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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 .
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
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
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.
Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers. Properties (respectively, ~) is a ...