2005/03/24 by Paul M. H. Wolters, Wolters, Paul M. H.
Mathematics · #19G38 #Algebraic Topology (math.AT) #FOS: Mathematics #Primary 11E70 #Rings and Algebras (math.RA) #Secondary 19G24 #math.AT #math.RA #msc:11E70 #msc:19G24 #msc:19G38
paper · pdf · doi:10.48550/arxiv.math/0503538
129 pages; september 1990 PhD thesis
arxiv created 2005/03/24 · arxiv updated 2009/12/01
In this thesis we consider two constructions generalizing the classical Arf invariant. In the first construction an ε-symmetric quadratic form over a ring with involution R is lifted to an ε(1+T)-symmetric quadratic form over the ring of formal power series R[[T]] with involution mapping T to (-T)/(1+T). The discriminant of this form can be viewed as the classical Arf invariant ω1 of the original form, and the Hasse-Witt invariant of this form gives rise to a `secondary' Arf invariant ω2, which is defined on the kernel of ω1. The second construction yields an invariant Υ which is defined on quadratic forms for which the underlying symmetric form is standard. It takes values in a quotient of quaternionic homology HQ1(R) which is defined using natural operations on HQ1. In the case of a commutative ring Υ agrees with (ω1,ω2). The invariant Υ is well suited for computations. In particular we prove that it is faithful if R is the group ring over GF(2) of a group with two ends.