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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 .
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).
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 group is given by Bott periodicity. It is always cyclic; and if r is positive, it is of order 2 if r is 0 or 1 modulo 8, infinite if r is 3 or 7 modulo 8, and order 1 otherwise (Switzer 1975, p. 488). In particular the image of the stable J-homomorphism is cyclic.
The phenomenon of periodicity named after Raoul Bott (see Bott periodicity theorem) can be formulated this way: ... Toggle the table of contents. Topological K-theory.
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.
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 ...