There is no equivariant coarse embedding of $L_{p}$ into $\ell_p$

On coarse embeddability into $\ell_p$-spaces and a conjecture of Dranishnikov

We show that the Hilbert space is coarsely embeddable into any $\ell_p$ for $1\le p<\infty$. In particular, this yields new characterizations of embeddability of separable metric spaces into the Hilbert space.

Coarse Quotient Mappings between Metric Spaces

We give a definition of coarse quotient mapping and show that several results for uniform quotient mapping also hold in the coarse setting. In particular, we prove that any Banach space that is a coarse quotient of $L_p\equiv L_p[0,1]$, $1<p<\infty$, is isomorphic to a linear quotient of $L_p$. It is also proved that $\ell_q$ is not a coarse quotient of $\ell_p$ for $1<p<q<\infty$ using Rolewicz's property ($β$).

Coarse embeddability into Banach spaces

The main purposes of this paper are (1) To survey the area of coarse embeddability of metric spaces into Banach spaces, and, in particular, coarse embeddability of different Banach spaces into each other; (2) To present new results on the problems: (a) Whether coarse non-embeddability into $\ell_2$ implies presence of expander-like structures? (b) To what extent $\ell_2$ is the most difficult space to embed into?

位形空间的粗上同调与粗嵌入

Coarse cohomology of configuration space and coarse embedding

本文引入了度量空间子空间补集的等变粗上同调概念，借此定义并计算位形空间的粗上同调，证明了其特定类可阻碍两度量空间间的粗嵌入。

A Bott periodicity theorem for $\ell^p$-spaces and the coarse Novikov conjecture at infinity

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…

左正合∞-范畴中的受控对象与Novikov猜想

Controlled objects in left-exact $\infty$-categories and the Novikov conjecture

本研究构造了带G作用的左正合∞-范畴的等变X受控对象范畴，结合代数K-理论得到新等变粗同调理论，将相关装配映射单射性结果应用于左正合∞-范畴的代数K-理论。

关于粗同调理论的单射结果

Injectivity results for coarse homology theories

本文通过等效粗同调理论和降维原理，研究了线性群及具有有限分解复杂度群的装配映射单射性问题。

膨胀图是 $\ell^p$ 粗 Baum-Connes 猜想的反例

Expanders are counterexamples to the $\ell^p$ coarse Baum-Connes conjecture

本文通过构造来自剩余有限双曲群的膨胀图，证明了 $\ell^p$ 粗 Baum-Connes 装配映射在 $1<p<\infty$ 时不是满射，从而给出该猜想的反例。