Cobordism (mathematics)

Cobordism (mathematics)
Name	Cobordism
Field	Topology
Introduced	1950s
Notable figures	René Thom, John Milnor, Michael Atiyah, Raoul Bott

Cobordism (mathematics) is a relation between manifolds introduced in the mid‑20th century that organizes manifolds into equivalence classes via bordisms and underpins generalized homology theories. It played a central role in the work of René Thom, influenced by developments in Élie Cartan's differential topology, and led to interactions with John Milnor, Michael Atiyah, and Raoul Bott through connections to index theory and homotopy theory.

Definition and basic concepts

A cobordism is a compact smooth manifold W whose boundary decomposes into the disjoint union of two closed smooth manifolds M and N, written ∂W = M ⊔ N, a notion formalized by René Thom and used by Hassler Whitney and Leray in manifold theory. Two n‑dimensional closed manifolds M and N are cobordant if there exists such a W; this equivalence relation yields cobordism classes studied by John Milnor, Lev Pontryagin, and Andrey Kolmogorov. Variants include oriented cobordism (related to Élie Cartan's orientation ideas), framed cobordism (used by Ralph Fox and Michel Kervaire), and unoriented, complex, spin, and string cobordism associated with structure groups like SO(n), O(n), U(n), Spin(n), and String(n). Cobordism connects to transversality results of Stephen Smale and embedding theorems of Whitney and to surgery techniques developed by C. T. C. Wall and William Browder.

Examples and low-dimensional cases

In dimension 0, cobordism classes correspond to parity counts of points, a case analyzed by Hermann Weyl and appearing in considerations by Emmy Noether. In dimension 1, circles and intervals lead to trivial cobordism due to classification results influenced by Riemann's and Bernhard Riemann's legacy in topology, while classic computations by René Thom and John Milnor show generators like the real projective plane RP^2 in unoriented 2‑dimensional cobordism. Low‑dimensional classifications use techniques from the works of William Thurston, Dennis Sullivan, and Michael Freedman to handle 3‑ and 4‑manifolds, with deep links to phenomena studied by Simon Donaldson and Edward Witten in four dimensions. Framed cobordism in low dimensions underlies the Pontryagin construction attributed to Lev Pontryagin and was central to examples in Kervaire–Milnor investigations.

Cobordism classes and groups

Cobordism classes form abelian groups under disjoint union, first organized by René Thom into graded rings and groups; unoriented cobordism gives the graded ring Ω_*^O, oriented cobordism Ω_*^{SO}, complex cobordism MU_*, and spin cobordism MSO_*, MSpin_*. The work of Michael Atiyah and Friedrich Hirzebruch related these groups to characteristic class formulas studied by Shiing-Shen Chern and Hirzebruch–Riemann–Roch contexts. Thom's classification theorem identifies cobordism groups with homotopy groups of Thom spaces, a viewpoint elaborated by J. F. Adams, Serre, and Frank Adams. Computations of torsion and free parts rely on algebraic topology techniques introduced by Henri Cartan and refined by Jean-Pierre Serre.

Thom spectrum and homology theories

The Thom spectrum construction produces spectra such as MO, MSO, MU, and MSpin which represent cobordism homology theories in the sense of Eilenberg–Steenrod axioms developed by Samuel Eilenberg and Norman Steenrod. The identification of MU as representing complex cobordism connects to the Landweber–Novikov operations and to formal group law work of Michael Hazewinkel and Daniel Quillen, the latter proving that MU_* is the universal complex oriented cohomology theory and linking to Quillen's algebraic geometry perspectives. Spectral sequence tools such as the Adams spectral sequence, pioneered by J. F. Adams and Hans Freudenthal, are used to compute homotopy groups of Thom spectra and to relate cobordism to stable homotopy groups of spheres studied by Douglas Ravenel and Mark Mahowald.

Ring and module structures; orientations and characteristic numbers

Cobordism rings carry multiplicative structures from Cartesian product of manifolds, a structure analyzed by Hirzebruch in terms of characteristic numbers like Stiefel–Whitney classes of Eduard Stiefel and Chern numbers of Shiing-Shen Chern. Orientations in cobordism theories—complex orientations in MU, spin orientations in MSpin—are formalized using work by Jean-Pierre Serre and Michael Atiyah and yield genera such as the Todd genus of Hirzebruch, the Â‑genus related to Atiyah–Singer index theorem by Isadore Singer, and elliptic genera connected to Edward Witten and Boris Dubrovin. Characteristic numbers give homomorphisms from cobordism rings to integers or other rings, central to classification results of René Thom and refinements by Milnor and Kervaire.

Calculations and classifications

Classical calculations by René Thom and Frank Adams determined unoriented and some oriented cobordism rings, with complex cobordism MU_* computed using formal group law techniques by Daniel Quillen. Work of J. F. Adams, John Milnor, Michel Kervaire, and Mark Mahowald produced results on torsion and exotic spheres via framed cobordism and the J‑homomorphism studied by Bott and Milnor. Advanced classification in high dimensions employs surgery theory of C. T. C. Wall, assembly maps appearing in the Novikov conjecture associated with Sergei Novikov, and computational input from the Adams–Novikov spectral sequence, a tool refined by Douglas Ravenel and Haynes Miller.

Applications and connections to other areas

Cobordism interfaces with index theory through the Atiyah–Singer index theorem of Michael Atiyah and Isadore Singer, with quantum field theory via anomalies studied by Edward Witten and Alain Connes, with string theory through the role of String structures and the MSpin theory in Edward Witten's work, and with algebraic geometry through complex genera and Grothendieck's Riemann–Roch traditions. It informs questions about exotic smooth structures in the work of Simon Donaldson and Michael Freedman, and it appears in modern homotopy theory in chromatic homotopy programs led by Douglas Ravenel and Jacob Lurie. Cobordism also underlies invariants used in low‑dimensional topology by William Thurston, Ciprian Manolescu, and Peter Kronheimer.

Category:Cobordism