cobordism

cobordism
Name	cobordism
Field	Topology
Introduced	1950s
Notable	René Thom, John Milnor, Michel Kervaire, Raoul Bott, Friedrich Hirzebruch

cobordism

Cobordism is a relation between closed manifolds studied in topology, introduced in the mid-20th century, that organizes manifolds into classes used to define algebraic invariants and compute classifications. It influenced major developments associated with figures and institutions such as René Thom, John Milnor, Michel Kervaire, Raoul Bott, and Friedrich Hirzebruch, and tools developed at places like Princeton University, École Normale Supérieure, and the Institute for Advanced Study.

Definition and Basic Concepts

A basic formulation identifies two closed n-dimensional manifolds as equivalent when there exists an (n+1)-dimensional manifold whose boundary decomposes into the disjoint union of the two, a notion formalized by René Thom and studied by John Milnor, Michel Kervaire, Shiing-Shen Chern, Raoul Bott, and Friedrich Hirzebruch. Foundational constructions and examples were developed in seminars at Princeton, Cambridge, and Paris, with contributions from Serge Lang, Alexander Grothendieck, Jean-Pierre Serre, and Henri Cartan shaping algebraic formulations. Later conceptual refinements involved tools from homotopy theory propagated by Daniel Quillen, J. Peter May, and Michael Hopkins, while computational frameworks were influenced by William Browder, James Milgram, Clarence Wilkerson, and Mark Mahowald.

Examples and First Constructions

Early examples used spheres and projective spaces studied in work by Élie Cartan, Évariste Galois (historical context for field ideas), Bernhard Riemann (geometric foundations), and Bernhard Bolzano’s successors; explicit low-dimensional constructions involved the circle and surfaces examined by Felix Klein, Henri Poincaré, and Emmy Noether. Thom’s work linked singularity theory developed by Vladimir Arnold, John Nash, and Stephen Smale to explicit cobordisms, while Milnor’s exotic spheres connected to research at Princeton, Harvard, and the University of California, Berkeley. Other early constructions referenced the contributions of Michael Atiyah, Isadore Singer, Nikolai Lobachevsky (geometric precedent), and Sophus Lie in symmetry considerations.

Cobordism Groups and Rings

Cobordism classes form abelian groups and graded rings; the algebraic structure was formalized by René Thom and expanded by Michel Kervaire, John Milnor, Friedrich Hirzebruch, and Raoul Bott. Work at institutions such as Columbia University, Stanford University, the University of Chicago, and the Institut des Hautes Études Scientifiques influenced development of formal ring structures, while algebraic inputs from Emmy Noether, David Hilbert, and Emmy Noether’s collaborators informed axiomatic methods. Connections to characteristic classes were illuminated through the work of Chern, Pontryagin, and Stiefel, and algebraic topology tools advanced by J. H. C. Whitehead, E. H. Spanier, and Norman Steenrod clarified the relationship between cobordism rings and cohomology theories. The development of formal group laws in the context of complex cobordism drew on ideas from Michel Lazard and Joseph Leech, with later expansions by Douglas Ravenel, Haynes Miller, and Mark Hovey.

Oriented, Unoriented, and Other Variants

Variants including oriented, unoriented, spin, spinc, complex, and framed versions were developed by leaders in topology such as John Milnor, Michel Kervaire, Michael Atiyah, Isadore Singer, Dennis Sullivan, and Edward Witten. Spin cobordism connected to work by Raoul Bott, Daniel Quillen, and Friedrich Hirzebruch, while spinc and complex cobordism were integrated into broader studies by Robert Stong, Goro Shimura, and Graeme Segal. Other refinements involving equivariant structures and bordism with singularities were pursued at research centers like the Max Planck Institute, the Clay Mathematics Institute, and the Mathematical Sciences Research Institute, with contributions from Graeme Segal, Maxim Kontsevich, and Vladimir Voevodsky.

Algebraic and Homotopy-Theoretic Approaches

Algebraic reformulations via spectra and stable homotopy theory were pioneered by J. Peter May, Daniel Quillen, Frank Adams, and John Frank Adams, with complex cobordism and the Landweber–Novikov algebra receiving attention from Lev Landweber, Sergei Novikov, and Douglas Ravenel. The Thom spectrum construction credited to René Thom was recast using spectra studied by Michael Boardman, Haynes Miller, and Mark Hovey; chromatic homotopy theory linked to work by Mike Hopkins, Jacob Lurie, and Paul Goerss provided modern computational frameworks. Operadic and higher-categorical perspectives involving Jacob Lurie, André Joyal, and James Dolan further related cobordism to topological quantum field theories studied by Edward Witten, Michael Atiyah, and Kevin Walker.

Applications and Connections

Cobordism has applications in singularity theory via Vladimir Arnold and John Mather, index theory through Michael Atiyah and Isadore Singer, and mathematical physics influenced by Edward Witten and Greg Moore. It also impacts enumerative geometry connected to Maxim Kontsevich, Gromov–Witten theory advanced by Mikhail Gromov and Rahul Pandharipande, and string topology developed by Dennis Sullivan and Chas–Sullivan collaborators. Intersections with algebraic geometry involve Alexander Grothendieck, Pierre Deligne, and Maxim Kontsevich; relations to surgery theory trace back to C. T. C. Wall, Andrew Ranicki, and William Browder. Computational influences came from the collaborative environments of the Institute for Advanced Study, Cambridge University, and the European Mathematical Society.

Computations and Classification Results

Key classification results include Thom’s computation of unoriented cobordism, Milnor and Kervaire’s analyses of exotic spheres, and Hirzebruch’s work on genera, with computational advances by Douglas Ravenel, Haynes Miller, and Mark Mahowald. Explicit calculations of complex cobordism rings involved further work by Lev Landweber, Sergei Novikov, and Daniel Quillen; modern computational techniques deployed by Mike Hopkins, Jacob Lurie, and Haynes Miller yield deeper structural results. Important classification theorems intersect with surgery theory from Andrew Ranicki and C. T. C. Wall, and with index-theoretic invariants developed by Michael Atiyah and Isadore Singer, while computational databases and collaborative projects at the Clay Mathematics Institute and the American Mathematical Society continue to support ongoing research.

Category:Topology