Generalized Arf invariants and reduced power operations in cyclic homology

Abstract

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 $\frac{-T}{1+T}$. The discriminant of this form can be viewed as the classical Arf invariant $ω_1$ of the original form, and the Hasse-Witt invariant of this form gives rise to a `secondary' Arf invariant $ω_2$, which is defined on the kernel of $ω_1$. 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 $HQ_1(R)$ which is defined using natural operations on $HQ_1$. In the case of a commutative ring $Υ$ agrees with $(ω_1,ω_2)$. 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.