J-homomorphism from the (unstable) homotopy groups of the (stable) classical groups to these stable homotopy groups . Originally described by George W. Whitehead, it became the subject of the famous Adams conjecture (1963) which was finally resolved in the affirmative by Daniel Quillen (1971). Bott's original results may be succinctly summarized in: Corollary: The (unstable) homotopy groups of the (infinite) classical groups are periodic: Note: The second and third of these isomorphisms intertwine to give the 8-fold periodicity results: Loop spaces and classifying spaces For the theory associated to the infinite unitary group, U, the space BU is the classifying space for stable complex vector bundles (a Grassmannian in infinite dimensions). One formulation of Bott periodicity describes the twofold loop space, of BU. Here, is the loop space functor, right adjoint to suspension and left adjoint to the classifying space construction. Bott periodicity states that this double loop space is essentially BU again; more precisely, is essentially (that is, homotopy equivalent to) the union of a countable number of copies of BU. An equivalent formulation is Either of these has the immediate effect of showing why (complex) topological K-theory is a 2-fold periodic theory. In the corresponding theory for the infinite orthogonal group, O, the space BO is the classifying space for stable real vector bundles. In this case, Bott periodicity states that, for the 8-fold loop space, or equivalently, which yields the consequence that KO-theory is an 8-fold periodic theory. Also, for the infinite symplectic group, Sp, the space BSp is the classifying space for stable quaternionic vector bundles, and Bott periodicity states that or equivalently Thus both topological real K-theory (also known as KO-theory) and topological quaternionic K-theory (also known as KSp-theory) are 8-fold periodic theories. Geometric model of loop spaces One elegant formulation of Bott periodicity makes use of the observation that there are natural embeddings (as closed subgroups) between the classical groups. The loop spaces in Bott periodicity are then homotopy equivalent to the symmetric spaces of successive quotients, with additional discrete factors of Z. Over the complex numbers: Over the real numbers and quaternions: These sequences corresponds to sequences in Clifford algebras – see classification of Clifford algebras; over the complex numbers: Over the real numbers and quaternions: where the division algebras indicate "matrices over that algebra". As they are 2-periodic/8-periodic, they can be arranged in a circle, where they are called the Bott periodicity