Generalized Arf invariants in algebraic L-theory

Generalized Arf invariants and reduced power operations in cyclic homology

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…

球面稳定同调群的几何方法 II；阿夫-凯瓦里不变量

Geometric approach to stable homotopy groups of spheres II; Arf-Kervaire Invariants

本文通过几何方法解决球面稳定同调群中的凯瓦里不变量问题，引入了新的对称结构并证明了相关定理。

Algebraic independence of generalized Morita-Miller-Mumford classes

The generalized Morita-Miller-Mumford classes of a smooth oriented manifold bundle are defined as the image of the characteristic classes of the vertical tangent bundle under the Gysin homomorphism. We show that if the dimension of the manifold is even, then all MMM-classes in rational cohomology are nonzero for some bundle. In odd dimensions, this is also true with one exception: the MMM-class associated with the Hirzebruch $\cL$-class is always zero. We also show a similar result for holomorphic fibre bundles.

关于广义代数纽结的切片亏格

On the slice genus of generalized algebraic knots

本文通过构造代数切片但四亏格为二的代数纽结线性组合，推广了Hedden–Kirk–Livingston的迭代环面纽结例子，并利用Casson–Gordon不变量与缆绳公式给出亏格下界。

A proof of the Generalized Jacobian conjecture

Based on the reduction of degree in polynomial mappings and some known results in algebraic geometry, by introducing the Brouwer degree, a tool from differential topology, algebraic topology and algebraic geometry, we completely prove the Generalized Jacobian conjecture in the field of real numbers, which implies the Generalized complex Jacobian conjecture. Also, for the strong real Jacobian conjecture, we present a newly sufficient and necessary condition.

Algebraic and combinatorial codimension-1 transversality

The Waldhausen construction of Mayer-Vietoris splittings of chain complexes over an injective generalized free product of group rings is extended to a combinatorial construction of Seifert-van Kampen splittings of CW complexes with fundamental group an injective generalized free product.