Cobordism (topology)

Cobordism (topology)
Name	Cobordism
Field	Topology
Introduced	1950s
Key people	René Thom, John Milnor, Vladimir Rokhlin, Michael Atiyah, Raoul Bott

Cobordism (topology) Cobordism is a relation between manifolds that organizes closed manifolds into equivalence classes via bounding manifolds; it links ideas from differential topology, algebraic topology, and geometric topology. Originating in work of René Thom and developed by John Milnor, Vladimir Rokhlin, and Michael Atiyah, cobordism plays a central role in the classification of manifolds, in surgery theory associated to C. T. C. Wall, and in the construction of generalized homology theories via Thom spectra.

Definition and basic concepts

Two closed n-dimensional manifolds M and N are cobordant if there exists an (n+1)-dimensional compact manifold W with boundary ∂W = M ⨿ N. The condition is traditionally studied in the smooth category introduced by René Thom and in the piecewise-linear category studied by John Milnor and Vladimir Rokhlin; variants include topological manifolds considered by Michael Freedman and Simon Donaldson. Cobordism is an equivalence relation producing sets of equivalence classes studied by algebraists such as Daniel Quillen and homotopy theorists such as J. Peter May. Fundamental constructions use transversality results of Stephen Smale and immersion theory of Morris Hirsch; foundational compactness and embedding results invoke work of Hassler Whitney and Shing-Tung Yau.

Examples and first properties

Basic examples include the null-cobordant n-sphere S^n as the boundary of the (n+1)-ball B^{n+1}; classical constructions by Henri Poincaré and Emmy Noether motivate invariants distinguishing nontrivial cobordism classes such as the signature introduced by Hassler Whitney and later algebraic formulations by John Tate. Products of manifolds produce multiplicative structures seen in the work of Raoul Bott and Michael Atiyah, while connected sum operations studied by William Browder and Frank Quinn affect cobordism classes. Low-dimensional classification results draw on the work of Vladimir Rokhlin on 3- and 4-manifolds and William Thurston's geometrization ideas.

Cobordism groups and ring structures

Cobordism classes form graded abelian groups Ω_n under disjoint union, with ring structures via Cartesian product yielding a graded ring Ω_*. René Thom identified Ω_* with homotopy groups of Thom spaces, a perspective extended by Daniel Quillen and J. F. Adams. Complex cobordism MU_* introduced by Raoul Bott and Douglas Ray provides a universality property used by Michael Hopkins and Haynes Miller in formal group law applications, while unoriented cobordism MO_* links to calculations by John Milnor and Sergei Novikov. Computations of Ω_* use spectral sequences developed by Jean-Pierre Serre and Frank Adams and relate to invariants studied by Isadore Singer and Edward Witten.

Orientations and variations (orientable, framed, complex, spin)

Variants of cobordism are defined by additional structure: oriented cobordism Ω^{SO}_* studied by René Thom and C. T. C. Wall; complex cobordism MU_* tied to the work of Friedrich Hirzebruch and Hirzebruch–Riemann–Roch contexts; spin cobordism MSpin_* related to Rokhlin's theorem and applications by Michael Atiyah and Isadore Singer in index theory; framed cobordism π_*^S connected to stable homotopy groups of spheres explored by J. H. C. Whitehead and Sergei Novikov. Each orientation restricts allowable bounding manifolds and interacts with characteristic classes such as Stiefel–Whitney classes introduced by Eduard Stiefel and Pontryagin classes studied by Lev Pontryagin; these classes were systematized by Friedrich Hirzebruch and Jean-Pierre Serre.

Thom spectra and homotopy-theoretic formulation

Thom's construction yields Thom spectra (MO, MSO, MU, MSpin) whose homotopy groups recover cobordism rings; this was pivotal in stable homotopy theory developed by J. H. C. Whitehead, J. Peter May, and Daniel Quillen. Quillen's work on formal group laws and complex oriented cohomology connects MU to algebraic geometry themes in Grothendieck's school and to the chromatic perspective of Douglas Ravenel, Jack Morava, and Michael Hopkins. Adams spectral sequence calculations by J. Frank Adams and Mark Mahowald provide computations of stable stems linked to framed cobordism, while the Hopkins–Miller theorem and work by Jacob Lurie integrate cobordism spectra into modern higher category and derived algebraic geometry frameworks championed by Alexander Grothendieck and William G. Dwyer.

Applications and relations to surgery and classification

Cobordism underpins surgery theory of C. T. C. Wall and Browder–Novikov–Sullivan–Wall approaches to manifold classification; it appears in classification results by William Browder and Andrew Casson, and in topological 4-manifold results by Michael Freedman and Simon Donaldson. Cobordism invariants feed into index theorems of Michael Atiyah and Isadore Singer, and into quantum field theoretic applications developed by Edward Witten and Graeme Segal. Relations to knot theory and low-dimensional topology draw on William Thurston and Vaughan Jones, while interactions with symplectic topology connect to Yasha Eliashberg and Maksim Kontsevich.

Historical development and key results

René Thom's foundational classification and the Thom transversality theorem initiated the subject in the 1950s, leading to the formulation of cobordism rings and computations by John Milnor, Vladimir Rokhlin, and Sergei Novikov. The identification of complex cobordism MU_* as universal complex-oriented theory by Daniel Quillen, and the subsequent chromatic program of Douglas Ravenel and Michael Hopkins, marked major advances. Key theorems include Rokhlin's theorem on signatures, the Hirzebruch–Riemann–Roch framework by Friedrich Hirzebruch, and the Atiyah–Singer index theorem linking analysis and topology by Michael Atiyah and Isadore Singer. Contemporary developments by Jacob Lurie, Mikhail Kapranov, and Graeme Segal place cobordism within higher category theory and quantum field theory programs influenced by Alexander Grothendieck and Vladimir Drinfeld.

Category:Algebraic topology