arXiv research

Proves Borel Conjecture for certain 3D spaces.

We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space.

Researchers prove a conjecture about a specific type of 3D space.

New 4D shapes found that defy smoothness rules.

We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and app…

Equivariant homotopy methods developed over the last 20 years lead to recent breakthroughs in the Borel isomorphism conjectures for Loday assembly maps in K- and L-theories. An important consequence of these algebraic conjectures is the topological rigidity of compact aspherical manifolds. Our goal is to strip the basi…

Study rigidity of real moment-angle manifolds using cubical geometry.

Consider the middle perversity intersection cohomology groups of various compactifications of a Hermitian locally symmetric space. Rapoport and independently Goresky and MacPherson have conjectured that these groups coincide for the reductive Borel-Serre compactification and the Baily-Borel-Satake compactification. Thi…

Proves summability of state integrals for specific hyperbolic knots.

In this short note we prove the Borel conjecture for a family of aspherical manifolds that includes higher graph manifolds.

We call a group FJ if it satisfies the $K$- and $L$-theoretic Farrell-Jones conjecture with coefficients in $\mathbb Z$. We show that if $G$ is FJ, then the simple Borel conjecture (in dimensions $\ge 5$) holds for every group of the form $G\rtimes\mathbb Z$. If in addition $Wh(G\times \mathbb Z)=0$, which is true for …

Proves resurgence properties for Habiro elements from radial limits of theta series.

A surgery classification theory is introduced for manifolds of bounded geometry up to quasi-isometry. The Borel conjecture for this theory is proven for flat Euclidean space.

The paper develops surgery theories for foliations and solves a problem posed by Weinberger.

Two 4-manifolds are stably diffeomorphic if they become diffeomorphic after connected sum with S^2 x S^2's. This paper shows that two closed, orientable, homotopy equivalent, smooth 4-manifolds are stably diffeomorphic, provided a certain map from the second homology of the fundamental group with coefficients in Z/2 to…

This article will explore the K- and L-theory of group rings and their applications to algebra, geometry and topology. The Farrell-Jones Conjecture characterizes K- and L-theory groups. It has many implications, including the Borel and Novikov Conjectures for topological rigidity. Its current status, and many of its co…

Quantum dilogarithm function proven from a linear difference equation.

In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on RP^n invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that t…

We show that in all dimensions >7 there are closed aspherical manifolds whose fundamental groups have nontrivial center but do not possess any topological circle actions. This disproves a conjectured converse (proposed by Conner and Raymond) to a classical theorem of Borel.

We obtain a topological and weakly equivariant classification of closed three-dimensional Alexandrov spaces with an effective isometric circle action. As an application of the classification we prove a version of the Borel conjecture for closed three-dimensional Alexandrov spaces with circle symmetry.

We show, up to h-cobordism, that the existence and uniqueness of connected sum decompositions of oriented 4-dimensional manifolds is an invariant of homotopy equivalence, assuming that the fundamental group of each summand is "good" in the sense of Freedman and Quinn. On a separate note, we observe that the Borel Conje…

Let X be a locally symmetric space associated to a reductive algebraic group G defined over Q. L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of X; they were introduced in [math.RT/0112251]. That paper also introduced the micro-support of an L-module, a com…

The BPS decomposition theorem splits cohomology of symmetric stacks into invariant parts.

We introduce systems of objects and operators in linear monoidal categories called $\hat Ψ$-systems. A $\hat Ψ$-system satisfying several additional assumptions gives rise to a topological invariant of triples (a closed oriented 3-manifold $M$, a principal bundle over $M$, a link in $M$). This construction generalizes …

L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of a locally symmetric space. We define the micro-support of an L-module; it is a set of irreducible modules for the Levi quotients of the parabolic Q-subgroups associated to the strata. We prove a vanishing th…

The paper connects arithmetic invariants of hyperbolic 3-manifolds.

Paper studies Frobenius manifolds and quantum differential equations, proving Dubrovin Conjecture for Hirzebruch surfaces.

The study shows how certain ODEs and integrals are regular under Borel summation.

New findings on cusped Borel Anosov representations and their properties.

The article has been withdrawn by the author. Wolfgang Lueck and Peter Linnell pointed out that the proof of Lemma 3.8 does not apply to the unrestricted case of wreath product. It is not clear at this stage how to complete the proof of Theorem 3.1 using the present version of Lemma 3.8. The valid results originating f…

New tools analyze the complexity of left-ordering equivalence relations in groups and 3-manifolds.

The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate when a non-aspherical oriented connected closed manifold M is topological rigid in the following sense. If f: N --> M is an orientation preserving homotopy equivalence with a closed oriented manifold as target, …

Extends Borel invariant to measurable cocycles of 3-manifold groups.

Study shows Roller compactification's median graph has limited asymptotic dimension.

This expository article is an expanded version of talks given at the "Current Developments in Mathematics, 2002" conference. It gives an introduction to the (generalized) conjecture of Rapoport and Goresky-MacPherson which identifies the intersection cohomology of a real equal-rank Satake compactification of a locally …

In this project we further investigate the idea of reducing the dimensionality of datasets using a Borel isomorphism with the purpose of subsequently applying supervised learning algorithms, as originally suggested by my supervisor V. Pestov (in 2011 Dagstuhl preprint). Any consistent learning algorithm, for example kN…

Proves resurgent nature of a series solution to deformed Painlevé I equation.

Geometrically proves WKB solutions of Schrödinger equations are resurgent.

We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology.

The paper discusses rigidity results for inequalities on weighted Riemannian manifolds.

New equivariant formality concepts solve the toral rank conjecture.

Study proves hyperbolic groups have specific subgroup properties.

Study on Borel Anosov subgroups in SL(d,R) for d≠5,8k±1.

Maximal and Borel Anosov representations in $Sp(4,\mathbb{R})$ are proven to be Hitchin.

Study topological 4-manifolds with specific fundamental groups.

Our aim is to prove that two formal power series of importance to quantum topology are Gevrey. These series are the Kashaev invariant of a knot (reformulated by Huynh and the second author) and the Gromov norm of the LMO of an integral homology 3-sphere. It follows that the power series associated to a simple Lie algeb…

Let $X$ be an algebraic variety over $\mathbb{C}$. We say that $X$ is Borel hyperbolic if, for every finite type reduced scheme $S$ over $\mathbb{C}$, every holomorphic map $S^{an}\to X^{an}$ is algebraic. We use a transcendental specialization technique to prove that $X$ is Borel hyperbolic if and only if, for every s…

Develops measures for non-Borel Anosov groups on Furstenberg boundary.