Computing Borel's Regulator

We present an infinite series formula based on the Karoubi-Hamida integral, for the universal Borel class evaluated on H_{2n+1}(GL(\mathbb{C})). For a cyclotomic field F we define a canonical set of elements in K_3(F) and present a novel approach (based on a free differential calculus) to constructing them. Indeed, we are able to explicitly construct their images in H_{3}(GL(\mathbb{C})) under the Hurewicz map. Applying our formula to these images yields a value V_1(F), which coincides with the Borel regulator R_1(F) when our set is a basis of K_3(F) modulo torsion. For F= \mathbb{Q}(e^{2\pi i/3}) a computation of V_1(F) has been made based on our techniques.Comment: 29 pages, section on computational aspects changed (now in Appendix A), section 5 moved to Appendix B, introduction change

Hurewicz images in BP and related homology theories

In this paper BP-theory is used to give a proof that there exists astable homotopy element in ?S2n+1?2(RP?) with non-zero Hurewicz imagein ju-theory if and only if there exists an element of ?S2n+1?2(S0) which is represented by a framed manifold of Arf invariant one

Some topics in K-theory

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