Rank gradient versus stable integral simplicial volume

We observe that stable integral simplicial volume of closed manifolds gives an upper bound for the rank gradient of the corresponding fundamental groups.     

Measure homology and singular homology are isometrically isomorphic

Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CW-complexes. It is the aim of this paper to prove that this isomorphism is isometric with respect to the ℓ¹-seminorm on singular homology and the seminorm on measure homology induced by the total variation. This, in particular, implies that one can calculate the simplicial volume via measure homology – as already claimed by Thurston. For example, measure homology can be used to prove Gromov's proportionality principle of simplicial volume.     

Aspherical manifolds with relatively hyperbolic fundamental groups

We show that the aspherical manifolds produced via the relative strict hyperboli- zation of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automorphism group, absence of splitting over elementary subgroups, and acylindricity. In fact, some of these properties hold for any compact aspherical manifold with incompressible aspherical boundary components, provided the fundamental group is hyperbolic relative to fundamental groups of boundary components. We also show that no manifold obtained via the relative strict hyperbolization can be embedded into a compact Kähler manifold of the same dimension, except when the dimension is two.     

Degree theorems and Lipschitz simplicial volume for nonpositively curved manifolds of finite volume

We study a metric version of the simplicial volume on Riemannianmanifolds, the Lipschitz simplicial volume, with applicationsto degree theorems in mind. We establish a proportionality principleand a product inequality from which we derive an extension ofGromov's volume comparison theorem to products of negativelycurved manifolds or locally symmetric spaces of noncompact type.In contrast, we provide vanishing results for the ordinary simplicialvolume; for instance, we show that the ordinary simplicial volumeof noncompact locally symmetric spaces with finite volume of-rank at least 3 is zero. Received November 6, 2007. Revised August 20, 2008.     

Representability theorems for presheaves of spectra

Suppose that M is a simplicial model category and that F is a contravariant simplicial functor defined on M which takes values in pointed simplicial sets. This note displays conditions on the simplicial model category M and the functor F such that F is representable up to weak equivalence. The conditions on F are homotopy coherent versions of the classical conditions for Brown representability, while M should have the fundamental properties of the stable model structure for presheaves of spectra on a Grothendieck site.     

Exact integration formulas for the finite volume element method on simplicial meshes

This article considers the technological aspects of the finite volume element method for the numerical solution of partial differential equations on simplicial grids in two and three dimensions. We derive new classes of integration formulas for the exact integration of generic monomials of barycentric coordinates over different types of fundamental shapes corresponding to a barycentric dual mesh. These integration formulas constitute an essential component for the development of high‐order accurate finite volume element schemes. Numerical examples are presented that illustrate the validity of the technology. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Algebraic fundamental group and simplicial complexes

In this paper, we prove that the fundamental group of a simplicial complex is isomorphic to the algebraic fundamental group of its incidence algebra, and we derive some applications.     

Combinatorial Laplacians of matroid complexes

We combinatorially interpret the spectra of discrete Laplace operators from the boundary maps in the simplicial complex of independent sets of a matroid. The interpretation follows from a surprising orthogonal decomposition of the simplicial chain groups. This decomposition is in general finer than the spectral decomposition. As a consequence, the spectra are integral. One corollary to our combinatorial interpretation may be paraphrased as stating that one can ``hear" the characteristic polynomial of a matroid.

     

On the computability of the Steenrod squares

We give explicitely the formulas of a sequence of morphisms which measure the failure of commutativity of the cup product on the cochain level, provided that we work with simplicial sets; these formulas are established in terms of the component morphisms of a given Eilenberg-Zilber contraction. As a consequence, in the case in which the simplicial set is finite in each dimension, we obtain an algorithm for calculating Steenrod squares.     

Sabitov polynomials for volumes of polyhedra in four dimensions

In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain monic polynomial with coefficients depending on the combinatorial type and on edge lengths of the polyhedron only. Moreover, the coefficients of this polynomial are polynomials in edge lengths of the polyhedron. This result implies that the volume of a simplicial polyhedron with fixed combinatorial type and edge lengths can take only finitely many values. In particular, this yields that the volume of a flexible polyhedron in a 3-dimensional Euclidean space is constant. Until now it has been unknown whether these results can be obtained in dimensions greater than 3. In this paper we prove that all these results hold for polyhedra in a 4-dimensional Euclidean space.     

Sails and Hilbert bases

A Klein polyhedron is the convex hull of the nonzero integral points of a simplicial coneC⊂ ℝⁿ. There are relationships between these polyhedra and the Hilbert bases of monoids of integral points contained in a simplicial cone. In the two-dimensional case, the set of integral points lying on the boundary of a Klein polyhedron contains a Hilbert base of the corresponding monoid. In general, this is not the case if the dimension is greater than or equal to three (e.g., [2]). However, in the three-dimensional case, we give a characterization of the polyhedra that still have this property. We give an example of such a sail and show that our criterion does not hold if the dimension is four. CEREMADE, University Paris 9. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 43–49, April–June, 2000. Translated by J.-O. Moussafir     

On inner Kan complexes in the category of dendroidal sets

The category of dendroidal sets is an extension of that of simplicial sets, suitable for defining nerves of operads rather than just of categories. In this paper, we prove some basic properties of inner Kan complexes in the category of dendroidal sets. In particular, we extend fundamental results for simplicial sets of Boardman and Vogt, of Cordier and Porter, and of Joyal to dendroidal sets.     

Symmetry,oriented matroids and two conjectures of Michel Las Vergnas

The paper has two parts. In the first part we survey the existing results on the cube conjecture of Las Vergnas. This conjecture claims that the orientation of the matroid of the cube is determined by the symmetries of the underlying matroid. The second part deals with Euclidean representations of matroids as geometric simplicial complexes defined by symmetry properties abstracting those of zonotopes. Both sections involve arguments concerning simplicial regions illustrating, once more, the fundamental importance of the simplex conjecture of Las Vergnas.     

Moment-angle manifolds and connected sums of simplicial spheres

Science China Mathematics - The connected sum is a fundamental operation in geometric topology and combinatorics. In this paper, we study the connection between connected sums of simplicial spheres...     

Quandle homology and complex volume

We introduce a new homology theory of quandles, called simplicial quandle homology, which is quite different from quandle homology developed by Carter et al. We construct a homomorphism from a quandle homology group to a simplicial quandle homology group. As an application, we obtain a method for computing the complex volume of a hyperbolic link only from its diagram.     

Subdivisions of Toric Complexes

We introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and we define their associated face rings. Abstract simplicial complexes and rational fans can be considered as toric complexes, and the face ring for toric complexes extends Stanley and Reisner’s face ring for abstract simplicial complexes [20] and Stanley’s face ring for rational fans [21]. Given a toric complex with defining ideal I for the face ring we give a geometrical interpretation of the initial ideals of I with respect to weight orders in terms of subdivisions of the toric complex generalizing a theorem of Sturmfels in [23]. We apply our results to study edgewise subdivisions of abstract simplicial complexes.     

New examples of Beauville surfaces

We find new mixed and unmixed Beauville structures for groups constructed as 2-quotients, i.e. quotients of 2-power order, of the fundamental group of a certain simplicial complex.     

New examples of Beauville surfaces

We find new mixed and unmixed Beauville structures for groups constructed as 2-quotients, i.e. quotients of 2-power order, of the fundamental group of a certain simplicial complex.     

A volume estimate for piecewise smooth metrics on simplicial complexes

We prove L. Green's volume estimate in the case of piecewise smooth Riemannian metrics on simplicial complexes.     

Proportionality principle for the simplicial volume of families of \mathbb Q -rank 1 locally symmetric spaces

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for $\mathbb Q $ -rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the proportionality principle for the locally finite simplicial volume and the relative simplicial volume of $\mathbb Q $ -rank $1$ locally symmetric spaces with amenable cusp groups established by Löh and Sauer [26].