Complex bordism
Taras Panov (Talk | contribs) |
Taras Panov (Talk | contribs) |
||
| Line 3: | Line 3: | ||
== Introduction == | == Introduction == | ||
<wikitex>; | <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 $MU$, $\Omega_*^{U} \cong \pi_n(MU)$. | + | <!--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 groups laws and genera|formal group techniques]] and [[#Adams-Novikov spectral sequence]] were outlined in the pioneering work \cite{Novikov1968}. | ||
</wikitex> | </wikitex> | ||
== Stably complex structures == | == Stably complex structures == | ||
| + | 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. | ||
| + | |||
== Definition of bordism and cobordism == | == Definition of bordism and cobordism == | ||
| Line 18: | Line 30: | ||
== Multiplicative generators == | == Multiplicative generators == | ||
| − | == Formal group laws and genera | + | == Formal group laws and genera == |
| − | == Adams-Novikov spectral sequence | + | == Adams-Novikov spectral sequence == |
== References == | == References == | ||
Revision as of 16:31, 10 March 2010
|
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 [Novikov1968].
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 the complex structures. This leads directly to considering stably complex (also known as weakly almost complex, stably almost complex or quasicomplex) manifolds.
3 Definition of bordism and cobordism
4 Geometric cobordisms
5 Structure results
6 Multiplicative generators
7 Formal group laws and genera
8 Adams-Novikov spectral sequence
9 References
|
This page has not been refereed. The information given here might be incomplete or provisional. |
