Some topics in K-theory

• Main Author: Snaith, Victor Percy
• Published: University of Warwick 1969
• Subjects: 510; QA Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.594892

In part A (II) the uniqueness theorem for equivariant cohomology theories, (proved in part A (I)), is used to calculate the operation rings, Op(KG,KG^) and OP(KG^,KG^), (^ is I(G)-adic completion ; G is a finite group). Semi-groups, SG < KG, are introduced and Op(~SG(-), KG) is calculated in order to investigate (Op(KG), the ring of self-operations of KG. Finally Op(KG) is related to the other two rings of operations and any self-operation of KG is proved to be continuous with respect to the I(G)-adic topology. In Part B some higher order operations in K-theory, called Massey products, are defined and proved to be the differentials in the Equivariant Kunneth Formula spectral sequence in K-theory. In Part C the Rothenberg-Steenrod spectral sequences are used (i) to calculate the K-theory of conjugate bundles of Lie groups, (ii) to prove a small theorem on the K-theory of homogeneous spaces of Lie groups, and (iii) to calculate the homological dimension of R(H) as an R(G)-module, for an inclusion of Lie groups, H<G. As an example of (ii) the algebra K (SO(m)) and the operation ring, lim<--- K(SO(m)), are computed.