Relative differential K-characters

Relative (generalized) differential cohomology

Let h be a rationally even cohomology theory and h^ the natural differential refinement, as defined by Hopkins and Singer. We consider the possible definitions of the relative differential cohomology groups, generalizing the analogous picture for the Deligne cohomology, and we show the corresponding long exact sequence in each case.

Relative differential cohomology and generalized Cheeger-Simons characters

We provide a suitable axiomatic framework for differential cohomology in the relative case and we deduce the corresponding long exact sequences. We also construct the relative version of the generalized Cheeger-Simons characters and we define the integration map when the fibre has a boundary.

Relative Milnor $K$-groups and differential forms of split nilpotent extensions

Let $R$ be a commutative ring and $I\subset R$ be a nilpotent ideal such that the quotient $R/I$ splits out of $R$ as a ring. Let $N$ be a natural number such that ${I^N=0}$. We establish a canonical isomorphism between the relative Milnor $K$-group $K^{M}_{n+1}(R,I)$ and the quotient of the relative module of differential forms $Ω^n_{R,I}/d\,Ω^{n-1}_{R,I}$ assuming that $N!$ is invertible in $R$ and that the ring $R$ is weakly $5$-fold stable. The latter means that any $4$-tuple of elements in $R$ can be shifted by an invertible element to become a $4$-tuple of invertible elements.

高阶微分算子的相对 K-同调

Relative K-homology of higher-order differential operators

本文将谱三元组扩展为高阶相对谱三元组以描述带边界流形上的亚椭圆算子，并通过计算其生成的相对K-同调循环的边界映射，推广了Baum-Douglas-Taylor指标定理。

相对指标定理与Fredholm算子的刻画

The relative index theorem and a characterization of Fredholm operators

本文将相对指标定理推广到任意阶亚椭圆微分算子，并证明无穷远处可逆性是Fredholm性的充要条件，为非紧空间指标理论提供几何刻画和KK-理论工具。