Mapping Torus Research Articles

COHERENCE, LOCAL INDICABILITY AND NONPOSITIVE IMMERSIONS

AbstractWe examine 2-complexes $X$ with the property that for any compact connected $Y$, and immersion $Y\rightarrow X$, either $\unicode[STIX]{x1D712}(Y)\leqslant 0$ or $\unicode[STIX]{x1D70B}_{1}Y=1$. The mapping torus of an endomorphism of a free group has this property. Every irreducible 3-manifold with boundary has a spine with this property. We show that the fundamental group of any 2-complex with this property is locally indicable. We outline evidence supporting the conjecture that this property implies coherence. We connect the property to asphericity. Finally, we prove coherence for 2-complexes with a stricter form of this property. As a corollary, every one-relator group with torsion is coherent.

A dynamical construction of Liouville domains

We first present a general construction of Liouville domains as partial mapping tori. Then we study two examples where the (partial) monodromies exhibit certain hyperbolic behavior in the sense of Dynamical Systems. The first example is based on Smale's attractor, a.k.a., solenoid; and the second example is based on certain hyperbolic toral automorphisms.

3d-3d correspondence for mapping tori

One of the main challenges in 3d-3d correspondence is that no existent approach offers a complete description of 3d mathcal{N} = 2 SCFT T [M3] — or, rather, a “collection of SCFTs” as we refer to it in the paper — for all types of 3-manifolds that include, for example, a 3-torus, Brieskorn spheres, and hyperbolic surgeries on knots. The goal of this paper is to overcome this challenge by a more systematic study of 3d-3d correspondence that, first of all, does not rely heavily on any geometric structure on M3 and, secondly, is not limited to a particular supersymmetric partition function of T [M3]. In particular, we propose to describe such “collection of SCFTs” in terms of 3d mathcal{N} = 2 gauge theories with “non-linear matter” fields valued in complex group manifolds. As a result, we are able to recover familiar 3-manifold invariants, such as Turaev torsion and WRT invariants, from twisted indices and half-indices of T [M3], and propose new tools to compute more recent q-series invariants hat{Z} (M3) in the case of manifolds with b1> 0. Although we use genus-1 mapping tori as our “case study,” many results and techniques readily apply to more general 3-manifolds, as we illustrate throughout the paper.

Irreducible nonsurjective endomorphisms of Fn are hyperbolic

Previously, Reynolds showed that any irreducible nonsurjective endomorphism can be represented by an irreducible immersion on a finite graph. We give a new proof of this and also show a partial converse holds when the immersion has connected Whitehead graphs with no cut vertices. The next result is a characterization of finitely generated subgroups of the free group that are invariant under an irreducible nonsurjective endomorphism. Consequently, irreducible nonsurjective endomorphisms are fully irreducible. The characterization and Reynolds' theorem imply that the mapping torus of an irreducible nonsurjective endomorphism is word-hyperbolic.

On the topology of metric f\u2013K-contact manifolds

We observe that the class of metric f–K-contact manifolds, which naturally contains that of K-contact manifolds, is closed under forming mapping tori of automorphisms of the structure. We show that the de Rham cohomology of compact metric f–K-contact manifolds naturally splits off an exterior algebra, and relate the closed leaves of the characteristic foliation to its basic cohomology.

Virtual Betti numbers of mapping tori of 3-manifolds

Given a reducible $3$-manifold $M$ with an aspherical summand in its prime decomposition and a homeomorphism $f\colon M\to M$, we construct a map of degree one from a finite cover of $M\rtimes_f S^1$ to a mapping torus of a certain aspherical $3$-manifold. We deduce that $M\rtimes_f S^1$ has virtually infinite first Betti number, except when all aspherical summands of $M$ are virtual $T^2$-bundles. This verifies all cases of a conjecture of T.-J. Li and Y. Ni, that any mapping torus of a reducible $3$-manifold $M$ not covered by $S^2\times S^1$ has virtually infinite first Betti number, except when $M$ is virtually $(\#_n T^2\rtimes S^1)\#(\#_mS^2\times S^1)$. Li-Ni's conjecture was recently confirmed by Ni with a group theoretic result, namely, by showing that there exists a $\pi_1$-surjection from a finite cover of any mapping torus of a reducible $3$-manifold to a certain mapping torus of $\#_m S^2\times S^1$ and using the fact that free-by-cyclic groups are large when the free group is generated by more than one element.

The simplicial volume of mapping tori of 3-manifolds

We prove that any mapping torus of a closed 3-manifold has zero simplicial volume. When the fiber is a prime 3-manifold, classification results can be applied to show vanishing of the simplicial volume, however the case of reducible fibers is by far more subtle. We thus analyse the possible self-homeomorphisms of reducible 3-manifolds, and use this analysis to produce an explicit representative of the fundamental class of the corresponding mapping tori. To this end, we introduce a new technique for understanding self-homeomorphisms of connected sums in arbitrary dimensions on the level of classifying spaces and for computing the simplicial volume. In particular, we extend our computations to mapping tori of certain connected sums in higher dimensions. Our main result completes the picture for the vanishing of the simplicial volume of fiber bundles in dimension four. Moreover, we deduce that dimension four together with the trivial case of dimension two are the only dimensions where all mapping tori have vanishing simplicial volume. As a group theoretic consequence, we derive an alternative proof of the fact that the fundamental group $G$ of a mapping torus of a 3-manifold $M$ is Gromov hyperbolic if and only if $M$ is virtually a connected sum $\# S^2\times S^1$ and $G$ does not contain $\mathbb{Z}^2$.

The Conley index for discrete dynamical systems and the mapping torus

The classical Conley index for flows is defined as a certain homotopy type. In the case of a discrete dynamical system, one usually considers the shift equivalence class of the so-called index map. This equivalence relation is rarely used in other contexts and not well understood in general. Here we propose using a topological invariant of the shift equivalence definition: The homotopy type of the mapping torus of the index map. Using a homotopy type offers new ways for comparing Conley indices–theoretically and numerically. We present some basic properties and examples, compare it to the definition via shift equivalence and sketch an idea for its construction using rigorous numerics.

Reducible mapping class of the canonical Heegaard splitting in a mapping torus

Let S be an orientable closed surface with genus at least two. From S×I, for a given orientation-reversing homeomorphism f from S×{1} to S×{0}, there is an orientable closed 3-manifold Mf=S×I/f which is called a mapping torus. It is known that Mf admits a canonical Heegaard splitting H1∪ΣH2. By the construction of Namazi [H.Namazi, Topology Appl. 154 (2007), no. 16, 2939-2949], the mapping class group of this Heegaard splitting, denoted by Mod(Σ;H1,H2), contains a reducible mapping class which has infinitely order. So it is interesting to know that for a given element in Mod(Σ;H1,H2), whether it is reducible or not.Using the translation length of f in the curve complex, we prove that if f is the identity map or its translation length is at least 8, then each element of Mod(Σ;H1,H2) is reducible.

The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups

We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional. In an appendix we use the same techniques to give a cohomological proof of a lower bound, originally due to Brock, on the volume of the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed surface in terms of its Teichmüller translation distance.

Regular genus and gem-complexity of some mapping tori

In this article, we construct a crystallization of the mapping torus of some (PL) homeomorphisms $f:M \to M$ for a certain class of PL-manifolds $M$. These yield upper bounds for gem-complexity and regular genus of a large class of PL-manifolds. The bound for the regular genus is sharp for the mapping torus of some (PL) homeomorphisms $f:M \to M$, where $M$ is $\mathbb{RP}^2$, $\mathbb{RP}^2\#\mathbb{RP}^2$, $\mathbb{S}^1\times \mathbb{S}^1$, $\mathbb{RP}^3$, $\mathbb{S}^{2} \times \mathbb{S}^1$, $\mathbb{S}^{\hspace{.2mm}2} \mbox{$\times \hspace{-2.6mm}_{-}$} \, \mathbb{S}^{\hspace{.1mm}1}$ or $\mathbb{S}^d$. In particular, for $M=\mathbb{S}^{d-1} \times \mathbb{S}^1$ or $\mathbb{S}^{\hspace{.2mm}d-1} \mbox{$\times\hspace{-2.6mm}_{-}$} \, \mathbb{S}^{\hspace{.1mm}1}$, our construction gives a crystallization of a mapping torus of a (PL) homeomorphism $f:M \to M$ with regular genus $d^2-d$. As a consequence, we prove the existence of an orientable mapping torus of a (PL) homeomorphism $f:(\mathbb{S}^{2} \times \mathbb{S}^1)\to (\mathbb{S}^{2} \times \mathbb{S}^1)$ with regular genus 6. This disproves a conjecture of Spaggiari which states that regular genus six characterizes the topological product $\mathbb{RP}^3 \times \mathbb{S}^1$ among closed connected prime orientable PL $4$-manifolds.

Twisted monodromy homomorphisms and Massey products

Let ϕ:M→M be a diffeomorphism of a C∞ compact connected manifold, and X its mapping torus. There is a natural fibration p:X→S1, denote by ξ∈H1(X,Z) the corresponding cohomology class. Let ρ:π1(X,x0)→GL(n,C) be a representation (here x0∈M); denote by H⁎(X,ρ) the corresponding twisted cohomology of X. Denote by ρ0 the restriction of ρ to π1(M,x0), and by ρ0⁎ the antirepresentation conjugate to ρ0. We construct from these data the twisted monodromy homomorphismϕ⁎ of the group H⁎(M,ρ⁎0). This homomorphism is a generalization of the homomorphism induced by ϕ in the ordinary homology of M. The aim of the present work is to establish a relation between Massey products in H⁎(X,ρ) and Jordan blocks of ϕ⁎.We have a natural pairing H⁎(X,C)⊗H⁎(X,ρ)→H⁎(X,ρ); one can define Massey products of the form 〈ξ,…,ξ,x〉, where x∈H⁎(X,ρ). The Massey product containing r terms ξ will be denoted by 〈ξ,x〉r; we say that the length of this product is equal to r. Denote by Mk(ρ) the maximal length of a non-zero Massey product 〈ξ,x〉r for x∈Hk(X,ρ). Given a non-zero complex number λ define a representation ρλ:π1(X,x0)→GL(n,C) as follows: ρλ(g)=λξ(g)⋅ρ(g). Denote by Jk(ϕ⁎,λ) the maximal size of a Jordan block of eigenvalue λ of the automorphism ϕ⁎ in the homology of degree k.The main result of the paper says that Mk(ρλ)=Jk(ϕ⁎,λ). In particular, ϕ⁎ is diagonalizable, if a suitable formality condition holds for the manifold X. This is the case if X a compact Kähler manifold and ρ is a semisimple representation. The proof of the main theorem is based on the fact that the above Massey products can be identified with differentials in a Massey spectral sequence, which in turn can be explicitly computed in terms of the Jordan normal form of ϕ⁎.

Largest projections for random walks and shortest curves in random mapping tori

We show that the largest subsurface projection distance between a marking and its image under the nth step of a random walk grows logarithmically in n, with probability approaching 1 as n tends to infinity. Our setup is general and also applies to (relatively) hyperbolic groups and to $\mathrm{Out}(F_n)$. We then use this result to prove Rivin's conjecture that for a random walk $(w_n)$ on the mapping class group, the shortest geodesic in the hyperbolic mapping torus $M_{w_n}$ has length on the order of $1/ \log^2(n)$.

Asymptotic expansions of the Witten–Reshetikhin–Turaev invariants of mapping tori I

In this paper we engage in a general study of the asymptotic expansion of the Witten–Reshetikhin–Turaev invariants of mapping tori of surface mapping class group elements. We use the geometric construction of the Witten–Reshetikhin–Turaev topological quantum field theory via the geometric quantization of moduli spaces of flat connections on surfaces. We identify assumptions on the mapping class group elements that allow us to provide a full asymptotic expansion. In particular, we show that our results apply to all pseudo-Anosov mapping classes on a punctured torus and show by example that our assumptions on the mapping class group elements are strictly weaker than hitherto successfully considered assumptions in this context. The proof of our main theorem relies on our new results regarding asymptotic expansions of oscillatory integrals, which allows us to go significantly beyond the standard transversely cut out assumption on the fixed point set. This makes use of the Picard–Lefschetz theory for Laplace integrals.

Classification of closed manifolds with $${\mathrm{Sol}_1}^{4}$$ Sol 1 4 -geometry

Every closed 4-manifold with a geometric structure modeled on the Lie group $${\mathrm{Sol}_1}^{4}$$ , or equivalently, infra-solvmanifold of $${\mathrm{Sol}_1}^{4}$$ , is either a mapping torus of an infra-nilmanifold of the 3-dimensional geometry $${\mathrm{Nil}}$$ , or a union of twisted I-bundles over infra-nilmanifolds of $${\mathrm{Nil}}$$ . We determine ab initio presentations for the fundamental groups of the $${\mathrm{Sol}_1}^{4}$$ -geometry manifolds.

Normalized entropy versus volume for pseudo-Anosovs

Thanks to a recent result by Jean-Marc Schlenker, we establish an explicit linear inequality between the normalized entropies of pseudo-Anosov automorphisms and the hyperbolic volumes of their mapping tori. As its corollaries, we give an improved lower bound for values of entropies of pseudo-Anosovs on a surface with fixed topology, and a proof of a slightly weaker version of the result by Farb, Leininger and Margalit first, and by Agol later, on finiteness of cusped manifolds generating surface automorphisms with small normalized entropies. Also, we present an analogous linear inequality between the Weil-Petersson translation distance of a pseudo-Anosov map (normalized by multiplying the square root of the area of a surface) and the volume of its mapping torus, which leads to a better bound.

Invariant incompressible surfaces in reducible 3-manifolds

We study the effect of the mapping class group of a reducible 3-manifold $M$ on each incompressible surface that is invariant under a self-homeomorphism of $M$ . As an application of this study we answer a question of F. Rodriguez Hertz, M. Rodriguez Hertz, and R. Ures: a reducible 3-manifold admits an Anosov torus if and only if one of its prime summands is either the 3-torus, the mapping torus of $-\text{id}$ , or the mapping torus of a hyperbolic automorphism.

On the quantum Teichmüller invariants of fibred cusped 3-manifolds

We show that the reduced quantum hyperbolic invariants of pseudo-Anosov diffeomorphisms of punctured surfaces are intertwiners of local representations of the quantum Teichm\"uller spaces. We characterize them as the only intertwiners that satisfy certain natural cut-and-paste operations of topological quantum field theories and such that their traces define invariants of mapping tori.

Fibered commensurability and arithmeticity of random mapping tori

We consider a random walk on the mapping class group of a surface of finite type. We assume that the random walk is determined by a probability measure whose support is finite and generates a non-elementary subgroup H . We further assume that H is not consisting only of lifts with respect to any one covering. Then we prove that the probability that such a random walk gives a non-minimal mapping class in its fibered commensurability class decays exponentially. As an application of the minimality, we prove that for the case where a surface has at least one puncture, the probability that a random walk gives mapping classes with arithmetic mapping tori decays exponentially. We also prove that a random walk gives rise to asymmetric mapping tori with exponentially high probability for closed case.

Non ambiguous structures on 3-manifolds and quantum symmetry defects

The state sums defining the quantum hyperbolic invariants (QHI) of hyperbolic oriented cusped 3-manifolds can be split in a “symmetrization” factor and a “reduced” state sum. We show that these factors are invariants on their own, that we call “symmetry defects” and “reduced QHI,” provided the manifolds are endowed with an additional “non ambiguous structure,” a new type of combinatorial structure that we introduce in this paper. A suitably normalized version of the symmetry defects applies to compact 3-manifolds endowed with PSL _2 (\mathbb C -characters, beyond the case of cuspedmanifolds. Given a manifold M with non empty boundary, we provide a partial “holographic” description of the nonambiguous structures in terms of the intrinsic geometric topology of \partial M . Special instances of non ambiguous structures can be defined by means of taut triangulations, and the symmetry defects have a particularly nice behaviour on such “taut structures.” Natural examples of taut structures are carried by any mapping torus with punctured fibre of negative Euler characteristic, or by sutured manifold hierarchies. For a cusped hyperbolic 3-manifold M which bres over S^1 , we address the question of determining whether the fibrations over a same fibered face of the Thurston ball define the same taut structure. We describe a few examples in detail. In particular, they show that the symmetry defects or the reduced QHI can distinguish taut structures associated to different fibrations of M . To support the guess that all this is an instance of a general behaviour of state sum invariants of 3-manifolds based on some theory of 6 j -symbols, finally we describe similar results about reduced Turaev–Viro invariants.