arXiv research

Study of $p$-adic simplicial volumes and their properties.

Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical m…

We define the ideal simplicial volume for compact manifolds with boundary. Roughly speaking, the ideal simplicial volume of a manifold $M$ measures the minimal size of possibly ideal triangulations of $M$ "with real coefficients", thus providing a variation of the ordinary simplicial volume defined by Gromov in 1982, t…

We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated s…

We show that non-elliptic prime 3-manifolds satisfy integral approximation for the simplicial volume, i.e., that their simplicial volume equals the stable integral simplicial volume. The proof makes use of integral foliated simplicial volume and tools from ergodic theory.

We study the multiscale simplicial flat norm (MSFN) problem, which computes flat norm at various scales of sets defined as oriented subcomplexes of finite simplicial complexes in arbitrary dimensions. We show that the multiscale simplicial flat norm is NP-complete when homology is defined over integers. We cast the mul…

Alexander's conjecture extended to infinite simplicial complexes.

Proposes a method to learn representations of higher-dimensional simplicial complexes.

Local Kan conditions enable differentiation of simplicial manifolds.

Study simplicial volume in fiber bundles with connected groups.

The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact lo…

The study limits how many parts regular simplicial partitions can overlap.

The study examines conditions for minimal volume entropy of simplicial complexes.

We provide sharp lower bounds for the simplicial volume of compact $3$-manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of $3$-manifolds, including handlebodies and products of surfaces with the interval. Our results provide the firs…

Positive simplicial volume implies locally symmetric space structure.

Formula connects foliated simplicial volume with group cost.

Extends circle pattern theorem to quasi-simplicial triangulations.

Paper studies spaces of flattenings of simplicial spheres and their homotopy type.

We prove that cubical simplicial volume of oriented closed 3-manifolds is equal to one fifth of ordinary simplicial volume.

Mixes higher-order simplicial complexes for data augmentation.

Study shows simplicial volume of certain fiber bundles is zero.

Constructs simplified or complexified simplicial complexes.

Minimal volume entropy vanishes or is positive under certain fiber growth conditions.

Study simplicial volume of manifolds from reflection group trick.

Integral foliated simplicial volume is zero for certain amenable covers.

The paper proves simplicial volume positivity for certain nonpositively curved 4-manifolds with nonzero Euler characteristic.

Simplicial learning improves classification by generating compact sparse representations.

A notion of up and down Grover walks on simplicial complexes are proposed and their properties are investigated. These are abstract Szegedy walks, which is a special kind of unitary operators on a Hilbert space. The operators introduced in the present paper are usual Grover walks on graphs defined by using combinatoria…

Study shows range of simplicial volumes for open manifolds.

An additional minimal simplicial n-complex contains a non-splittable link in R^(2n).

The simplicial volume of oriented closed connected smooth manifolds that admit a non-trivial smooth $S^1$-action vanishes. In the present work we prove a version of this result for the integral foliated simplicial volume of aspherical manifolds: The integral foliated simplicial volume of aspherical oriented closed conn…

Let $M$ be the interior of a connected, oriented, compact manifold $V$ of dimension at least 2. If each path component of $\partial V$ has amenable fundamental group, then we prove that the simplicial volume of $M$ is equal to the relative simplicial volume of $V$ and also to the geometric (Lipschitz) simplicial volume…

We establish a functor $Kan$ from local Kan simplicial manifolds to weak Kan simplicial manifolds. It gives a solution to the problem of extending local Lie groupoids to Lie 2-groupoids.

We construct non-constructible simplicial $d$-spheres with $d+10$ vertices and non-constructible, non-realizable simplicial $d$-balls with $d+9$ vertices for $d\geq 3$.

New techniques prove bounded acyclicity results for semi-simplicial sets.

Algorithm calculates Hopf invariant for simplicial mappings.

The study shows simplicial volume finiteness for certain manifolds with amenable fundamental groups.

The simplicial volume of non-R^3 contractible 3-manifolds is infinite.

We show that there exist closed manifolds with arbitrarily small transcendental simplicial volumes. Moreover, we exhibit an explicit uncountable family of (transcendental) real numbers that are not realised as the simplicial volume of a closed manifold.

Hypernetworks are simplified simplicial complexes with curvature.

In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations are homeomorphic to the product of spheres. As a corollary, we obtain the chara…

Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal …

Unified probabilistic foundation for fuzzy simplicial sets in dimensionality reduction.

The simplicial complexity is an invariant for finitely presentable groups that was recently introduced by Babenko, Balacheff and Bulteau to study systolic area. The simplicial complexity $κ(G)$ was proved to be a good approximation of the systolic area $σ(G)$ for large values of $κ(G)$. In this paper we compute the sim…

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively c…

Positive simplicial volume found for certain non-positively curved manifolds with specific submanifolds.

Minimal simplicial complexes in high dimensions always contain complex links.

Study essentiality and simplicial volume of manifolds fibered over spheres.