Bott–Tu

Bott–Tu
Name	Bott–Tu
Subject	Mathematics
Authors	Raoul Bott; Loring W. Tu
Fields	Algebraic topology; Differential geometry; Symplectic geometry
Notable works	"Differential Forms in Algebraic Topology"

Bott–Tu is the colloquial designation for the ideas and results centered on the collaboration and synthesis by Raoul Bott and Loring W. Tu, particularly exemplified in the text "Differential Forms in Algebraic Topology". The body of work connects constructions in Morse theory, de Rham cohomology, characteristic classes, and equivariant cohomology, providing computational tools and conceptual bridges between Henri Poincaré-style topology and modern geometric methods. Influential in the development of interactions among Atiyah–Bott fixed-point theorem, Chern–Weil theory, and the study of fiber bundles, it has shaped research in Symplectic geometry, Algebraic geometry, and Mathematical physics.

History and development

The origin traces to the work of Raoul Bott on periodicity and index phenomena and the pedagogical synthesis by Loring W. Tu in the late 20th century. Influences include earlier contributions by Élie Cartan on differential forms, Shiing-Shen Chern on characteristic classes, and the structural insights of Andrey Kolmogorov-era topology. The development entwines with milestones such as the Atiyah–Singer index theorem, the Lefschetz fixed-point theorem, and the advent of equivariant localization techniques promoted by Mikita Brodsky and later popularized by Nigel Hitchin and Edward Witten. Workshops at institutions like Institute for Advanced Study and conferences at International Congress of Mathematicians helped disseminate these ideas.

Mathematical foundations

Foundational tools draw from classical constructions: de Rham cohomology as formulated by Georges de Rham, singular cohomology via Samuel Eilenberg and Norman Steenrod, and differential topology developed by John Milnor and Marston Morse. The theory uses vector bundles classified by maps to Grassmannian manifolds and encoded by Chern classes and Pontryagin classes, with calculations leveraging spectral sequences introduced by Jean Leray and refined by Jean-Pierre Serre. Critical analytical inputs include elliptic operator theory stemming from Atiyah and Isadore Singer and heat kernel techniques related to Daniel Quillen and Patodi-type formulas.

Bott–Tu theorem and main results

The central compilation provides concrete theorems linking differential form representatives to topological invariants, offering versions of the de Rham isomorphism compatible with fiber integration and transfer maps originally framed by Edwin Spanier and Hermann Weyl. It formalizes the computation of characteristic classes through differential forms via the Chern–Weil homomorphism and articulates functorial properties in the presence of group actions, resonating with the Atiyah–Bott fixed-point theorem and the Berline–Vergne localization formula. Key results include explicit de Rham representatives for Thom classes and Gysin sequences comparable to those used in the work of Raoul Bott with Michael Atiyah and in later treatments by Bott and Tu that enable calculations for bundles over complex projective space and flag varietys.

Applications and examples

Applications span computations in Characteristic class theory for tangent bundles of manifolds like real projective space, complex projective space, and Grassmannians, as well as evaluations of integrals appearing in enumerative geometry and moduli space analyses used by Michael Atiyah, Nigel Hitchin, and Edward Witten. Examples include explicit form representatives for the Euler class on oriented sphere bundles, constructions for the Thom isomorphism underpinning Poincaré duality calculations on compact manifolds such as K3 surfaces and Calabi–Yau manifolds, and equivariant computations relevant to fixed-point formulas in the studies of Hamiltonian group actions investigated by Frances Kirwan and Victor Guillemin.

Proof sketch and methods

Proof techniques combine homological algebra via spectral sequence manipulations, analytic methods from elliptic operator theory as in the Atiyah–Singer index theorem, and differential form constructions inspired by Élie Cartan’s moving frame. Key steps use partition of unity arguments from the work of H. Whitney to construct global forms, transgression and fiber integration as in Dennis Sullivan’s approaches to characteristic classes, and equivariant extension constructions comparable to those used in Berline–Vergne and Witten’s localization proofs. Computations often reduce to model calculations on tubular neighborhoods and standard bundles such as the universal bundle over a Grassmannian.

Generalizations and related theories

Generalizations extend to equivariant de Rham models for noncompact groups studied by Victor Guillemin and Shlomo Sternberg, to derived and homotopical refinements in derived algebraic geometry influenced by Jacob Lurie and Bertrand Toën, and to interactions with index theory and noncommutative geometry advanced by Alain Connes. Related frameworks include Chern–Simons theory in mathematical physics as developed by Edward Witten and the role of characteristic forms in mirror symmetry contexts examined by Maxim Kontsevich and Kontsevich–Soibelman-style wall-crossing. Ongoing research connects these methods to computations in Floer homology and Gromov–Witten invariants pursued by researchers such as Paul Seidel and Dusa McDuff.

Category:Mathematical theorems