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 $\epsilon$-symmetric quadratic form over a ring with involution $R$ is lifted to an $\epsilon(1+T)$-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 $\omega_1$ of the original form, and the Hasse-Witt invariant of this form gives rise to a `secondary' Arf invariant $\omega_2$, which is defined on the kernel of $\omega_1$. The second construction yields an invariant $\Upsilon$ 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 $\Upsilon$ agrees with $(\omega_1,\omega_2)$. The invariant $\Upsilon$ 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.