Bordism groups

Bordism groups
Name	Bordism groups
Field	Algebraic topology
Introduced	early 20th century
Notable	René Thom, John Milnor, Sergei Novikov

Bordism groups are algebraic invariants of manifolds introduced to classify manifolds up to the relation of being boundaries of higher-dimensional manifolds. They organize equivalence classes of closed manifolds under bordism into graded abelian groups that interact with cobordism, homotopy theory, and characteristic classes. The development of bordism theory involved contributions from René Thom, John Milnor, Sergei Novikov, Michael Atiyah, and Raoul Bott and influenced results associated with the Hirzebruch–Riemann–Roch theorem, Pontryagin classes, and the Atiyah–Singer index theorem.

Definition and basic concepts

A bordism relation is defined by considering two closed n-dimensional manifolds M and N to be bordant if there exists an (n+1)-dimensional manifold W with boundary ∂W ≅ M ⨿ N. Early systematic study was initiated by René Thom in his classification work related to the Thom spectrum and ∞-cobordism ideas appearing in David Hilbert's era problems. The set of equivalence classes of closed n-manifolds under bordism forms an abelian group via disjoint union; this group is traditionally denoted by symbols tied to orientations or structures studied by Hermann Weyl and later generalized by John Milnor. Bordism connects to concrete constructions such as the mapping torus, the product manifold, and to geometric notions appearing in the work of Sophus Lie and Bernhard Riemann.

Types of bordism (oriented, unoriented, complex, spin, framed, etc.)

Several variants arise by imposing additional structure on manifolds before forming classes: unoriented bordism (no extra structure), oriented bordism (orientation data), complex bordism (stably complex structures), spin bordism (spin structures), and framed bordism (stable trivializations of the tangent bundle). Important names associated with these variants include Ludwig Schläfli for orientability discussions, Friedrich Hirzebruch for complex-oriented genera, Vladimir Rokhlin for spin-related invariants, and Andrey Kolmogorov for structural analogies. Complex bordism is organized by the MU spectrum introduced by J. Frank Adams and studied by Adams-Novikov spectral sequence contributors like Novikov and Douglas Ravenel. Framed bordism ties directly to the Pontryagin–Thom construction developed by Lev Pontryagin and elaborated by René Thom and John Milnor.

Algebraic structure and operations

Bordism groups carry graded ring structures under cartesian product and module structures over complex bordism; these operations relate to formal group laws studied by Michel Demazure and Hirzebruch. The ring MU_* is a universal complex-oriented cohomology ring connected to work by Quillen on formal group laws and Michel Lazard's classification. Operations such as the external product, cap product, and transfer maps appear in the literature of Atiyah–Hirzebruch spectral sequence contexts and are used in computations by Serre and Jean-Pierre Serre-inspired methods. Cohomology operations and the structure of the Steenrod algebra, developed by Norman Steenrod and Samuel Eilenberg, act on bordism-type theories and yield rich algebraic structure exploited in the Adams spectral sequence approach by J. F. Adams.

Computation and examples

Classical computations include the unoriented bordism ring computed by René Thom showing it is a polynomial algebra over Z/2 generated by manifolds represented by projective spaces such as real projective space examples studied by Élie Cartan. Oriented bordism groups are more intricate; early calculations by Ralph Fox and John Milnor revealed torsion phenomena linked to Stiefel–Whitney classes, Pontryagin classes, and Chern classes of complex vector bundles. Complex bordism MU_* is a polynomial ring over Z on generators in even dimensions, with computations refined by Douglas Ravenel and Haynes Miller using the Adams–Novikov spectral sequence. Low-dimensional examples include classes represented by spheres, tori, and projective spaces such as complex projective space CP^n; framed bordism of spheres connects to the stable homotopy groups of spheres investigated by Georges Péter and modern researchers like Mark Mahowald.

Bordism and homology/cohomology theories

Bordism theories are examples of generalized homology and cohomology theories in the sense of Eilenberg–Steenrod axioms generalized by Brown representability theorem; they are represented by spectra such as MU, MSO, and MSpin. Quillen's work connected complex bordism to formal group laws and algebraic geometry-style techniques, linking to the Landweber exact functor theorem used by Paul Landweber and further developed in chromatic homotopy theory by Douglas Ravenel. The relation between bordism and ordinary homology is mediated by characteristic numbers and genera such as the Todd genus, Â-genus (Atiyah–Singer context), and the Hirzebruch signature theorem due to Friedrich Hirzebruch.

Applications and connections (cobordism theorem, surgery, manifold invariants)

Bordism underpins key classification results like the h-cobordism theorem proved by Stephen Smale and the s-cobordism theorem used in surgery theory by William Browder and C. T. C. Wall. Surgery theory employs bordism and L-theory developed by Andrew Ranicki to classify high-dimensional manifolds, involving invariants studied by Bott and Atiyah. Cobordism invariants such as the Rohlin invariant, the Kervaire invariant (named after Michel Kervaire), and characteristic numbers play roles in proofs associated with the Atiyah–Singer index theorem and results by Michael Freedman in four-manifold topology. Modern interactions include connections to string theory and conformal field theory where spin and complex bordism inform anomaly cancellation and classification of topological phases studied at institutions like Institute for Advanced Study.

Category:Algebraic topology