This is the second in a series of papers studying the relationship between Rohlin's theorem and gauge theory. We discuss an invariant of a homology S^1 cross S^3 defined by Furuta and Ohta as an analogue of Casson's invariant for homology 3-spheres. Our main result is a calculation of the Furuta-Ohta invariant for the mapping torus of a finite-order diffeomorphism of a homology sphere. The answer is the equivariant Casson invariant (Collin-Saveliev 2001) if the action has fixed points, and a version of the Boyer-Nicas (1990) invariant if the action is free. We deduce, for finite-order mapping tori, the conjecture of Furuta and Ohta that their invariant reduces mod 2 to the Rohlin invariant of a manifold carrying a generator of the third homology group. Under some transversality assumptions, we show that the Furuta-Ohta invariant coincides with the Lefschetz number of the action on Floer homology. Comparing our two answers yields an example of a diffeomorphism acting trivially on the representation variety but non-trivially on Floer homology.
This is the first in a series of papers exploring the relationship between the Rohlin invariant and gauge theory. We discuss the Casson-type invariant of a 3-manifold with the integral homology of a torus, given by counting projectively flat connections. We show that its mod 2 evaluation is given by the triple cup product in cohomology, and so it coincides with a sum of Rohlin invariants. Our counting argument makes use of a natural action of the first cohomology on the moduli space of projectively flat connections; along the way we construct perturbations that are equivariant with respect to …
This is the third in our series of papers relating gauge theoretic invariants of certain 4-manifolds with invariants of 3-manifolds derived from Rohlin's theorem. Such relations are well-known in dimension three, starting with Casson's integral lift of the Rohlin invariant of a homology sphere. We consider two invariants of a spin 4-manifold that has the integral homology of a 4-torus. The first is a degree zero Donaldson invariant, counting flat connections on a certain SO(3)-bundle. The second, which depends on the choice of a 1-dimensional cohomology class, is a combination of Rohlin invari…
This is a survey of our recent work with Tom Mrowka on Seiberg-Witten gauge theory and index theory for manifolds with periodic ends. We explain how this work leads to a new invariant, which is related to the classical Rohlin invariant of homology 3-spheres and to the Furuta-Ohta invariant originating in Yang-Mills gauge theory. We give some new calculations of our invariant for 4-dimensional mapping tori.
We extend the techniques in a previous paper to calculate the Heegaard Floer homology groups for fibered 3-manifolds M whose monodromy is a power of a Dehn twist about a genus-1 separating circle on a surface of genus g > 1. We only consider non-torsion Spin^c-structures on M.
Given a closed hyperbolic 3-manifold T_ψthat fibers over the circle with monodromy ψ: S -> S, the monodromy $ψ$ determines an isometry of Teichmuller space with its Weil-Petersson metric whose translation distance ||ψ||_WP is positive. We show there is a constant K >= 1 depending only on the topology of S so that the volume of T_ψsatisfies ||ψ||_WP/K <= vol(T_ψ) <= K ||ψ||_WP.
We show that the graph TQFT for Heegaard Floer homology satisfies a strong version of Atiyah's duality axiom for a TQFT. As an application, we compute some Heegaard Floer mixed invariants of 4-dimensional mapping tori in terms of Lefschetz numbers on $HF^+$.
We prove that for every P there is a bound B depending only on P so that the mapping torus of every P--small irreducible train-track map can be obtained by surgery from one of B mapping tori. We show that given an integer P>0 there is a bound $M$ depending only on P, so that there exists a presentation of the fundamental group of the mapping torus of a P--small irreducible train-track map with less than M generators and M relations.
We show how to construct an ideal triangulation of a mapping torus of a pseudo-Anosov map punctured along the singular fibers. This gives rise to a new conjugacy invariant of mapping classes, and a new proof of a theorem of Farb-Leininger-Margalit. The approach in this paper is based on ideas of Hamenstadt.
We show that if the monodromy of a 3-manifold M that fibers over the circle has large translation distance in the curve complex, then the rank of the fundamental group of M is 2g+1, where g is the genus of the fiber.
We investigate the relation between the topological entropy of pseudo-Anosov maps on surfaces with punctures and the rank of the first homology of their mapping tori. On the surface $S$ of genus $g$ with $n$ punctures, we show that the entropy of a pseudo-Anosov map is bounded from above by $\dfrac{(k+1)\log(k+3)}{|χ(S)|}$ up to a constant multiple when the rank of the first homology of the mapping torus is $k+1$ and $k, g, n$ satisfy a certain assumption. This is a partial generalization of precedent works of Tsai and Agol-Leininger-Margalit.
The invariant measured foliations of a pseudo-Anosov homeomorphism induce a natural (singular) Sol structure on mapping tori of surfaces with pseudo-Anosov monodromy. We show that when the pseudo-Anosov $ϕ:S\rightarrow S$ has orientable foliations and does not have 1 as an eigenvalue of the induced cohomology action on the closed surface, then the Sol structure can be deformed to nearby cone hyperbolic structures, in the sense of projective structures. The cone angles can be chosen to be decreasing from multiples of $2π$.
We consider the pseudo-Anosov elements of the mapping class group of a surface of genus g that fix a rank k subgroup of the first homology of the surface. We show that the smallest entropy among these is comparable to (k+1)/g. This interpolates between results of Penner and of Farb and the second and third authors, who treated the cases of k=0 and k=2g, respectively, and answers a question of Ellenberg. We also show that the number of conjugacy classes of pseudo-Anosov mapping classes as above grows (as a function of g) like a polynomial of degree k.
We use the geometry of the Farey graph to give an alternative proof of the fact that if $A \in GL_2\mathbb Z$ and $G_A=\mathbb Z^2 \rtimes_A \mathbb Z$ is generated by two elements, there is a single Nielsen equivalence class of $2$-element generating sets for $G_A$ unless $A$ is conjugate to $\pm \left(\begin {smallmatrix} 2 & 1 \\ 1 & 1 \end {smallmatrix}\right )$, in which case there are two.
In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over $S^1$ with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive Kodaira dimension, then the fiber itself is a sphere or torus bundle over $S^1$. In a different direction, we prove that if the 3-dimensional fiber of such a 4-manifold is virtually fibered then the 4-manifold is virtually symplectic unless its virtual first Betti number is 1.
We present the mathematical background of a software package that computes triangulations of mapping tori of surface homeomorphisms, suitable for Jeff Weeks's program SnapPea. It consists of two programs. jmt computes triangulations and prints them in a human-readable format. jsnap converts this format into SnapPea's triangulation file format and may be of independent interest because it allows for quick and easy generation of input for SnapPea. As an application, we obtain a new solution to the restricted conjugacy problem in the mapping class group.
Let $G$ be the mapping torus of a polynomially growing automorphism of a finitely generated free group. We determine which epimorphisms from $G$ to $\mathbb{Z}$ have finitely generated kernel, and we compute the rank of the kernel. We thus describe all possible ways of expressing $G$ as the mapping torus of a free group automorphism. This is similar to the case for 3--manifold groups, and different from the case of mapping tori of exponentially growing free group automorphisms. The proof uses a hierarchical decomposition of $G$ and requires determining the Bieri-Neumann-Strebel invariant of th…
We calculate the Heegaard Floer homologies$HF^+(M,s) for mapping tori M associated to certain surface diffeomorphisms, where s is any Spin^c structure on M whose first Chern class is non-torsion. Let gamma and delta be a pair of geometrically dual nonseparating curves on a genus g Riemann surface Sigma_g, and let sigma be a curve separating Sigma_g into components of genus 1 and g-1. Write t-gamma, t_delta, and t_sigma for the right-handed Dehn twists about each of these curves. The examples we consider are the mapping tori of the diffeomorphisms t_gamma^m circ t_delta^n for m,n in Z and that …
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 ran…
Let X_n be a cycle of n projective lines, and T_n a symplectic torus with n punctures. In this paper we review results appeared in arXiv:1103.2462 and in arXiv:1109.6615, which establish a version of homological mirror symmetry relating X_n and T_n, and define on D^b(Coh(X_n)) an action of the pure mapping class group of T_n.
Let $M$ be a closed orientable manifold. We introduce two numerical invariants, called filling volumes, on the mapping class group $\mathrm{MCG}(M)$ of $M$, which are defined in terms of filling norms on the space of singular boundaries on $M$, both with real and with integral coefficients. We show that filling volumes are length functions on $\mathrm{MCG}(M)$, we prove that the real filling volume of a mapping class $f$ is equal to the simplicial volume of the corresponding mapping torus $E_f$, while the integral filling volume of $f$ is not smaller than the stable integral simplicial volume …
Given an irreducible, end-periodic homeomorphism f of a surface S with finitely many ends, all accumulated by genus, the mapping torus is the interior of a compact, irreducible, atoroidal 3-manifold with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of the compactified mapping torus in terms of the translation length of f on the pants graph of S. This builds on work of Brock and Agol in the finite-type setting. We also construct a broad class of examples of irreducible, end-periodic homeomorphisms and use them to show that our bound is asymptotical…
Let $G$ be a finitely presented group that can be written as an extension \[ 1 \longrightarrow K \longrightarrow G \longrightarrow F_2 \longrightarrow 1 \] where $K$ is either the finitely generated free group $F_n$, $n > 2$ or the fundamental group of a closed surface of genus $g > 1$. We prove that if the image of the monodromy map $ρ\colon F_2 \to \operatorname{Out(K)}$ contains an element $φ\in \operatorname{Out(K)}$ such that the mapping torus $K \rtimes_φ \Bbb{Z}$ is virtually residually finite rationally solvable (for instance whenever the mapping torus is hyperbolic), then $G$ is not c…
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 clas…
We state Asymptotic Expansion and Growth Rate conjectures for the Witten-Reshetikhin-Turaev invariants of arbitrary framed links in 3-manifolds, and we prove these conjectures for the natural links in mapping tori of finite-order automorphisms of marked surfaces. Our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface, which we show coincides with the construction of the Witten-Reshetikhin-Turaev invariants using conformal field theory, as was recently completed by Andersen and Ueno.
Let K \subset Y be a knot in a three manifold which admits a longitude-framed surgery such that the surgered manifold has first Betti number greater than that of Y. We find a formula which computes the twisted Floer homology of the surgered manifold, in terms of twisted knot Floer homology. Using this, we compute the twisted Heegaard Floer homology \underline{HF}^+ of the mapping torus of a diffeomorphism of a closed Riemann surface whose mapping class is periodic, giving an almost complete description of the structure of these groups. When the mapping class is nontrivial, we find in particula…
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 $1/ \log^2(n)$.
We identify the leading order term of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants for finite order mapping tori with classical invariants for all simple and simply-connected compact Lie groups. The square root of the Reidemeister torsion is used as a density on the moduli space of flat connections and the leading order term is identified with the integral over this moduli space of this density weighted by a certain phase for each component of the moduli space. We also identify this phase in terms of classical invariants such as Chern-Simons invariants, eta invariants, …
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}_{-}$} \, \mathbb{S}^{\hspace{.1mm}1}$ o…
We study random elements of subgroups (and cosets) of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping torus (with rational, integer, and finite field coefficients, the hyperbolic volume (whenever the manifold is hyperbolic), the dilatation of the monodromy, the injectivity radius, and the bottom eigenvalue of the Laplacian on these mapping tori. We also study mapping tori of punctured surface bundles, and various invariants of their Dehn fillings. We also study …
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…
Let $ϕ: M\to M$ be a diffeomorphism of a $C^\infty$ compact connected manifold, and $X$ its mapping torus. There is a natural fibration $p:X\to S^1$, denote by $ξ\in H^1(X, \mathbb{Z})$ the corresponding cohomology class. Let $λ\in \mathbb{Z}^*$. Consider the endomorphism $ϕ_k^*$ induced by $ϕ$ in the cohomology of $M$ of degree $k$, and denote by $J_k(λ)$ the maximal size of its Jordan block of eigenvalue $λ$. Define a representation $ρ_λ: π_1(X)\to\mathbb{C}^*$ by $ρ_λ(g) = λ^{p_*(g)}.$ Let $H^*(X,ρ_λ)$ be the corresponding twisted cohomology of $X$. We prove that $J_k(λ)$ is equal to the ma…
One of the main challenges in 3d-3d correspondence is that no existent approach offers a complete description of 3d $N=2$ SCFT $T[M_3]$ --- 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 $M_3$ and, secondly, is not limited to a particular supersymmetric partition function of $T[…
We prove the Farrell-Jones Conjecture for mapping tori of automorphisms of virtually torsion-free hyperbolic groups. The proof uses recently developed geometric methods for establishing the Farrell-Jones Conjecture by Bartels-Lück-Reich, as well as the structure theory of mapping tori by Dahmani-Krishna.
In this note, we investigate genera for the slopes of a knotted torus in the 4-sphere analogous to the genus of a classical knot. We compare various formulations of this notion, and use this notion to study the extendable subgroup of the mapping class group of the knotted torus.
