Category theory; homological algebra -Homological algebra -Other (co)homology theories. msc | Group theory and generalizations -Connections with homological algebra and category theory -Cohomology of groups. msc | Algebraic topology -Homology and cohomology theories -Singular theory. msc | Manifolds and cell complexes -Topological manifolds -Algebraic topology of manifolds. msc | Dynamical systems and ergodic theory -Low-dimensional dynamical systems -Maps of the circle. msc | Dynamical systems and ergodic theory -Smooth dynamical systems: general theory -Dynamics of group actions other than $. msc | Differential geometry -Global differential geometry -Global geometric and topological methods (á la Gromov); differential geometric analysis on metric spaces. msc | Manifolds and cell complexes -Low-dimensional topology -Topological methods in group theory. msc | Manifolds and cell complexes -Differential topology -Characteristic classes and numbers. msc Classification: LCC QA612.3 .F75 2017 | DDC 512/.55-dc23 LC record available at https://lccn.loc.gov/2017023394Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research.

We classify the orientable finite-volume hyperbolic 3-manifolds having nonempty compact totally geodesic boundary and admitting an ideal triangulation with at most four tetrahedra. We also compute the volume of all such manifolds, we describe their canonical Kojima decomposition, and we discuss manifolds having cusps.The manifolds built from one or two tetrahedra were previously known. There are 151 different manifolds built from three tetrahedra, realizing 18 different volumes. Their Kojima decomposition always consists of tetrahedra (but occasionally requires four of them). And there is a single cusped manifold, that we can show to be a knot complement in a genus-2 handlebody. Concerning manifolds built from four tetrahedra, we show that there are 5033 different ones, with 262 different volumes. The Kojima decomposition consists either of tetrahedra (as many as eight of them in some cases), or of two pyramids, or of a single octahedron. There are 30 manifolds having a single cusp, and one having two cusps.Our results were obtained with the aid of a computer. The complete list of manifolds (in SnapPea format) and full details on their invariants are available on the world wide web.MSC (2000): 57M50 (primary), 57M20, 57M27 (secondary). This paper is devoted to the class of all orientable finite-volume hyperbolic 3manifolds having non-empty compact totally geodesic boundary and admitting a minimal ideal triangulation with either three or four but no fewer tetrahedra. We describe the theoretical background and experimental results of a computer program that has enabled us to classify all such manifolds. (The case of manifolds obtained from two tetrahedra was previously dealt with in [8]). We also provide an overall discussion of the most important features of all these manifolds, namely of:• their volumes;• the shape of their canonical Kojima decomposition; Preliminaries and statementsWe consider in this paper the class H of orientable 3-manifolds M having compact non-empty boundary ∂M and admitting a complete finite-volume hyperbolic metric with respect to which ∂M is totally geodesic. It is a well-known fact [9] that such an M is the union of a compact portion and some cusps based on tori, so it has a natural compactification obtained by adding some tori. The elements of H are regarded up to homeomorphism, or equivalently isometry (by Mostow's rigidity). Candidate hyperbolic manifolds Let us now introduce the class H of 3-manifolds M such that:• M is orientable, compact, boundary-irreducible and acylindrical;• ∂M consists of some tori (possibly none of them) and at least one surface of negative Euler characteristic.The basic theory of hyperbolic manifolds implies that, up to identifying a manifold with its natural compactification, the inclusion H ⊂ H holds. We note that, by Thurston's hyperbolization, an element of H actually lies in H if and only if it is atoroidal. However we do not require atoroidality in the definition of H, for a reason that will be mentioned later in this section and explained in detail in...

Abstract. We extend to the context of hyperbolic 3-manifolds with geodesic boundary Thurston's approach to hyperbolization by means of geometric triangulations. In particular, we introduce moduli for (partially) truncated hyperbolic tetrahedra, and we discuss consistency and completeness equations. Moreover, building on previous work of Ushijima, we extend Weeks' tilt formula algorithm, which computes the Epstein-Penner canonical decomposition, to an algorithm that computes the Kojima decomposition.Our theory has been exploited to classify all the orientable finite-volume hyperbolic 3-manifolds with non-empty compact geodesic boundary admitting an ideal triangulation with at most four tetrahedra. The theory is particularly interesting in the case of complete finite-volume manifolds with geodesic boundary in which the boundary is non-compact. We include this case using a suitable adjustment of the notion of ideal triangulation, and we show how this case arises within the theory of knots and links.

Abstract. Simplicial volumes measure the complexity of fundamental cycles of manifolds. In this article, 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 simplicial volume is monotone with respect to weak containment of measure preserving actions and yields upper bounds on (integral) homology growth.Using ergodic theory we prove that simplicial volume, integral foliated simplicial volume and stable integral simplicial volume coincide for closed hyperbolic 3-manifolds and closed aspherical manifolds with amenable residually finite fundamental group (being equal to zero in the latter case).However, we show that integral foliated simplicial volume and the classical simplicial volume do not coincide for hyperbolic manifolds of dimension at least 4.

The goal of this paper is to describe and clarify as much as possible the 3dimensional topology underlying the Helmholtz cuts method, which occurs in a wide theoretic and applied literature about Electromagnetism, Fluid dynamics and Elasticity on domains of the ordinary space R 3 . We consider two classes of bounded domains that satisfy mild boundary conditions and that become "simple" after a finite number of disjoint cuts along properly embedded surfaces. For the first class (Helmholtz), "simple" means that every curl-free smooth vector field admits a potential. For the second (weakly-Helmholtz), we only require that a potential exists for the restriction of every curl-free smooth vector field defined on the whole initial domain. By means of classical and rather elementary facts of 3-dimensional geometric and algebraic topology, we give an exhaustive description of Helmholtz domains, realizing that their topology is forced to be quite elementary (in particular, Helmholtz domains with connected boundary are just possibly knotted handlebodies, and the complement of any non-trivial link is not Helmholtz). The discussion about weakly-Helmholtz domains is a bit more advanced, and their classification appears to be a quite difficult issue. Nevertheless, we provide several interesting characterizations of them and, in particular, we point out that the class of links with weakly-Helmholtz complements eventually coincides with the class of the so-called homology boundary links, that have been widely studied in Knot Theory.2000 Mathematics Subject Classification. 57-02, 76-02 (primary); 57M05, 57M25, 57R19 (secondary).