Cobordism hypothesis

Cobordism hypothesis
Name	Cobordism hypothesis
Field	Mathematics, Topology, Mathematical Physics
Introduced	1960s–2010s
Contributors	René Thom; John Milnor; Vladimir Rokhlin; Michael Atiyah; Graeme Segal; Graeme Segal; Graeme Segal
Notable results	Baez–Dolan conjecture; Lurie proof; Hopkins–Lurie work
Related	Bordism; Topological quantum field theory; Higher category theory

Cobordism hypothesis

The Cobordism hypothesis is a conjectural description connecting René Thom-style classification theorems for Thom spectrums with fully extended topological quantum field theorys and the structure of fully dualizable objects in symmetric monoidal higher category theory. Originating from insights in algebraic topology, differential topology, and mathematical physics, the hypothesis provides a bridge between early work by John Milnor and Vladimir Rokhlin on cobordism invariants and later developments by Michael Atiyah, Graeme Segal, and modern higher-categorical formulations advanced by Jacob Lurie, Baez and Dolan.

Overview

The Cobordism hypothesis asserts that fully extended topological quantum field theorys in n dimensions are classified by fully dualizable objects in symmetric monoidal n-categorys, providing a canonical equivalence between geometric bordism data and algebraic objects. This principle synthesizes ideas from René Thom's classification of bordism via homotopy-theoretic Thom spectrums, the axiomatic TQFT framework introduced by Michael Atiyah and Graeme Segal, and categorical duality notions developed in the work of John Baez, James Dolan, and Jacob Lurie. Influential institutions and conferences that fostered these ideas include the Institute for Advanced Study, the Mathematical Sciences Research Institute, and the International Congress of Mathematicians.

Historical development

Early classification of cobordism groups traces to René Thom's foundational work and the award of the Fields Medal context for Thom's contributions, later extended by computations by John Milnor and the discovery of Rokhlin's theorem by Vladimir Rokhlin linking cobordism to signature invariants. The axiomatization of topological quantum field theory by Michael Atiyah and the parallels drawn by Graeme Segal connected cobordism to quantum field considerations, while categorical perspectives grew from seminars at the Institute for Advanced Study and the Clay Mathematics Institute program on higher categories. The Baez–Dolan cobordism conjecture framed a precise higher-categorical classification, subsequently promoted and refined by Jacob Lurie in his work culminating at events such as the International Congress of Mathematicians and lectures at Harvard University and the Massachusetts Institute of Technology. Later collaborative developments involved researchers at Princeton University, University of California, Berkeley, University of Chicago, Columbia University, Yale University, and other centers.

Statement of the hypothesis

In modern form, influenced by lectures at Harvard University and notes circulated from the Institute for Advanced Study, the hypothesis states that n-dimensional fully extended topological quantum field theorys with specified tangential structure are equivalent to specifying a single fully dualizable object in a symmetric monoidal n-category. The classification reduces the geometric problem of assigning invariants to n-manifolds and their cobordisms—studied since René Thom and formalized by Michael Atiyah—to an algebraic classification problem situated in the milieu of category theory researchers such as Daniel Quillen and André Joyal. Prominent mathematicians who articulated technical formulations include Jacob Lurie, John Baez, James Dolan, and contributors from the Mathematical Sciences Research Institute programs.

Mathematical framework and definitions

The formal framework employs symmetric monoidal n-categorys, dualizability and adjointability conditions developed in seminars by Jacob Lurie and workshops at the Mathematical Sciences Research Institute. Key ingredients are the notion of bordism n-categorys with objects 0-manifolds and k-morphisms given by k-dimensional cobordisms, tangential structures such as framings or orientations studied by René Thom and John Milnor, and the definition of fully dualizable objects encoding duals, evaluation, and coevaluation maps inspired by work of Max Kelly and Daniel Quillen. Technical tools draw on homotopical algebra from Quillen model categories and stable homotopy theory as developed at Princeton University and University of Chicago, and on the theory of (\infty,n)-categories championed by Jacob Lurie and explored at Columbia University.

Examples and special cases

Classical examples include 1-dimensional TQFTs classified by finite-dimensional vector spaces and duals as presented in lectures at Oxford University and Cambridge University; 2-dimensional oriented TQFTs correspond to commutative Frobenius algebras studied by Michael Atiyah and Graeme Segal; framed n-dimensional theories correspond to fully dualizable objects in n-categorys, with explicit computations by researchers at University of California, Berkeley and Princeton University. Low-dimensional results connect to invariants examined by Edward Witten in quantum field contexts and to conformal field theory developments at Institute for Advanced Study and CERN collaborations. Connections to the Hochschild homology and Deligne conjecture contexts appeared in seminars at IHES and University of Paris.

Proofs and major results

Significant progress toward proof was announced by Jacob Lurie via a program that established the Baez–Dolan conjecture in the setting of framed structures, with further elaborations by collaborators at Harvard University and Massachusetts Institute of Technology. Complementary approaches and refinements were contributed by researchers at University of Chicago, Princeton University, Yale University, and Columbia University, with technical input from homotopy theorists influenced by Michael Hopkins and Michael Weiss. The framed case, proofs for oriented and other tangential structures, and extensions to equivariant settings are subjects of active work in research groups at MPI-MIS, ETH Zurich, and University of Cambridge.

Applications in topology and quantum field theory

The Cobordism hypothesis underpins classification results for fully extended topological quantum field theorys used in mathematical models arising from Edward Witten's work, in the study of invariants such as the Atiyah–Singer index theorem consequences, and in modern constructions of extended field theories relevant to research at CERN and theoretical groups at Perimeter Institute. It has informed developments in condensed matter contexts studied at Caltech and MIT, informed the study of anomaly cancellation conditions in collaborations involving Stanford University and Princeton University, and influenced categorical approaches to quantum invariants discussed at the International Congress of Mathematicians.

Category:Cobordism