Gennadi Kasparov, Guoliang Yu
The coarse geometric Novikov conjecture provides an algorithm to determine when the higher index of an elliptic operator on a noncompact space is nonzero. The purpose of this paper is to prove the coarse geometric Novikov conjecture for spaces which admit a (coarse) uniform embedding into a uniformly convex Banach space.
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The Coarse Novikov Conjecture for Finite Products of Fibred Coarsely Embeddable Spaces
本文证明了有界几何真度量空间的有限乘积满足粗Novikov猜想，通过构造粗真代数和多阶段Roe代数及指标映射，建立了连接迭代相对指标理论与全局猜想的约化定理。
The equivariant coarse Novikov conjecture provides an algorithm for determining nonvanishing of equivariant higher index of elliptic differential operators on noncompact manifolds. In this article, we prove the equivariant coarse Novikov conjecture under certain coarse embeddability conditions. More precisely, if a discrete group $Γ$ acts on a bounded geometric space $X$ properly, isometrically, and with bounded distortion, $X/Γ$ and $Γ$ admit coarse embeddings into Hilbert space, then the $Γ$-equivariant coarse Novikov conjecture holds for $X$. Here bounded distortion means that for any $γ\in…
We formulate and prove a Bott periodicity theorem for an $\ell^p$-space ($1\leq p<\infty$). For a proper metric space $X$ with bounded geometry, we introduce a version of $K$-homology at infinity, denoted by $K_*^{\infty}(X)$, and the Roe algebra at infinity, denoted by $C^*_{\infty}(X)$. Then the coarse assembly map descents to a map from $\lim_{d\to\infty}K_*^{\infty}(P_d(X))$ to $K_*(C^*_{\infty}(X))$, called the coarse assembly map at infinity. We show that to prove the coarse Novikov conjecture, it suffices to prove the coarse assembly map at infinity is an injection. As a result, we show…
Strong relative Novikov conjecture for coarsely embeddable groups
本文证明了粗可嵌入Hilbert空间的任意群对满足强相对Novikov猜想，推进了群论与几何拓扑中的相关猜想研究。
In this paper, we introduce a strongly quasi-local version of the coarse Novikov conjecture, which states that certain assembly map from the coarse $K$-homology of a metric space to the $K$-theory of its strongly quasi-local algebra is injective. We prove that the conjecture holds for metric spaces with bounded geometry which can be coarsely embedded into Banach spaces with Property (H), introduced by Kasparov and Yu. Besides, we also generalise the notion of strong quasi-locality to proper metric spaces and provide a (strongly) quasi-local picture for $K$-homology.
Quantitative index, Novikov conjecture and coarse decomposability
本文定义考虑伪微分传播的定量装配映射估计，建立其与 Novikov 猜想及粗分解的联系，并几何证明有限分解复杂性群的 Novikov 猜想。
Let $1 \to N \to G \to G/N \to 1$ be a short exact sequence of countable discrete groups and let $B$ be any $G$-$C^*$-algebra. In this paper, we show that the strong Novikov conjecture with coefficients in $B$ holds for such a group $G$ when the normal subgroup $N$ and the quotient group $G/N$ are coarsely embeddable into Hilbert spaces. As a result, the group $G$ satisfies the Novikov conjecture under the same hypothesis on $N$ and $G/N$.
Hilbert-Hadamard spaces and the equivariant coarse Novikov conjecture
本文研究可等变粗嵌入到可容许 Hilbert-Hadamard 空间的度量空间的等变粗 Novikov 猜想，先证明无挠群情形下等变粗强 Novikov 猜想有理成立，再引入有理解析等变粗 Novikov 猜想并去掉无挠假设，证明相应装配映射为有理单射。
Let $(1\to N_n\to G_n\to Q_n \to 1)_{n\in \mathbb{N}}$ be a sequence of extensions of countable discrete groups. Endow $(G_n)_{n\in \mathbb{N}}$ with metrics associated to proper length functions on $(G_n)_{n\in \mathbb{N}}$ respectively such that the sequence of metric spaces $(G_n)_{n\in \mathbb{N}}$ have uniform bounded geometry. We show that if $(N_n)_{n\in \mathbb{N}}$ and $(Q_n)_{n\in \mathbb{N}}$ are coarsely embeddable into Hilbert space, then the coarse Novikov conjecture holds for the sequence $(G_n)_{n\in \mathbb{N}}$, which may not admit a coarse embedding into Hilbert space.
The connection between the coarse geometry of metric spaces and analytic properties of topological groupoids is well known. One of the main results of Skandalis, Tu and Yu is that a space admits a coarse embedding into Hilbert space if and only if a certain associated topological groupoid is a-T-menable. This groupoid characterisation then reduces the proof that the coarse Baum-Connes conjecture holds for a coarsely embeddable space to known results for a-T-menable groupoids. The property of admitting a fibred coarse embedding into Hilbert space was introduced by Chen, Wang and Yu to provide a…
In this article we formulate a version of the analytic Novikov conjecture for semigroups rather than groups, and show that the descent argument from coarse geometry generalises effectively to this new situation.
Coarse geometry of Hecke pairs and the Baum-Connes conjecture
本研究借助Hecke对陪集空间与Schlichting完备化的粗几何，证明了余Haagerup性质下Baum-Connes和Novikov猜想的新稳定性结果，推广了既有结论并给出满足带系数Baum-Connes猜想的S-算术群新实例。
A groupoid approach to the equivariant coarse Baum--Connes conjecture
本文通过群体群论方法研究等价粗 Baum--Connes 猜想，引入等价粗群体群并证明其与等价粗 Baum--Connes 猜想的等价性，进一步证明了当空间 X 可以粗嵌入到 Hilbert 空间时，等价粗 Novikov 猜想成立。
Relative higher index theory on quotients of Roe algebras and positive scalar curvature at infinity
本文利用Roe代数商的相对高阶指标理论，研究非紧流形无穷远处正数量曲率度量的存在性障碍，提出相对粗Baum-Connes猜想并验证其部分情形，应用于扩展图乘积。
Higher index theory for spaces with an FCE-by-FCE structure
本文研究有限群扩张序列形成的 FCE-by-FCE 结构，并在核序列与商序列均可纤维化粗嵌入 Hilbert 空间的条件下，证明相应空间的粗 Novikov 猜想成立。
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