The simpler K-theories of a space are often related to vector bundles over the space, and different sorts of K-theories correspond to different structures that can be put on a vector bundle. Real K-theory Spectrum: KO Coefficient ring: The coefficient groups πi(KO) have period 8 in i, given by the sequence Z, Z2, Z2,0, Z, 0, 0, 0, repeated. As a ring, it is generated by a class η in degree 1, a class x4 in degree 4, and an invertible class v14 in degree 8, subject to the relations that 2η = η3 = ηx4 = 0, and x42 = 4v14. KO0(X) is the ring of stable equivalence classes of real vector bundles over X. Bott periodicity implies that the K-groups have period 8. Complex K-theory Spectrum: KU (even terms BU or Z × BU, odd terms U). Coefficient ring: The coefficient ring K*(point) is the ring of Laurent polynomials in a generator of degree 2. K0(X) is the ring of stable equivalence classes of complex vector bundles over X. Bott periodicity implies that the K-groups have period 2. Quaternionic K-theory Spectrum: KSp Coefficient ring: The coefficient groups πi(KSp) have period 8 in i, given by the sequence Z, 0, 0, 0,Z, Z2, Z2,0, repeated. KSp0(X) is the ring of stable equivalence classes of quaternionic vector bundles over X. Bott periodicity implies that the K-groups have period 8. K theory with coefficients Spectrum: KG G is some abelian group; for example the localization Z(p) at the prime p. Other K-theories can also be given coefficients. Self conjugate K-theory Spectrum: KSC Coefficient ring: to be written... The coefficient groups πi(KSC) have period 4 in i, given by the sequence Z, Z2, 0, Z, repeated. Introduced by D. Anderson in his unpublished 1964 Berkeley PhD dissertation, "A new cohomology theory". Connective K-theories Spectrum: ku for connective K-theory, ko for connective real K-theory. Coefficient ring: For ku, the coefficient ring is the ring of polynomials over Z on a single class v1 in dimension 2. For ko, the coefficient ring is the quotient of a polynomial ring on three generators, η in dimension 1, x4 in dimension 4, and v14 in dimension 8, the periodicity generator, modulo the relations that 2η = 0, x42 = 4v14, η3 = 0, and ηx = 0. Roughly speaking, this is K-theory with the negative dimensional parts killed off. KR-theory This is a cohomology theory defined for spaces with involution, from which many of the other K-theories can be derived. |

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