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Stephan Tillmann

Publications and source records attributed to Stephan Tillmann.

At least 37 records · Page 2Linked to original sources
arXiv2019

Traversing three-manifold triangulations and spines

A celebrated result concerning triangulations of a given closed 3-manifold is that any two triangulations with the same number of vertices are connected by a sequence of so-called 2-3 and 3-2 moves. A similar result is known for ideal triangulations of topologically finite non-compact 3-manifolds. These results build on classical work that goes back to Alexander, Newman, Moise, and Pachner. The key special case of 1-vertex triangulations of closed 3-manifolds was independently proven by Matveev and Piergallini. The general result for closed 3-manifolds can be found in work of Benedetti and Petronio, and Amendola gives a proof for topologically finite non-compact 3-manifolds. These results (and their proofs) are phrased in the dual language of spines. The purpose of this note is threefold. We wish to popularise Amendola's result; we give a combined proof for both closed and non-compact manifolds that emphasises the dual viewpoints of triangulations and spines; and we give a proof replacing a key general position argument due to Matveev with a more combinatorial argument inspired by the theory of subdivisions.

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arXiv2018

Determining the trisection genus of orientable and non-orientable PL 4-manifolds through triangulations

Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. This paper improves and implements an algorithm due to Bell, Hass, Rubinstein and Tillmann to compute trisections using triangulations, and extends it to non-orientable 4-manifolds. Lower bounds on trisection genus are given in terms of Betti numbers and used to determine the trisection genus of all standard simply connected PL 4-manifolds. In addition, we construct trisections of small genus directly from the simplicial structure of triangulations using the Budney-Burton census of closed triangulated 4-manifolds. These experiments include the construction of minimal genus trisections of the non-orientable 4-manifolds $S^3 \tilde{\times} S^1$ and $\mathbb{R}P^4$.

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arXiv2018

Reflections on trisection genus

The Heegaard genus of a 3-manifold, as well as the growth of Heegaard genus in its finite sheeted cover spaces, has extensively been studied in terms of algebraic, geometric and topological properties of the 3-manifold. This note shows that analogous results concerning the trisection genus of a smooth, orientable 4-manifold have more general answers than their counterparts for 3-manifolds. In the case of hyperbolic 4-manifolds, upper and lower bounds are given in terms of volume and a trisection of the Davis manifold is described.

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arXiv2018

Ideal points of character varieties, algebraic non-integral representations, and undetected closed essential surfaces in 3-manifolds

Closed essential surfaces in a three-manifold can be detected by ideal points of the character variety or by algebraic non-integral representations. We give examples of closed essential surfaces not detected in either of these ways. For ideal points, we use Chesebro's module-theoretic interpretation of Culler-Shalen theory. As a corollary, we construct an infinite family of closed hyperbolic Haken 3-manifolds with no algebraic non-integral representations into PSL(2, C), resolving a question of Shanuel and Zhang.

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arXiv2018

Deforming convex projective manifolds

We study a properly convex real projective manifold with (possibly empty) compact, strictly convex boundary, and which consists of a compact part plus finitely many convex ends. We extend a theorem of Koszul which asserts that for a compact manifold without boundary the holonomies of properly convex structures form an open subset of the representation variety. We also give a relative version for non-compact (G,X)-manifolds of the openess of their holonomies.

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arXiv2018

Moduli spaces of real projective structures on surfaces: Notes on a paper by V.V. Fock and A.B. Goncharov

These notes grew out of our learning and applying the methods of Fock and Goncharov concerning moduli spaces of real projective structures on surfaces with ideal triangulations. We give a self-contained treatment of Fock and Goncharov's description of the moduli space of framed marked properly convex projective structures with minimal or maximal ends, and deduce results of Marquis and Goldman as consequences. We also discuss the Poisson structure on moduli space and its relationship to Goldman's Poisson structure on the character variety.

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arXiv2017

Z2-Thurston Norm and Complexity of 3-Manifolds, II

In this sequel to earlier papers by three of the authors, we obtain a new bound on the complexity of a closed 3--manifold, as well as a characterisation of manifolds realising our complexity bounds. As an application, we obtain the first infinite families of minimal triangulations of Seifert fibred spaces modelled on Thurston's geometry $\widetilde{\text{SL}_2(\mathbb{R})}.$

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arXiv2017

Multisections of piecewise linear manifolds

Recently Gay and Kirby described a new decomposition of smooth closed $4$-manifolds called a trisection. This paper generalises Heegaard splittings of $3$-manifolds and trisections of $4$-manifolds to all dimensions, using triangulations as a key tool. In particular, we prove that every closed piecewise linear $n$-manifold has a multisection, i.e. can be divided into $k+1$ $n$-dimensional $1$-handlebodies, where $n=2k+1$ or $n=2k$, such that intersections of the handlebodies have spines of small dimensions. Several applications, constructions and generalisations of our approach are given.

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arXiv2017

Computing trisections of 4-manifolds

Algorithms that decompose a manifold into simple pieces reveal the geometric and topological structure of the manifold, showing how complicated structures are constructed from simple building blocks. This note describes a way to algorithmically construct a trisection, which describes a $4$-dimensional manifold as a union of three $4$-dimensional handlebodies. The complexity of the $4$-manifold is captured in a collection of curves on a surface, which guide the gluing of the handelbodies. The algorithm begins with a description of a manifold as a union of pentachora, or $4$-dimensional simplices. It transforms this description into a trisection. This results in the first explicit complexity bounds for the trisection genus of a $4$-manifold in terms of the number of pentachora ($4$-simplices) in a triangulation.

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arXiv2017

Unravelling the Dodecahedral Spaces

The hyperbolic dodecahedral space of Weber and Seifert has a natural non-positively curved cubulation obtained by subdividing the dodecahedron into cubes. We show that the hyperbolic dodecahedral space has a 6-sheeted irregular cover with the property that the canonical hypersurfaces made up of the mid-cubes give a very short hierarchy. Moreover, we describe a 60-sheeted cover in which the associated cubulation is special. We also describe the natural cubulation and covers of the spherical dodecahedral space (aka Poincaré homology sphere).

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arXiv2017

Tessellating the moduli space of strictly convex projective structures on the once-punctured torus

We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of framed strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Goncharov, the action of the mapping class group as well as algorithmic real algebraic geometry. We also show that the decorated moduli space of framed strictly convex real projective structures of finite volume on the thrice-punctured sphere has a natural cell decomposition.

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arXiv2017

Linear representations of 3-manifold groups over rings

The fundamental groups of compact 3-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by only allowing finite groups of the form $PGL(2,R),$ where $R$ is some finite commutative ring with identity. We give an equivalent formulation of Luo's conjecture via faithful representations and provide various examples and a counterexample.

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arXiv2016

Groups and polytopes

In a series of papers the authors associated to an $L^2$-acyclic group $Γ$ an invariant $\mathcal{P}(Γ)$ that is a formal difference of polytopes in the vector space $H_1(Γ;\Bbb{R})$. This invariant is in particular defined for most 3-manifold groups, for most 2-generator 1-relator groups and for all free-by-cyclic groups. In most of the above cases the invariant can be viewed as an actual polytope. In this survey paper we will recall the definition of the polytope invariant $\mathcal{P}(Γ)$ and we state some of the main properties. We conclude with a list of open problems.

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arXiv2016

Pseudo-developing maps for ideal triangulations II: Positively oriented ideal triangulations of cone-manifolds

We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescribed cone angles at all edges is (if non-empty) a smooth complex manifold of dimension the sum of the genera of the vertex links. Moreover, we show that the complex lengths of a collection of peripheral elements give a local holomorphic parameterisation of this manifold.

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arXiv2015

A new combinatorial class of 3-manifold triangulations

We define a new combinatorial class of triangulations of closed 3-manifolds, satisfying a weak version of 0-efficiency combined with a weak version of minimality, and study them using twisted squares. As an application, we obtain strong restrictions on the topology of a 3-manifold from the existence of non-smooth maxima of the volume function on the space of circle-valued angle structures.

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arXiv2015

A birationality result for character varieties

Let M be an orientable, cusped hyperbolic 3-manifold of finite volume. We show that the restriction map from a Dehn surgery component in the PSL(2,C)-character variety of M to the character variety of the boundary of M is a birational isomorphism onto its image. This generalises a result by Nathan Dunfield. A key step in our proof is the exactness of Craig Hodgson's volume differential on the eigenvalue variety.

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arXiv2015

Thurston norm via Fox calculus

In 1976 Thurston associated to a $3$-manifold $N$ a marked polytope in $H_1(N;\mathbb{R}),$ which measures the minimal complexity of surfaces representing homology classes and determines all fibered classes in $H^1(N;\mathbb{R})$. Recently the first and the last author associated to a presentation $π$ with two generators and one relator a marked polytope in $H_1(π;\mathbb{R})$ and showed that it determines the Bieri-Neumann-Strebel invariant of $π$. In this paper, we show that if the fundamental group of a 3-manifold $N$ admits such a presentation $π$, then the corresponding marked polytopes in $H_1(N;\mathbb{R})=H_1(π;\mathbb{R})$ agree.

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