complex cobordism

complex cobordism
Name	Complex cobordism
Field	Algebraic topology
Introduced	1960s

complex cobordism

Complex cobordism is a generalized cohomology theory arising from the study of manifolds equipped with stable complex structures. It organizes information about manifolds, vector bundles, and characteristic classes into an algebraic framework that connects to the work of René Thom, John Milnor, Sergei Novikov, Dale Husemoller, and J. F. Adams. The theory underpins deep interactions among homotopy theory, algebraic geometry, number theory, and the classification problems studied by the Institute for Advanced Study and major topology groups at institutions such as Princeton University and Massachusetts Institute of Technology.

Introduction

Complex cobordism emerged from attempts to classify differentiable manifolds using cobordism relations studied by René Thom and further developed by John Milnor and Donald C. Spencer. The subject crystallized when complex structures on stable normal bundles were incorporated, leading to the definition of the complex cobordism ring and the associated spectrum MU studied intensively by Michael Boardman, J. Peter May, J. F. Adams, and Douglas C. Ravenel. Major milestones include the discovery of connections to the Adams–Novikov spectral sequence and to formal group laws considered by Michel Lazard and Jean-Pierre Serre.

Definitions and basic properties

One defines complex cobordism via equivalence classes of closed, smooth manifolds with stable complex structures on their tangent bundles, following foundational ideas of René Thom and constructions used by John Milnor. Two such manifolds are cobordant if there exists a compact manifold with boundary whose induced stable complex structure restricts to the given structures, a notion refined in the work of Milnor and C. T. C. Wall. The resulting graded abelian group MU_* admits a ring structure via cartesian product, reflecting multiplicative structures explored by J. F. Adams and J. P. May. Orientations in complex cobordism generalize classical orientations studied by Hermann Weyl and are compatible with pushforward maps analogous to those considered at Harvard University and Cambridge University topology seminars.

Complex cobordism ring and formal group law

The complex cobordism ring MU_* is a polynomial algebra on generators discovered by Milnor and analyzed by Donald Quillen, who proved that MU_* is universal for complex-oriented theories. Quillen identified a canonical formal group law on MU^*, linking to work of Michel Lazard on the universal formal group and to Pierre Deligne's perspectives in algebraic geometry. This universal formal group law makes MU central to the study of elliptic genera explored by Witten, Edward Witten, and Gerd Faltings. Connections to the Modular group and Atkin–Lehner theory arise through genera computations influenced by Serge Lang and researchers at Institute for Advanced Study.

Cobordism spectra and MU-theory

The representing spectrum MU (sometimes denoted M U) encodes complex cobordism as a multiplicative generalized cohomology theory, a perspective developed by J. P. May and H. R. Miller. MU is a commutative ring spectrum whose homotopy groups equal MU_*, linking to spectra studied at Princeton University and University of Chicago topology groups. The study of MU-modules and MU-algebras interacts with structured ring spectra initiated by J. F. Adams and expanded in the context of Brave New Algebra by researchers including Michael Hopkins and Mark Behrens. The multiplicative structure supports the Adams–Novikov spectral sequence, a computational tool refined by Sergei Novikov and J. F. Adams.

Calculations and structure theorems

Explicit calculations in MU_* and MU^* have been advanced by John Milnor, Donald Quillen, and Douglas C. Ravenel, yielding structure theorems describing MU_* as a polynomial ring over the integers localized at primes analyzed by Jean-Pierre Serre and J. P. Serre. The Landweber exact functor theorem, developed by Peter Landweber and applied by Mark Hovey and Neil Strickland, provides criteria for constructing homology theories from MU by flatness conditions similar to arguments in Algebraic Geometry seminars at University of California, Berkeley. The chromatic filtration and nilpotence results due to Devinatz, Hopkins, and Smith place MU at the center of the chromatic picture elucidated in the Ravenel conjectures.

Relations to other cohomology theories

Complex cobordism maps naturally to ordinary cohomology, K-theory developed by Atiyah and Max Karoubi, and to elliptic cohomology studied by Landweber, Hopkins, and Ando. The universality of MU for complex-oriented theories implies that many genera, such as the Todd genus of Friedrich Hirzebruch and the Witten genus explored by Edward Witten, factor through MU. Connections to Morava K-theory and Brown–Peterson cohomology refine computations at primes, as investigated by Paul Balmer and Michael Hopkins in collaboration with topology groups at Northwestern University and University of Chicago.

Applications and examples

Applications of complex cobordism span stable homotopy computations pioneered at Princeton University and classification problems for manifolds addressed by John Milnor and C. T. C. Wall. Explicit examples include calculations of cobordism classes for complex projective spaces studied by Blaschke-style enumerations and genera computations for complex surfaces analyzed by Freeman Dyson and Kunihiko Kodaira techniques. The theory informs modern work on topological modular forms pursued by Hopkins, Haynes Miller, and Mark Behrens, as well as interactions with mathematical physics influenced by Edward Witten and research at institutions such as Institute for Advanced Study and Princeton University.

Category:Algebraic topology