\xi\oplus\underline{\mathbb C}$$

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 groups of complex bordisms and cobordisms of a space $X$ are defined 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 $EU(k)\to BU(k)$. 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.

4 Geometric cobordisms

5 Structure results

6 Multiplicative generators

7 Formal group laws and genera

8 Adams-Novikov spectral sequence

9 References

• [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
• [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010