Cobordism theory

Cobordism theory
Name	Cobordism theory
Field	Algebraic topology
Introduced	1950s
Introduced by	René Thom

Cobordism theory Cobordism theory studies equivalence relations among compact manifolds via cobordism and provides algebraic invariants that connect manifold topology to stable homotopy, characteristic classes, and global analysis. It unites work of René Thom, John Milnor, and Samuel Eilenberg with techniques from Henri Poincaré, Solomon Lefschetz, and Georges de Rham, and it interfaces with machinery developed by Daniel Quillen, Michael Atiyah, Friedrich Hirzebruch, and Edward Witten. The subject has deep ties to the Pontryagin–Thom construction, Adams spectral sequence, and index theorems such as the Atiyah–Singer theorem.

Introduction

Cobordism originates in the classification of manifolds by considering when two manifolds bound a manifold of one higher dimension; this idea was formalized by René Thom in his work on the classification of differentiable manifolds and the structure of singularities. Key developments involved interactions with John Milnor's discoveries about exotic spheres, Hirzebruch's work on characteristic classes and genera, and Quillen's use of complex-oriented cohomology theories. Influential institutions and seminars at the Institut des Hautes Études Scientifiques, Princeton University, and the University of Cambridge fostered early progress, while later connections with mathematical physics emerged via Edward Witten and string theory research groups.

Historical development

Origins trace to Henri Poincaré's studies of manifolds and Euler characteristic, followed by Solomon Lefschetz's fixed point investigations and Hassler Whitney's embedding results. René Thom systematized cobordism in the 1950s, proving the fundamental theorem that framed cobordism relates to stable homotopy groups; contemporaries included John Milnor, Raoul Bott, and Georges de Rham. In the 1960s Michael Atiyah and Friedrich Hirzebruch connected cobordism with K-theory, while Daniel Quillen in the 1970s used formal group laws to recast complex cobordism as a universal complex-oriented cohomology theory. Later work by Sir Michael Atiyah, Ilya Piatetski-Shapiro, and Isadore Singer connected cobordism with index theory and global analysis, and Edward Witten's insights linked genera to quantum field theory and string dualities studied at CERN and the Institute for Advanced Study.

Definitions and basic concepts

A cobordism between two closed n-dimensional manifolds M and N is an (n+1)-dimensional manifold W with boundary equal to the disjoint union of M and N. Fundamental contributors to formal definitions include René Thom, John Milnor, and Marston Morse; foundational techniques draw on Whitney embedding theorems and Pontryagin classes developed by Lev Pontryagin. Variants such as oriented cobordism, unoriented cobordism, complex cobordism, and framed cobordism are distinguished by additional structure inspired by Hermann Weyl, David Hilbert, and Shiing-Shen Chern. The Pontryagin–Thom construction, pioneered by Lev Pontryagin and René Thom, identifies cobordism classes with stable homotopy classes of maps into Thom spaces, linking to spectra introduced by J. H. C. Whitehead and stable homotopy theory advanced by J. F. Adams.

Cobordism groups and rings

Cobordism classes form graded abelian groups and, with Cartesian product, graded rings; the unoriented cobordism ring was computed by René Thom, while complex cobordism MU_* was elucidated by Milnor and Novikov. Daniel Quillen identified the complex cobordism ring with the Lazard ring of formal group laws studied by Jean-Pierre Serre and Michel Demazure, enabling algebraic geometry techniques from Alexander Grothendieck and Pierre Deligne to be applied. Computations use spectral sequences such as the Adams spectral sequence developed by J. F. Adams and the Adams–Novikov spectral sequence introduced by Sergei Novikov, with input from homological algebra traditions of Samuel Eilenberg and Saunders Mac Lane.

Orientations, characteristic numbers, and genera

Orientations in cobordism—unoriented, oriented, Spin, Spinc, complex, and string—reflect structures studied by Élie Cartan and Shiing-Shen Chern and relate to characteristic classes like Stiefel–Whitney, Chern, Pontryagin, and Todd classes. Hirzebruch defined genera such as the Â-genus and Todd genus and proved signature theorems linking to index theory of Atiyah and Singer. Quillen and Novikov connected genera to formal group laws and complex-oriented cohomology, while Witten interpreted elliptic genera using ideas from Michael Freedman, Anton Kapustin, and Cumrun Vafa within quantum field theory and manifold invariants used in string theory.

Relations to homotopy theory and spectra

The Pontryagin–Thom isomorphism identifies cobordism groups with homotopy groups of Thom spectra such as MO, MSO, MU, and MString; these spectra were formalized in the work of J. H. C. Whitehead and stable homotopy pioneers including J. F. Adams and Mark Mahowald. Quillen's theorem situates MU as a universal object in the category studied by Vladimir Voevodsky and Jacob Lurie, enabling connections to motivic homotopy theory and higher category theory developed at the Clay Mathematics Institute and the Institut des Hautes Études Scientifiques. Computational tools rely on the Adams–Novikov spectral sequence and the machinery of Elmendorf–Krishna–May and others working on structured ring spectra and operads.

Applications and examples

Cobordism classifications explain phenomena discovered by John Milnor such as exotic spheres and inform constructions in differential topology studied at Princeton and ETH Zurich. The signature theorem of Hirzebruch and Atiyah–Singer links cobordism invariants to analytic indices used by Isadore Singer, Daniel Freed, and Pierre Deligne in geometry and mathematical physics. Applications appear in string theory and conformal field theory research at CERN and the Institute for Advanced Study, and in low-dimensional topology contributions by William Thurston, Clifford Taubes, and Simon Donaldson. Examples include computations of unoriented cobordism by Thom, complex cobordism rings by Milnor and Novikov, and relationships to K-theory explored by Michael Atiyah and Graeme Segal.

Generalizations and variants

Variants include equivariant cobordism investigated by Glen Bredon and Tom Dieck, Spin and Spinc cobordism related to Kazhdan and Lusztig theory, and bordism theories in algebraic geometry such as algebraic cobordism developed by Levine and Morel at the Fields Institute. Higher-categorical and motivic generalizations involve the work of Jacob Lurie, Vladimir Voevodsky, and Peter May, while connections to quantum field theory and topological quantum field theories were developed by Edward Witten, Graeme Segal, and Daniel Freed. Recent research by Dustin Clausen, Akhil Mathew, and Peter Scholze explores chromatic phenomena and descent techniques linking cobordism to modern arithmetic geometry and higher algebra.

Category:Algebraic topology