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In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott (1957, 1959), which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres.
Raoul Bott (September 24, 1923 – December 20, 2005) [1] was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem , the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem .
Morse–Bott functions are useful because generic Morse functions are difficult to work with; the functions one can visualize, and with which one can easily calculate, typically have symmetries. They often lead to positive-dimensional critical manifolds. Raoul Bott used Morse–Bott theory in his original proof of the Bott periodicity theorem.
The abstract theory of Clifford modules was founded by a paper of M. F. Atiyah, R. Bott and Arnold S. Shapiro. A fundamental result on Clifford modules is that the Morita equivalence class of a Clifford algebra (the equivalence class of the category of Clifford modules over it) depends only on the signature p − q (mod 8).
Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups; Periodic function, a function whose output contains values that repeat periodically; Periodic mapping
In abstract algebra, the Virasoro group or Bott–Virasoro group (often denoted by Vir) [1] is an infinite-dimensional Lie group defined as the universal central extension of the group of diffeomorphisms of the circle. The corresponding Lie algebra is the Virasoro algebra, which has a key role in conformal field theory (CFT) and string theory.
It is built on the earlier Borel–Weil theorem of Armand Borel and André Weil, dealing just with the space of sections (the zeroth cohomology group), the extension to higher cohomology groups being provided by Raoul Bott. One can equivalently, through Serre's GAGA, view this as a result in complex algebraic geometry in the Zariski topology.
The Atiyah–Bott theorem is an equation in which the LHS must be the outcome of a global topological (homological) calculation, and the RHS a sum of the local contributions at fixed points of f. Counting codimensions in M × M {\displaystyle M\times M} , a transversality assumption for the graph of f and the diagonal should ensure that the ...