This article deals with topological assumptions under which the minimal volume entropy of a closed manifold M, and more generally of a finite simplicial complex X, vanishes or is positive. These topological conditions are expressed in terms of the growth of the fundamental group of the fibers of maps from a given finite simplicial complex X to lower dimensional simplicial complexes P. We also give examples of finite simplicial complexes with zero simplicial volume and arbitrarily large minimal volume entropy.

Let $G$ be a finitely presented group. A new complexity called \textit{Karoubi-Weibel complexity} or \textit{covering type}, is defined for $G$. The construction is inspired by recent work of Karoubi and Weibel \cite{KW}, initially applied to topological spaces. We introduce a similar notion in combinatorial form in order to apply it to finitely presentable groups. Some properties of this complexity as well as a few examples of calculation/estimation for certain classes of finitely presentable groups are considered. Finally we give a few applications of complexity to some geometric problems, namely to the systolic area and the volume entropy of groups.

We introduce the volume entropy semi-norm and the systolic volume semi-norm in real homology and show that they satisfy functorial properties similar to the ones of the simplicial volume. Along the way, we also establish a roughly optimal upper bound on the systolic volume of the multiples of any homology class. Finally, we prove that the volume entropy semi-norm, the systolic volume semi-norm and the simplicial volume semi-norm are equivalent in every dimension.

This article deals with topological assumptions under which the minimal volume entropy of a closed manifold, and more generally of a finite simplicial complex, vanishes or is positive. In the first part of the article, we present complementing topological conditions expressed in terms of the growth of the fundamental group of the fibers of maps from a given finite simplicial complex to simplicial complexes of lower dimension which ensure that the minimal volume entropy of the simplicial complex either vanishes or is positive. We also give examples of finite simplicial complexes with zero simplicial volume and arbitrarily large minimal volume entropy. In the second part of the article, we present topological assumptions related to the exponential growth of certain subgroups in the fundamental group of a finite simplicial complex and to the topology of the loop space of its classifying space under which the minimal volume entropy is positive. Several examples are presented throughout the text.

We introduce the volume entropy semi-norm in real homology and show that it satisfies functorial properties similar to the ones of the simplicial volume. Answering a question of M. Gromov, we prove that the volume entropy semi-norm is equivalent to the simplicial volume semi-norm in every dimension. We also establish a roughly optimal upper bound on the systolic volume of the multiples of any homology class.

Twenty years ago Gromov asked about how large is the set of isomorphism classes of groups whose systolic area is bounded from above. This article introduces a new combinatorial invariant for finitely presentable groups called simplicial complexity that allows to obtain a quite satisfactory answer to his question. Using this new complexity, we also derive new results on systolic area for groups that specify its topological behaviour.

We define Dirichlet type series associated with homology length spectra of Riemannian, or Finsler, manifolds, or polyhedra, and investigate some of their analytical properties. As a consequence we obtain an inequality analogous to Gromov's classical intersystolic inequality, but taking the whole homology length spectrum into account rather than just the systole.

Given an integer homology class of a finitely presentable group, the systolic volume quantifies how tight could be a geometric realization of this class. In this paper, we study various aspects of this numerical invariant showing that it is a complex and powerful tool to investigate topological properties of homology classes of finitely presentable groups.

Given a pair (G,a) where G is a finitely presentable group and a is an integer homology class of this group, Gromov defined in his article "Filling Riemannian manifolds" a new numerical invariant associated to this pair called systolic volume. Our goal is to propose a systematic study of systolic volume as a function of the two variables G and a. In particular we focus on the distribution of the values of the systolic volume on the real line.

We study the stable norm on the first homology of a Riemannian polyhedron. In the one-dimensional case (metric graphs), the geometry of the unit ball of this norm is completely described by the combinatorial structure of the graph. For a smooth manifold of dimension ≥3 and using polyhedral techniques, we show that a large class of polytopes appears as unit ball of the stable norm associated to some metric conformal to a given one.

Nous étudions la constante systolique de la somme connexe de n exemplaires d’une variété M en fonction de ce nombre. Le comportement asymptotique de cette constante, connu dans le cas deux dimensionnel, demeure un problème ouvert dans les dimensions plus grandes que deux. Nous exhibons une borne supérieure, montrant ainsi que la croissance de la constante systolique en fonction de n est toujours plus lente que la croissance linéaire. La méthode utilisée est appliquée à l’étude du comportement systolique des revêtements cycliques en fonction du nombre de feuilles.

géométrie » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php).

Systols of dimension k for a Riemannian manifold of dimension n were introduced by M. Berger in 1972. The problem of intersystolic freedom (or (k,n-k)-freedom) deals with the supremum of the product of two supplementary dimensional systols, say k and n-k, when metric of M runs in the class of metrics with unit volume. Intersystolic freedom means that this supremum is equal to ∞. A few partial results in this direction were recently obtained by M. Katz, A. Suciu and the author. In the article we present a general theorem about the strong intersystolic freedom of arbitrary Riemannian polyhedrons. This result implies in particular the intersystolic freedom for any closed manifold.

Examples of nonformal simply connected symplectic manifolds are constructed.

A smooth manifold M is called symplectic if it carries a nondegenerate closed 2-form ω which is called a symplectic form. In this event a symplectic manifold means a pair (M,ω). Since the skew-symmetric form ω is nondegenerate, M is even-dimensional and moreover such manifold always carries an almost complex structure. Topology of symplectic manifolds is being intensively developed now ([18]) but still there are no many examples of compact symplectic manifolds. The simplest of them are Kähler manifolds and there are three constructions of new manifolds from old ones. These are a symplectic fibration ([25]), a blow up ([17]) and a fiber connect sum ([8]). The latter two constructions were first outlined in [11]. By results of Gromov ([10]) and Tischler ([26]), any compact symplectic manifold is diffeomorphic to a symplectic submanifold of a complex projective space. The problem posed by Weinstein was to find compact symplectic manifolds not carrying Kähler structure. First example of such manifold was found by Thurston ([25]) and the first simply connected example of such manifold was constructed by McDuff ([17]). Later Gompf constructed simply connected examples of the minimal possible dimension which equals four ([8]). An important property of Kähler manifolds is their formality established by Deligne, Griffiths, Morgan, and Sullivan in [5], which means that the rational homotopy type is completely determined by the rational cohomology ring. Formality of Kähler manifolds was used for distinguishing non simply connected symplectic manifolds with no Kähler structure in [2, 14]. For non simply connected spaces the notion of formality is only defined for nilpotent spaces, whereas for simply connected spaces it is always defined. Meanwhile the problem of existence of nonformal simply connected symplectic manifolds was open until now and moreover there was a conjecture that such manifolds do not exist (the Lupton–Oprea conjecture, [27]). We disprove it as follows

A smooth manifold M is called symplectic if it carries a nondegenerate closed 2-form ω which is called a symplectic form. In this event a symplectic manifold means a pair (M,ω). Since the skew-symmetric form ω is nondegenerate, M is even-dimensional and moreover such a manifold always carries an almost complex structure. Topology of symplectic manifolds is being intensively developed now ([18]) but still there are not many examples of compact symplectic manifolds. The simplest ones are Kähler manifolds and there are three constructions of new manifolds from old ones. These are a symplectic fibration ([25]), a blow up ([17]) and a fiber connect sum ([8]). The latter two constructions were first outlined in [11]. By results of Gromov ([10]) and Tischler ([26]), any compact symplectic manifold is diffeomorphic to a symplectic submanifold of a complex projective space. The problem posed by Weinstein was to find compact symplectic manifolds not carrying a Kähler structure. The first example of such a manifold was found by Thurston ([25]) and the first simply connected example of such a manifold was constructed by McDuff ([17]). Later Gompf constructed simply connected examples of the minimal possible dimension which equals four ([8]). An important property of Kähler manifolds is their formality established by Deligne, Griffiths, Morgan, and Sullivan in [5], which means that the rational homotopy type is completely determined by the rational cohomology ring. Formality of Kähler manifolds was used for distinguishing non simply connected symplectic manifolds with no Kähler structure in [2, 14]. For non simply connected spaces the notion of formality is only defined for nilpotent spaces, whereas for simply connected spaces it is always defined. Meanwhile the problem of existence of nonformal simply connected symplectic manifolds was open until now and moreover there was a conjecture that such manifolds do not exist (the Lupton–Oprea conjecture, [27]). We disprove it as follows