arXiv subjects
Publications and source records attributed to Bruno Martelli.
The geometrisation theorem of 3-manifolds was conjectured by Thurston the 1980s and proved by Perelman in the 2000s. This is an overview on the subject. We explain the content of the theorem and describe its effects in various situations.
This paper is an introduction to Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries. It consists of essentially two parts that could be read independently. In the first we introduce non-obtuse polyhedra in the spherical, Euclidean, and hyperbolic spaces, and prove various fundamental theorems originated from Andreev, Coxeter, and Vinberg. In the second we introduce Coxeter polyhedra and use them to describe regular, semiregular, and uniform polyhedra and tessellations, mostly via the Wythoff construction.
We know from previous work with Italiano and Migliorini that there exists some hyperbolic 5-manifold that fibers over the circle. Here we build one example where the monodromy is a "pseudo-Anosov homeomorphism" of the 4-dimensional fiber, in a way that is surprisingly similar to the familiar and beautiful two-dimensional picture of Nielsen and Thurston for surfaces. This fact has various consequences: (1) There is a compact smooth 4-manifold $M$ such that no non-trivial class in $H_2(M)$ is represented by immersed tori, and infinitely many classes are represented by smoothly embedded genus two surfaces. (2) There is a compact locally CAT(0) space $Y$ such that $\pi_1(Y)$ is not hyperbolic and does not contain $\mathbb Z \times \mathbb Z$. The latter answers a question of Gromov, known as the Closing Flat Problem.
The Davis hyperbolic four-manifold $\mathcal{D}$ is not almost-complex, so that its Seiberg-Witten invariants corresponding to zero-dimensional moduli spaces are vanishing by definition. In this paper, we show that all the Seiberg-Witten invariants involving higher-dimensional moduli spaces also vanish. Our proof involves the adjunction inequalities corresponding to 864 genus two totally geodesic surfaces embedded inside $\mathcal{D}$.
Ivansic proved that there is a link $L$ of five tori in $S^4$ with hyperbolic complement. We describe $L$ explicitly with pictures, study its properties, and discover that $L$ is in many aspects similar to the Borromean rings in $S^3$. In particular the following hold: (1) Any two tori in $L$ are unlinked, but three are not; (2) The complement $M = S^4 \setminus L$ is integral arithmetic hyperbolic; (3) The symmetry group of $L$ acts $k$-transitively on its components for all $k$; (4) The double branched covering over $L$ has geometry $\mathbb H^2 \times \mathbb H^2$; (5) The fundamental group of $M$ has a nice presentation via commutators; (6) The Alexander ideal has an explicit simple description; (7) Every class $x \in H^1(M,Z) = Z^5$ with non-zero xi is represented by a perfect circle-valued Morse function; (8) By longitudinal Dehn surgery along $L$ we get a closed 4-manifold with fundamental group $Z^5$; (9) The link $L$ can be put in perfect position. This leads also to the first descriptions of a cusped hyperbolic 4-manifold as a complement of tori in $\mathbb{RP}^4$ and as a complement of some explicit Lagrangian tori in the product of two surfaces of genus two.
Let $N$ be a complete finite-volume hyperbolic $n$-manifold. An efficient cycle for $N$ is the limit (in an appropriate measure space) of a sequence of fundamental cycles whose $\ell^1$-norm converges to the simplicial volume of $N$. Gromov and Thurston's smearing construction exhibits an explicit efficient cycle, and Jungreis and Kuessner proved that, in dimension $n\geq 3$, such cycle actually is the unique efficient cycle for a huge class of finite volume hyperbolic manifolds, including all the closed ones. In this paper we prove that, for $n\geq 3$, the class of finite-volume hyperbolic manifolds for which the uniqueness of the efficient cycle does not hold is exactly the commensurability class of the figure-8 knot complement (or, equivalently, of the Gieseking manifold).
We study how the spin structures on finite-volume hyperbolic n-manifolds restrict to cusps. When a cusp cross-section is a (n-1)-torus, there are essentially two possible behaviours: the spin structure is either bounding or Lie. We show that in every dimension n there are examples where at least one cusp is Lie, and in every dimension n <= 8 there are examples where all the cusps are bounding. By work of C. Bar, this implies that the spectrum of the Dirac operator is R in the first case, and discrete in the second. We therefore deduce that there are cusped hyperbolic manifolds whose spectrum of the Dirac operator is R in all dimensions, and whose spectrum is discrete in all dimensions n <= 8.
We construct some cusped finite-volume hyperbolic $n$-manifolds $M_n$ that fiber algebraically in all the dimensions $5\leq n \leq 8$. That is, there is a surjective homomorphism $π_1(M_n) \to \mathbb Z$ with finitely generated kernel. The kernel is also finitely presented in the dimensions $n=7, 8$, and this leads to the first examples of hyperbolic $n$-manifolds $\widetilde M_n$ whose fundamental group is finitely presented but not of finite type. These $n$-manifolds $\widetilde M_n$ have infinitely many cusps of maximal rank and hence infinite Betti number $b_{n-1}$. They cover the finite-volume manifold $M_n$. We obtain these examples by assigning some appropriate colours and states to a family of right-angled hyperbolic polytopes $P_5, \ldots, P_8$, and then applying some arguments of Jankiewicz, Norin, Wise and Bestvina, Brady. We exploit in an essential way the remarkable properties of the Gosset polytopes dual to $P_n$, and the algebra of integral octonions for the crucial dimensions $n=7,8$.
We prove that every complete finite-volume hyperbolic 3-manifold $M$ that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold $W$, which is also tessellated into right-angled regular polytopes (120-cells and ideal 24-cells). If $M$ is connected, then Vol($W$) < $2^{49}$Vol($M$). This applies for instance to the Borromean link complement. As a consequence, the Borromean link complement bounds geometrically a hyperbolic 4-manifold.
This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds. It contains complete proofs of Mostow's rigidity, the thick-thin decomposition, Thurston's classification of the diffeomorphisms of surfaces (via Bonahon's geodesic currents), the prime and JSJ decomposition, the topological and geometric classification of Seifert manifolds, and Thurston's hyperbolic Dehn filling Theorem.
We give examples of hyperbolic groups which contain subgroups that are of type $\mathscr{F}_{3}$ but not of type $\mathscr{F}_{4}$. These groups are obtained by Dehn filling starting from a non-uniform lattice in ${\rm PO}(8,1)$ which was previously studied by Italiano, Martelli and Migliorini.
We exhibit some finite-volume cusped hyperbolic 5-manifolds that fiber over the circle. These include the smallest hyperbolic 5-manifold known, discovered by Ratcliffe and Tschantz. As a consequence, we build a finite type subgroup of a hyperbolic group that is not hyperbolic.
We exhibit some (compact and cusped) finite-volume hyperbolic four-manifolds M with perfect circle-valued Morse functions, that is circle-valued Morse functions $f\colon M \to S^1$ with only index 2 critical points. We construct in particular one example where every generic circle-valued function is homotopic to a perfect one. An immediate consequence is the existence of infinitely many finite-volume (compact and cusped) hyperbolic 4-manifolds $M$ having a handle decomposition with bounded numbers of 1- and 3-handles, so with bounded Betti numbers $b_1(M)$, $b_3(M)$ and rank of $\pi_1(M)$.
A closed connected hyperbolic $n$-manifold bounds geometrically if it is isometric to the geodesic boundary of a compact hyperbolic $(n+1)$-manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many manifolds that bound geometrically in every dimension. We construct here infinitely many explicit examples in dimension $n=3$ using right-angled dodecahedra and $120$-cells and a simple colouring technique introduced by M. Davis and T. Januszkiewicz. Namely, for every $k\geqslant 1$, we build an orientable compact closed $3$-manifold tessellated by $16k$ right-angled dodecahedra that bounds a $4$-manifold tessellated by $32k$ right-angled $120$-cells. A notable feature of this family is that the ratio between the volumes of the $4$-manifolds and their boundary components is constant and, in particular, bounded.
We show that every plumbing of disc bundles over surfaces whose genera satisfy a simple inequality may be embedded as a convex submanifold in some closed hyperbolic four-manifold. In particular its interior has a geometrically finite hyperbolic structure that covers a closed hyperbolic four-manifold.
We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n \geq 4$. The core of the argument is the construction of a compact orientable hyperbolic $4$-manifold $M$ that contains a surface $S$ of genus $3$ with self intersection $1$. The $4$-manifold $M$ has an odd intersection form and is hence not spin. It is built by carefully assembling some right angled $120$-cells along a pattern inspired by the minimum trisection of $\mathbb{C}\mathbb{P}^2$. The manifold $M$ is also the first example of a compact orientable hyperbolic $4$-manifold satisfying any of these conditions: 1) $H_2(M,\mathbb{Z})$ is not generated by geodesically immersed surfaces. 2) There is a covering $\tilde{M}$ that is a non-trivial bundle over a compact surface.
We classify all the exceptional Dehn surgeries on the minimally twisted chain links with six and seven components.
We study the set of all closed oriented smooth 4-manifolds experimentally, according to a suitable complexity defined using Turaev's shadows. This complexity roughly measures how complicated the 2-skeleton of the 4-manifold is. We characterise here all the closed oriented 4-manifolds that have complexity at most one. They are generated by a certain set of 20 blocks, that are some basic 4-manifolds with boundary consisting of copies of $S^2 \times S^1$, plus connected sums with some copies of $\mathbb{CP}^2$ with either orientation. All the manifolds generated by these blocks are doubles. Many of these are doubles of 2-handlebodies and are hence efficiently encoded using finite presentations of groups. In contrast to the complexity zero case, in complexity one there are also plenty of doubles that are not doubles of 2-handlebodies, like for instance $\mathbb{RP}^3 \times S^1$.