Generalized Arf invariants and reduced power operations in cyclic homology

الأصل · EN

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.