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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.
The connection is explained by the geometric model of loop spaces approach to Bott periodicity: their 2-fold/8-fold periodic embeddings of the classical groups in each other (corresponding to isomorphism groups of Clifford algebras), and their successive quotients are symmetric spaces which are homotopy equivalent to the loop spaces of the ...
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 symmetry classes are ordered according to the Bott clock (see below) so that the same values repeat in the diagonals. [5] An X in the table of "Symmetries" indicates that the Hamiltonian of the symmetry is broken with respect to the given operator. A value of ±1 indicates the value of the operator squared for that system. [5]
Bott periodicity computes the homotopy of the stable unitary group and stable orthogonal group. The Whitehead group of a ring (the first K-group) can be defined in terms of (). Stable homotopy groups of spheres are the stable groups associated with the suspension functor.
Coefficient ring: The coefficient groups π i (KSp) have period 8 in i, given by the sequence Z, 0, 0, 0,Z, Z 2, Z 2,0, repeated. KSp 0 (X) is the ring of stable equivalence classes of quaternionic vector bundles over X. Bott periodicity implies that the K-groups have period 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.
This period 8 pattern is known as Bott periodicity, and it is reflected in the stable homotopy groups of spheres via the image of the J-homomorphism which is: a cyclic group of order 2 if k is congruent to 0 or 1 modulo 8; trivial if k is congruent to 2, 4, 5, or 6 modulo 8; and