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Guoliang Yu

Publications and source records attributed to Guoliang Yu.

At least 55 records · Page 3Linked to original sources
arXiv2021

A Lichnerowicz Vanishing Theorem for the Maximal Roe Algebra

We show that if a countable discrete group acts properly and isometrically on a spin manifold of bounded Riemannian geometry and uniformly positive scalar curvature, then, under a suitable condition on the group action, the maximal higher index of the Dirac operator vanishes in K-theory of the maximal equivariant Roe algebra. The group action is not assumed to be cocompact. A key step in the proof is to establish a functional calculus for the Dirac operator in the maximal equivariant uniform Roe algebra. This allows us to prove vanishing of the index of the Dirac operator in K-theory of this algebra, which in turn yields the result for the maximal higher index.

math.KT↗
arXiv2021

Functoriality for higher rho invariants of elliptic operators

Let $N$ be a closed spin manifold with positive scalar curvature and $D_N$ the Dirac operator on $N$. Let $M_1$ and $M_2$ be two Galois covers of $N$ such that $M_2$ is a quotient of $M_1$. Then the quotient map from $M_1$ to $M_2$ naturally induces maps between the geometric $C^*$-algebras associated to the two manifolds. We prove, by a finite-propagation argument, that the \emph{maximal} higher rho invariants of the lifts of $D_N$ to $M_1$ and $M_2$ behave functorially with respect to the above quotient map. This can be applied to the computation of higher rho invariants, along with other related invariants.

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arXiv2020

Additivity of higher rho invariants and nonrigidity of topological manifolds

Let $X$ be a closed oriented connected topological manifold of dimension $n\geq 5$. The structure group of $X$ is the abelian group of equivalence classes of all pairs $(f, M)$ such that $M$ is a closed oriented manifold and $f\colon M \to X$ is an orientation-preserving homotopy equivalence. The main purpose of this article is to prove that a higher rho invariant defines a group homomorphism from the topological structure group of $X$ to the $C^*$-algebraic structure group of $X$. In fact, we introduce a higher rho invariant map on the homology manifold structure group of a closed oriented connected $\textit{topological}$ manifold, and prove its additivity. This higher rho invariant map restricts to the higher rho invariant map on the topological structure group. More generally, the same techniques developed in this paper can be applied to define a higher rho invariant map on the homology manifold structure group of a closed oriented connected $\textit{homology}$ manifold. As an application, we use the additivity of the higher rho invariant map to study non-rigidity of topological manifolds. More precisely, we give a lower bound for the free rank of the $\textit{algebraically reduced}$ structure group of $X$ by the number of torsion elements in $π_1 X$. Here the algebraic reduced structure group of $X$ is the quotient of the topological structure group of $X$ modulo a certain action of self-homotopy equivalences of $X$. We also introduce a notion of homological higher rho invariant, which can be used to detect many elements in the structure group of a closed oriented topological manifold, even when the fundamental group of the manifold is torsion free. In particular, we apply this homological higher rho invariant to show that the structure group is not finitely generated for a class of manifolds.

math.KT↗
arXiv2020

The equivariant coarse Novikov conjecture and coarse embedding

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 $γ\inΓ$, $\sup_{x\in Y} d(γx,x)<\infty$, where $Y$ is a fundamental domain of the $Γ$-action on $X$.

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arXiv2019

Delocalized eta invariants, cyclic cohomology and higher rho invariants

The first main result of this paper is to prove that the convergence of Lott's delocalized eta invariant holds for all differential operators with a sufficiently large spectral gap at zero. Furthermore, to each delocalized cyclic cocycle, we define a higher analogue of Lott's delocalized eta invariant and prove its convergence when the delocalized cyclic cocycle has at most exponential growth. Our second main result is to obtain an explicit formula of the delocalized Connes-Chern character of all $C^\ast$-algebraic secondary invariants for word hyperbolic groups. Equivalently, we give an explicit formula for the pairing between $C^\ast$-algebraic secondary invariants and delocalized cyclic cocycles of the group algebra. When the $C^\ast$-algebraic secondary invariant is a $K$-theoretic higher rho invariant of an invertible differential operator, we show this pairing is precisely the higher analogue of Lott's delocalized eta invariant alluded to above. Our work uses Puschnigg's smooth dense subalgebra for word hyperbolic groups in an essential way. We emphasize that our construction of the delocalized Connes-Chern character is at $C^\ast$-algebra $K$-theory level. This is of essential importance for applications to geometry and topology. As a consequence, we compute the paring between delocalized cyclic cocycles and $C^\ast$-algebraic Atiyah-Patodi-Singer index classes for manifolds with boundary, when the fundamental group of the given manifold is hyperbolic.

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arXiv2019

The Novikov Conjecture

We give a survey on recent development of the Novikov conjecture and its applications to topological rigidity and non-rigidity. .

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arXiv2019

Delocalized eta invariants, algebraicity, and $K$-theory of group $C^*$-algebras

In this paper, we establish a precise connection between higher rho invariants and delocalized eta invariants. Given an element in a discrete group, if its conjugacy class has polynomial growth, then there is a natural trace map on the $K_0$-group of its group $C^\ast$-algebra. For each such trace map, we construct a determinant map on secondary higher invariants. We show that, under the evaluation of this determinant map, the image of a higher rho invariant is precisely the corresponding delocalized eta invariant of Lott. As a consequence, we show that if the Baum-Connes conjecture holds for a group, then Lott's delocalized eta invariants take values in algebraic numbers. We also generalize Lott's delocalized eta invariant to the case where the corresponding conjugacy class does not have polynomial growth, provided that the strong Novikov conjecture holds for the group.

math.KT↗
arXiv2018

Finite decomposition complexity and the integral Novikov conjecture for higher algebraic K-theory

Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension. We prove a vanishing result for the continuously controlled algebraic K-theory of bounded geometry metric spaces with finite decomposition complexity. This leads to a proof of the integral K-theoretic Novikov conjecture, regarding split injectivity of the K-theoretic assembly map, for groups with finite decomposition complexity and finite CW models for their classifying spaces. By work of Guentner, Tessera, and Yu, this includes all (geometrically finite) linear groups.

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arXiv2017

K-theory of group Banach algebras and Banach property RD

We investigate Banach algebras of convolution operators on the $L^p$ spaces of a locally compact group, and their K-theory. We show that for a discrete group, the corresponding K-theory groups depend continuously on $p$ in an inductive sense. Via a Banach version of property RD, we show that for a large class of groups, the K-theory groups of the Banach algebras are independent of $p$.

math.FA↗
arXiv2016

Quantitative K-theory and the K{ü}nneth formula for operator algebras

In this paper, we apply quantitative operator K-theory to develop an algorithm for computing K-theory for the class of filtered C *-algebras with asymptotic finite nuclear decomposition. As a consequence, we prove the K{ü}nneth formula for C *-algebras in this class. Our main technical tool is a quantitative Mayer-Vietoris sequence for K-theory of filtered C *-algebras.

math.OA↗
arXiv2015

Dynamic Asymptotic Dimension: relation to dynamics, topology, coarse geometry, and $C^*$-algebras

We introduce dynamic asymptotic dimension, a notion of dimension for actions of discrete groups on locally compact spaces, and more generally for locally compact étale groupoids. We study our notion for minimal actions of the integer group, its relation with conditions used by Bartels, Lück, and Reich in the context of controlled topology, and its connections with Gromov's theory of asymptotic dimension. We also show that dynamic asymptotic dimension gives bounds on the nuclear dimension of Winter and Zacharias for C*-algebras associated to dynamical systems. Dynamic asymptotic dimension also has implications for K-theory and manifold topology: these will be drawn out in subsequent work.

math.DS↗
arXiv2014

Higher rho invariants and the moduli space of positive scalar curvature metrics

Given a closed smooth manifold M which carries a positive scalar curvature metric, one can associate an abelian group P(M) to the space of positive scalar curvature metrics on this manifold. The group of all diffeomorphisms of the manifold naturally acts on P(M). The moduli group of positive scalar curvature metrics is defined to be the quotient abelian group of this action, i.e. the coinvariant of the action. The moduli group measures the size of the moduli space of positive scalar curvature metrics on M. In this paper, we use the higher rho invariant and the finite part of the K-theory of the group C*-algebra of the fundamental group of M to give a lower bound of the rank of the moduli group.

math.OA↗
arXiv2014

An Analytic Grothendieck Riemann Roch Theorem

We extend the Boutet de Monvel Toeplitz index theorem to complex manifold with isolated singularities following the relative $K$-homology theory of Baum, Douglas, and Taylor for manifold with boundary. We apply this index theorem to study the Arveson-Douglas conjecture. Let $\ball^m$ be the unit ball in $\mathbb{C}^m$, and $I$ an ideal in the polynomial algebra $\mathbb{C}[z_1, \cdots, z_m]$. We prove that when the zero variety $Z_I$ is a complete intersection space with only isolated singularities and intersects with the unit sphere $\mathbb{S}^{2m-1}$ transversely, the representations of $\mathbb{C}[z_1, \cdots, z_m]$ on the closure of $I$ in $L^2_a(\ball^m)$ and also the corresponding quotient space $Q_I$ are essentially normal. Furthermore, we prove an index theorem for Toeplitz operators on $Q_I$ by showing that the representation of $\mathbb{C}[z_1, \cdots, z_m]$ on the quotient space $Q_I$ gives the fundamental class of the boundary $Z_I\cap \mathbb{S}^{2m-1}$. In the appendix, we prove with Kai Wang that if $f\in L^2_a(\ball^m)$ vanishes on $Z_I\cap \ball ^m$, then $f$ is contained inside the closure of the ideal $I$ in $L^2_a(\ball^m)$.

math.OA↗
arXiv2014

Positive scalar curvature, higher rho invariants and localization algebras

In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac operator on a spin manifold with boundary to the higher rho invariant of the Dirac operator on the boundary, where the boundary is endowed with a positive scalar curvature metric. Our result extends a theorem of Piazza and Schick.

math.KT↗
arXiv2014

Geometric Property (T)

This paper discusses `geometric property (T)'. This is a property of metric spaces introduced in earlier work of the authors for its applications to K-theory. Geometric property (T) is a strong form of `expansion property': in particular for a sequence of finite graphs $(X_n)$, it is strictly stronger than $(X_n)$ being an expander in the sense that the Cheeger constants $h(X_n)$ are bounded below. We show here that geometric property (T) is a coarse invariant, i.e. depends only on the large-scale geometry of a metric space $X$. We also discuss the relationships between geometric property (T) and amenability, property (T), and various coarse geometric notions of a-T-menability. In particular, we show that property (T) for a residually finite group is characterised by geometric property (T) for its finite quotients.

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