Geometry & Topology Articles (Project Euclid)
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The latest articles from Geometry & Topology on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 19 Oct 2017 14:27 EDTThu, 19 Oct 2017 14:27 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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On representation varieties of $3$–manifold groups
https://projecteuclid.org/euclid.gt/1508437634
<strong>Michael Kapovich</strong>, <strong>John Millson</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 4, 1931--1968.</p><p><strong>Abstract:</strong><br/>
We prove universality theorems (“Murphy’s laws”) for representation varieties of fundamental groups of closed [math] –dimensional manifolds. We show that germs of [math] –representation schemes of such groups are essentially the same as germs of schemes over [math] of finite type.
</p>projecteuclid.org/euclid.gt/1508437634_20171019142734Thu, 19 Oct 2017 14:27 EDTCentral limit theorems for mapping class groups and $\mathrm{Out}(F_N)$https://projecteuclid.org/euclid.gt/1513774911<strong>Camille Horbez</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 105--156.</p><p><strong>Abstract:</strong><br/>
We prove central limit theorems for the random walks on either the mapping class group of a closed, connected, orientable, hyperbolic surface, or on [math] , each time under a finite second moment condition on the measure (either with respect to the Teichmüller metric, or with respect to the Lipschitz metric on outer space). In the mapping class group case, this describes the spread of the hyperbolic length of a simple closed curve on the surface after applying a random product of mapping classes. In the case of [math] , this describes the spread of the length of primitive conjugacy classes in [math] under random products of outer automorphisms. Both results are based on a general criterion for establishing a central limit theorem for the Busemann cocycle on the horoboundary of a metric space, applied to either the Teichmüller space of the surface or to the Culler–Vogtmann outer space.
</p>projecteuclid.org/euclid.gt/1513774911_20171220080154Wed, 20 Dec 2017 08:01 ESTDynamics on flag manifolds: domains of proper discontinuity and cocompactnesshttps://projecteuclid.org/euclid.gt/1513774912<strong>Michael Kapovich</strong>, <strong>Bernhard Leeb</strong>, <strong>Joan Porti</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 157--234.</p><p><strong>Abstract:</strong><br/>
For noncompact semisimple Lie groups [math] with finite center, we study the dynamics of the actions of their discrete subgroups [math] on the associated partial flag manifolds [math] . Our study is based on the observation, already made in the deep work of Benoist, that they exhibit also in higher rank a certain form of convergence-type dynamics. We identify geometrically domains of proper discontinuity in all partial flag manifolds. Under certain dynamical assumptions equivalent to the Anosov subgroup condition, we establish the cocompactness of the [math] –action on various domains of proper discontinuity, in particular on domains in the full flag manifold [math] . In the regular case (eg of [math] –Anosov subgroups), we prove the nonemptiness of such domains if [math] has (locally) at least one noncompact simple factor not of the type [math] , [math] or [math] by showing the nonexistence of certain ball packings of the visual boundary.
</p>projecteuclid.org/euclid.gt/1513774912_20171220080154Wed, 20 Dec 2017 08:01 ESTA mathematical theory of the gauged linear sigma modelhttps://projecteuclid.org/euclid.gt/1513774913<strong>Huijun Fan</strong>, <strong>Tyler Jarvis</strong>, <strong>Yongbin Ruan</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 235--303.</p><p><strong>Abstract:</strong><br/>
We construct a mathematical theory of Witten’s Gauged Linear Sigma Model (GLSM). Our theory applies to a wide range of examples, including many cases with nonabelian gauge group.
Both the Gromov–Witten theory of a Calabi–Yau complete intersection [math] and the Landau–Ginzburg dual (FJRW theory) of [math] can be expressed as gauged linear sigma models. Furthermore, the Landau–Ginzburg/Calabi–Yau correspondence can be interpreted as a variation of the moment map or a deformation of GIT in the GLSM. This paper focuses primarily on the algebraic theory, while a companion article will treat the analytic theory.
</p>projecteuclid.org/euclid.gt/1513774913_20171220080154Wed, 20 Dec 2017 08:01 ESTChord arc properties for constant mean curvature diskshttps://projecteuclid.org/euclid.gt/1513774914<strong>William Meeks</strong>, <strong>Giuseppe Tinaglia</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 305--322.</p><p><strong>Abstract:</strong><br/>
We prove a chord arc type bound for disks embedded in [math] with constant mean curvature that does not depend on the value of the mean curvature. This bound is inspired by and generalizes the weak chord arc bound of Colding and Minicozzi in Proposition 2.1 of Ann. of Math. 167 (2008) 211–243 for embedded minimal disks. Like in the minimal case, this chord arc bound is a fundamental tool for studying complete constant mean curvature surfaces embedded in [math] with finite topology or with positive injectivity radius.
</p>projecteuclid.org/euclid.gt/1513774914_20171220080154Wed, 20 Dec 2017 08:01 ESTGromov–Witten invariants of the Hilbert schemes of points of a K3 surfacehttps://projecteuclid.org/euclid.gt/1513774915<strong>Georg Oberdieck</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 323--437.</p><p><strong>Abstract:</strong><br/>
We study the enumerative geometry of rational curves on the Hilbert schemes of points of a K3 surface.
Let [math] be a K3 surface and let [math] be the Hilbert scheme of [math] points of [math] . In the case of elliptically fibered K3 surfaces [math] , we calculate genus-0 Gromov–Witten invariants of [math] , which count rational curves incident to two generic fibers of the induced Lagrangian fibration [math] . The generating series of these invariants is the Fourier expansion of a power of the Jacobi theta function times a modular form, hence of a Jacobi form.
We also prove results for genus-0 Gromov–Witten invariants of [math] for several other natural incidence conditions. In each case, the generating series is again a Jacobi form. For the proof we evaluate Gromov–Witten invariants of the Hilbert scheme of two points of [math] , where [math] is an elliptic curve.
Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on [math] with respect to primitive curve classes. The conjecture is presented in terms of natural operators acting on the Fock space of [math] . We prove the conjecture in the first nontrivial case [math] . As a corollary, we find that the full genus-0 Gromov–Witten theory of [math] in primitive classes is governed by Jacobi forms.
We present two applications. A conjecture relating genus-1 invariants of [math] to the Igusa cusp form was proposed in joint work with R Pandharipande. Our results prove the conjecture when [math] . Finally, we present a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through two general points.
</p>projecteuclid.org/euclid.gt/1513774915_20171220080154Wed, 20 Dec 2017 08:01 ESTDetecting sphere boundaries of hyperbolic groupshttps://projecteuclid.org/euclid.gt/1513774916<strong>Benjamin Beeker</strong>, <strong>Nir Lazarovich</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 439--470.</p><p><strong>Abstract:</strong><br/>
We show that a one-ended simply connected at infinity hyperbolic group [math] with enough codimension- [math] surface subgroups has [math] . By work of Markovic (2013), our result gives a new characterization of virtually fundamental groups of hyperbolic [math] –manifolds.
</p>projecteuclid.org/euclid.gt/1513774916_20171220080154Wed, 20 Dec 2017 08:01 ESTEquivariant characteristic classes of external and symmetric products of varietieshttps://projecteuclid.org/euclid.gt/1513774917<strong>Laurenţiu Maxim</strong>, <strong>Jörg Schürmann</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 471--515.</p><p><strong>Abstract:</strong><br/>
We obtain refined generating series formulae for equivariant characteristic classes of external and symmetric products of singular complex quasiprojective varieties. More concretely, we study equivariant versions of Todd, Chern and Hirzebruch classes for singular spaces, with values in delocalized Borel–Moore homology of external and symmetric products. As a byproduct, we recover our previous characteristic class formulae for symmetric products and obtain new equivariant generalizations of these results, in particular also in the context of twisting by representations of the symmetric group.
</p>projecteuclid.org/euclid.gt/1513774917_20171220080154Wed, 20 Dec 2017 08:01 ESTHyperbolic extensions of free groupshttps://projecteuclid.org/euclid.gt/1513774918<strong>Spencer Dowdall</strong>, <strong>Samuel Taylor</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 517--570.</p><p><strong>Abstract:</strong><br/>
Given a finitely generated subgroup [math] of the outer automorphism group of the rank- [math] free group [math] , there is a corresponding free group extension [math] . We give sufficient conditions for when the extension [math] is hyperbolic. In particular, we show that if all infinite-order elements of [math] are atoroidal and the action of [math] on the free factor complex of [math] has a quasi-isometric orbit map, then [math] is hyperbolic. As an application, we produce examples of hyperbolic [math] –extensions [math] for which [math] has torsion and is not virtually cyclic. The proof of our main theorem involves a detailed study of quasigeodesics in Outer space that make progress in the free factor complex. This may be of independent interest.
</p>projecteuclid.org/euclid.gt/1513774918_20171220080154Wed, 20 Dec 2017 08:01 ESTComplete minimal surfaces densely lying in arbitrary domains of $\mathbb{R}^n$https://projecteuclid.org/euclid.gt/1513774919<strong>Antonio Alarcón</strong>, <strong>Ildefonso Castro-Infantes</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 571--590.</p><p><strong>Abstract:</strong><br/>
In this paper we prove that, given an open Riemann surface [math] and an integer [math] , the set of complete conformal minimal immersions [math] with [math] forms a dense subset in the space of all conformal minimal immersions [math] endowed with the compact-open topology. Moreover, we show that every domain in [math] contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface.
Our method of proof can be adapted to give analogous results for nonorientable minimal surfaces in [math] , complex curves in [math] , holomorphic null curves in [math] , and holomorphic Legendrian curves in [math] .
</p>projecteuclid.org/euclid.gt/1513774919_20171220080154Wed, 20 Dec 2017 08:01 ESTIntrinsic structure of minimal discs in metric spaceshttps://projecteuclid.org/euclid.gt/1513774920<strong>Alexander Lytchak</strong>, <strong>Stefan Wenger</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 591--644.</p><p><strong>Abstract:</strong><br/>
We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used to control the shapes of all curves and therefore the geometry and topology of the original metric space. The class of spaces arising in this way as intrinsic minimal discs is a natural generalization of the class of Ahlfors regular discs, well-studied in analysis on metric spaces.
</p>projecteuclid.org/euclid.gt/1513774920_20171220080154Wed, 20 Dec 2017 08:01 ESTThe Hilbert scheme of a plane curve singularity and the HOMFLY homology of its linkhttps://projecteuclid.org/euclid.gt/1517454106<strong>Alexei Oblomkov</strong>, <strong>Jacob Rasmussen</strong>, <strong>Vivek Shende</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 2, 645--691.</p><p><strong>Abstract:</strong><br/>
We conjecture an expression for the dimensions of the Khovanov–Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture (2012) relating the HOMFLY polynomial to the Euler numbers of the same spaces upon setting [math] . By generalizing results of Piontkowski on the structure of compactified Jacobians to the case of Hilbert schemes of points, we give an explicit prediction of the HOMFLY homology of a [math] torus knot as a certain sum over diagrams.
The Hilbert scheme series corresponding to the summand of the HOMFLY homology with minimal “ [math] ” grading can be recovered from the perverse filtration on the cohomology of the compactified Jacobian. In the case of [math] torus knots, this space furnishes the unique finite-dimensional simple representation of the rational spherical Cherednik algebra with central character [math] . Up to a conjectural identification of the perverse filtration with a previously introduced filtration, the work of Haiman and Gordon and Stafford gives formulas for the Hilbert scheme series when [math] .
</p>projecteuclid.org/euclid.gt/1517454106_20180131220213Wed, 31 Jan 2018 22:02 ESTDetecting periodic elements in higher topological Hochschild homologyhttps://projecteuclid.org/euclid.gt/1517454107<strong>Torleif Veen</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 2, 693--756.</p><p><strong>Abstract:</strong><br/>
Given a commutative ring spectrum [math] , let [math] be the Loday functor constructed by Brun, Carlson and Dundas. Given a prime [math] , we calculate [math] and [math] for [math] , and use these results to deduce that [math] in the [math] connective Morava K-theory of [math] is nonzero and detected in the homotopy fixed-point spectral sequence by an explicit element, whose class we name the Rognes class.
To facilitate these calculations, we introduce multifold Hopf algebras. Each axis circle in [math] gives rise to a Hopf algebra structure on [math] , and the way these Hopf algebra structures interact is encoded with a multifold Hopf algebra structure. This structure puts several restrictions on the possible algebra structures on [math] and is a vital tool in the calculations above.
</p>projecteuclid.org/euclid.gt/1517454107_20180131220213Wed, 31 Jan 2018 22:02 ESTLong-time behavior of $3$–dimensional Ricci flow: introductionhttps://projecteuclid.org/euclid.gt/1517454108<strong>Richard H Bamler</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 2, 757--774.</p><p><strong>Abstract:</strong><br/>
In the following series of papers we analyze the long-time behavior of [math] –dimensional Ricci flows with surgery. Our main result will be that if the surgeries are performed correctly, then only finitely many surgeries occur and after some time the curvature is bounded by [math] . This result confirms a conjecture of Perelman. In the course of the proof, we also obtain a qualitative description of the geometry as [math] .
</p>projecteuclid.org/euclid.gt/1517454108_20180131220213Wed, 31 Jan 2018 22:02 ESTLong-time behavior of $3$–dimensional Ricci flow, A: Generalizations of Perelman's long-time estimateshttps://projecteuclid.org/euclid.gt/1517454109<strong>Richard H Bamler</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 2, 775--844.</p><p><strong>Abstract:</strong><br/>
This is the first of a series of papers on the long-time behavior of [math] –dimensional Ricci flows with surgery. We first fix a notion of Ricci flows with surgery, which will be used in this and the following three papers. Then we review Perelman’s long-time estimates and generalize them to the case in which the underlying manifold is allowed to have a boundary. Eventually, making use of Perelman’s techniques, we prove new long-time estimates, which hold whenever the metric is sufficiently collapsed.
</p>projecteuclid.org/euclid.gt/1517454109_20180131220213Wed, 31 Jan 2018 22:02 ESTLong-time behavior of $3$–dimensional Ricci flow, B: Evolution of the minimal area of simplicial complexes under Ricci flowhttps://projecteuclid.org/euclid.gt/1517454112<strong>Richard H Bamler</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 2, 845--892.</p><p><strong>Abstract:</strong><br/>
In this second part of a series of papers on the long-time behavior of Ricci flows with surgery, we establish a bound on the evolution of the infimal area of simplicial complexes inside a [math] –manifold under the Ricci flow. This estimate generalizes an area estimate of Hamilton, which we will recall in the first part of the paper.
We remark that in this paper we will mostly be dealing with nonsingular Ricci flows. The existence of surgeries will not play an important role.
</p>projecteuclid.org/euclid.gt/1517454112_20180131220213Wed, 31 Jan 2018 22:02 ESTLong-time behavior of $3$–dimensional Ricci flow, C: $3$–manifold topology and combinatorics of simplicial complexes in $3$–manifoldshttps://projecteuclid.org/euclid.gt/1517454113<strong>Richard H Bamler</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 2, 893--948.</p><p><strong>Abstract:</strong><br/>
In the third part of this series of papers, we establish several topological results that will become important for studying the long-time behavior of Ricci flows with surgery. In the first part of this paper we recall some elementary observations in the topology of [math] –manifolds. The main part is devoted to the construction of certain simplicial complexes in a given [math] –manifold that exhibit useful intersection properties with embedded, incompressible solid tori.
This paper is purely topological in nature and Ricci flows will not be used.
</p>projecteuclid.org/euclid.gt/1517454113_20180131220213Wed, 31 Jan 2018 22:02 ESTLong-time behavior of $3$–dimensional Ricci flow, D: Proof of the main resultshttps://projecteuclid.org/euclid.gt/1517454114<strong>Richard H Bamler</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 2, 949--1068.</p><p><strong>Abstract:</strong><br/>
This is the fourth and last part of a series of papers on the long-time behavior of [math] –dimensional Ricci flows with surgery. In this paper, we prove our main two results. The first result states that if the surgeries are performed correctly, then the flow becomes nonsingular eventually and the curvature is bounded by [math] . The second result provides a qualitative description of the geometry as [math] .
</p>projecteuclid.org/euclid.gt/1517454114_20180131220213Wed, 31 Jan 2018 22:02 ESTPixton's double ramification cycle relationshttps://projecteuclid.org/euclid.gt/1517454115<strong>Emily Clader</strong>, <strong>Felix Janda</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 2, 1069--1108.</p><p><strong>Abstract:</strong><br/>
We prove a conjecture of Pixton, namely that his proposed formula for the double ramification cycle on [math] vanishes in codimension beyond [math] . This yields a collection of tautological relations in the Chow ring of [math] . We describe, furthermore, how these relations can be obtained from Pixton’s [math] –spin relations via localization on the moduli space of stable maps to an orbifold projective line.
</p>projecteuclid.org/euclid.gt/1517454115_20180131220213Wed, 31 Jan 2018 22:02 ESTTopology of closed hypersurfaces of small entropyhttps://projecteuclid.org/euclid.gt/1517454116<strong>Jacob Bernstein</strong>, <strong>Lu Wang</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 2, 1109--1141.</p><p><strong>Abstract:</strong><br/>
We use a weak mean curvature flow together with a surgery procedure to show that all closed hypersurfaces in [math] with entropy less than or equal to that of [math] , the round cylinder in [math] , are diffeomorphic to [math] .
</p>projecteuclid.org/euclid.gt/1517454116_20180131220213Wed, 31 Jan 2018 22:02 ESTMarkov numbers and Lagrangian cell complexes in the complex projective planehttps://projecteuclid.org/euclid.gt/1517454117<strong>Jonathan David Evans</strong>, <strong>Ivan Smith</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 2, 1143--1180.</p><p><strong>Abstract:</strong><br/>
We study Lagrangian embeddings of a class of two-dimensional cell complexes [math] into the complex projective plane. These cell complexes, which we call pinwheels , arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type [math] (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into [math] then [math] is a Markov number and we completely characterise [math] . We also show that a collection of Lagrangian pinwheels [math] , [math] , cannot be made disjoint unless [math] and the [math] form part of a Markov triple. These results are the symplectic analogue of a theorem of Hacking and Prokhorov, which classifies complex surfaces with quotient singularities admitting a [math] –Gorenstein smoothing whose general fibre is [math] .
</p>projecteuclid.org/euclid.gt/1517454117_20180131220213Wed, 31 Jan 2018 22:02 ESTAffine representability results in $\mathbb{A}^1$–homotopy theory, II: Principal bundles and homogeneous spaceshttps://projecteuclid.org/euclid.gt/1517454118<strong>Aravind Asok</strong>, <strong>Marc Hoyois</strong>, <strong>Matthias Wendt</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 2, 1181--1225.</p><p><strong>Abstract:</strong><br/>
We establish a relative version of the abstract “affine representability” theorem in [math] –homotopy theory from part I of this paper. We then prove some [math] –invariance statements for generically trivial torsors under isotropic reductive groups over infinite fields analogous to the Bass–Quillen conjecture for vector bundles. Putting these ingredients together, we deduce representability theorems for generically trivial torsors under isotropic reductive groups and for associated homogeneous spaces in [math] –homotopy theory.
</p>projecteuclid.org/euclid.gt/1517454118_20180131220213Wed, 31 Jan 2018 22:02 ESTSmooth factors of projective actions of higher-rank lattices and rigidityhttps://projecteuclid.org/euclid.gt/1517454119<strong>Alexander Gorodnik</strong>, <strong>Ralf Spatzier</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 2, 1227--1266.</p><p><strong>Abstract:</strong><br/>
We study smooth factors of the standard actions of lattices in higher-rank semisimple Lie groups on flag manifolds. Under a mild condition on the existence of a single differentiable sink, we show that these factors are [math] –conjugate to the standard actions on flag manifolds.
</p>projecteuclid.org/euclid.gt/1517454119_20180131220213Wed, 31 Jan 2018 22:02 ESTGoldman algebra, opers and the swapping algebrahttps://projecteuclid.org/euclid.gt/1522461617<strong>François Labourie</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1267--1348.</p><p><strong>Abstract:</strong><br/>
We define a Poisson algebra called the swapping algebra using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra, called the algebra of multifractions , as an algebra of functions on the space of cross ratios and thus as an algebra of functions on the Hitchin component as well as on the space of [math] –opers with trivial holonomy. We relate this Poisson algebra to the Atiyah–Bott–Goldman symplectic structure and to the Drinfel’d–Sokolov reduction. We also prove an extension of the Wolpert formula.
</p>projecteuclid.org/euclid.gt/1522461617_20180330220033Fri, 30 Mar 2018 22:00 EDTDeforming convex projective manifoldshttps://projecteuclid.org/euclid.gt/1522461618<strong>Daryl Cooper</strong>, <strong>Darren Long</strong>, <strong>Stephan Tillmann</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1349--1404.</p><p><strong>Abstract:</strong><br/>
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 noncompact [math] manifolds of the openness of their holonomies.
</p>projecteuclid.org/euclid.gt/1522461618_20180330220033Fri, 30 Mar 2018 22:00 EDTOrderability and Dehn fillinghttps://projecteuclid.org/euclid.gt/1522461619<strong>Marc Culler</strong>, <strong>Nathan M Dunfield</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1405--1457.</p><p><strong>Abstract:</strong><br/>
Motivated by conjectures relating group orderability, Floer homology and taut foliations, we discuss a systematic and broadly applicable technique for constructing left-orders on the fundamental groups of rational homology [math] –spheres. Specifically, for a compact [math] –manifold [math] with torus boundary, we give several criteria which imply that whole intervals of Dehn fillings of [math] have left-orderable fundamental groups. Our technique uses certain representations from [math] into [math] , which we organize into an infinite graph in [math] called the translation extension locus. We include many plots of such loci which inform the proofs of our main results and suggest interesting avenues for future research.
</p>projecteuclid.org/euclid.gt/1522461619_20180330220033Fri, 30 Mar 2018 22:00 EDTMirror theorem for elliptic quasimap invariantshttps://projecteuclid.org/euclid.gt/1522461620<strong>Bumsig Kim</strong>, <strong>Hyenho Lho</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1459--1481.</p><p><strong>Abstract:</strong><br/>
We propose and prove a mirror theorem for the elliptic quasimap invariants of smooth Calabi–Yau complete intersections in projective spaces. This theorem, combined with the wall-crossing formula of Ciocan-Fontanine and Kim, implies mirror theorems of Zinger and Popa for the elliptic Gromov–Witten invariants of those varieties. This paper and the wall-crossing formula provide a unified framework for the mirror theory of rational and elliptic Gromov–Witten invariants.
</p>projecteuclid.org/euclid.gt/1522461620_20180330220033Fri, 30 Mar 2018 22:00 EDTCounting problem on wind-tree modelshttps://projecteuclid.org/euclid.gt/1522461621<strong>Angel Pardo</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1483--1536.</p><p><strong>Abstract:</strong><br/>
We study periodic wind-tree models, that is, billiards in the plane endowed with [math] –periodically located identical connected symmetric right-angled obstacles. We give asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to [math] –translations) on the wind-tree billiard. We also explicitly compute the associated Siegel–Veech constant for generic wind-tree billiards depending on the number of corners on the obstacle.
</p>projecteuclid.org/euclid.gt/1522461621_20180330220033Fri, 30 Mar 2018 22:00 EDTGroup trisections and smooth $4$–manifoldshttps://projecteuclid.org/euclid.gt/1522461622<strong>Aaron Abrams</strong>, <strong>David T Gay</strong>, <strong>Robion Kirby</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1537--1545.</p><p><strong>Abstract:</strong><br/>
A trisection of a smooth, closed, oriented [math] –manifold is a decomposition into three [math] –dimensional [math] –handlebodies meeting pairwise in [math] –dimensional [math] –handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the [math] –dimensional handlebodies, the [math] –dimensional handlebodies and the closed [math] –manifold, with homomorphisms between them induced by inclusion, form a commutative diagram of epimorphisms, which we call a trisection of the [math] –manifold group. A trisected [math] –manifold thus gives a trisected group; here we show that every trisected group uniquely determines a trisected [math] –manifold. Together with Gay and Kirby’s existence and uniqueness theorem for [math] –manifold trisections, this gives a bijection from group trisections modulo isomorphism and a certain stabilization operation to smooth, closed, connected, oriented [math] –manifolds modulo diffeomorphism. As a consequence, smooth [math] –manifold topology is, in principle, entirely group-theoretic. For example, the smooth [math] –dimensional Poincaré conjecture can be reformulated as a purely group-theoretic statement.
</p>projecteuclid.org/euclid.gt/1522461622_20180330220033Fri, 30 Mar 2018 22:00 EDTSymmetric products and subgroup latticeshttps://projecteuclid.org/euclid.gt/1522461623<strong>Markus Hausmann</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1547--1591.</p><p><strong>Abstract:</strong><br/>
Let [math] be a finite group. We show that the rational equivariant homotopy groups of symmetric products of the [math] –equivariant sphere spectrum are naturally isomorphic to the rational homology groups of certain subcomplexes of the subgroup lattice of [math] .
</p>projecteuclid.org/euclid.gt/1522461623_20180330220033Fri, 30 Mar 2018 22:00 EDTConvex projective structures on nonhyperbolic three-manifoldshttps://projecteuclid.org/euclid.gt/1522461624<strong>Samuel A Ballas</strong>, <strong>Jeffrey Danciger</strong>, <strong>Gye-Seon Lee</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1593--1646.</p><p><strong>Abstract:</strong><br/>
Y Benoist proved that if a closed three-manifold [math] admits an indecomposable convex real projective structure, then [math] is topologically the union along tori and Klein bottles of finitely many submanifolds each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial results in the direction of a potential converse to Benoist’s theorem. We show that a cusped hyperbolic three-manifold may, under certain assumptions, be deformed to convex projective structures with totally geodesic torus boundary. Such structures may be convexly glued together whenever the geometry at the boundary matches up. In particular, we prove that many doubles of cusped hyperbolic three-manifolds admit convex projective structures.
</p>projecteuclid.org/euclid.gt/1522461624_20180330220033Fri, 30 Mar 2018 22:00 EDTHyperbolic Dehn filling in dimension fourhttps://projecteuclid.org/euclid.gt/1522461625<strong>Bruno Martelli</strong>, <strong>Stefano Riolo</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1647--1716.</p><p><strong>Abstract:</strong><br/>
We introduce and study some deformations of complete finite-volume hyperbolic four-manifolds that may be interpreted as four-dimensional analogues of Thurston’s hyperbolic Dehn filling.
We construct in particular an analytic path of complete, finite-volume cone four-manifolds [math] that interpolates between two hyperbolic four-manifolds [math] and [math] with the same volume [math] . The deformation looks like the familiar hyperbolic Dehn filling paths that occur in dimension three, where the cone angle of a core simple closed geodesic varies monotonically from [math] to [math] . Here, the singularity of [math] is an immersed geodesic surface whose cone angles also vary monotonically from [math] to [math] . When a cone angle tends to [math] a small core surface (a torus or Klein bottle) is drilled, producing a new cusp.
We show that various instances of hyperbolic Dehn fillings may arise, including one case where a degeneration occurs when the cone angles tend to [math] , like in the famous figure-eight knot complement example.
The construction makes an essential use of a family of four-dimensional deforming hyperbolic polytopes recently discovered by Kerckhoff and Storm.
</p>projecteuclid.org/euclid.gt/1522461625_20180330220033Fri, 30 Mar 2018 22:00 EDTSemidualities from products of treeshttps://projecteuclid.org/euclid.gt/1522461626<strong>Daniel Studenmund</strong>, <strong>Kevin Wortman</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1717--1758.</p><p><strong>Abstract:</strong><br/>
Let [math] be a global function field of characteristic [math] , and let [math] be a finite-index subgroup of an arithmetic group defined with respect to [math] and such that any torsion element of [math] is a [math] –torsion element. We define semiduality groups, and we show that [math] is a [math] –semiduality group if [math] acts as a lattice on a product of trees. We also give other examples of semiduality groups, including lamplighter groups, Diestel–Leader groups, and countable sums of finite groups.
</p>projecteuclid.org/euclid.gt/1522461626_20180330220033Fri, 30 Mar 2018 22:00 EDTBrane actions, categorifications of Gromov–Witten theory and quantum K–theoryhttps://projecteuclid.org/euclid.gt/1522461627<strong>Etienne Mann</strong>, <strong>Marco Robalo</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1759--1836.</p><p><strong>Abstract:</strong><br/>
Let [math] be a smooth projective variety. Using the idea of brane actions discovered by Toën, we construct a lax associative action of the operad of stable curves of genus zero on the variety [math] seen as an object in correspondences in derived stacks. This action encodes the Gromov–Witten theory of [math] in purely geometrical terms and induces an action on the derived category [math] which allows us to recover the quantum K–theory of Givental and Lee.
</p>projecteuclid.org/euclid.gt/1522461627_20180330220033Fri, 30 Mar 2018 22:00 EDTRicci flow on asymptotically Euclidean manifoldshttps://projecteuclid.org/euclid.gt/1522461628<strong>Yu Li</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 3, 1837--1891.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove that if an asymptotically Euclidean manifold with nonnegative scalar curvature has long-time existence of Ricci flow, the ADM mass is nonnegative. We also give an independent proof of the positive mass theorem in dimension three.
</p>projecteuclid.org/euclid.gt/1522461628_20180330220033Fri, 30 Mar 2018 22:00 EDTFrom operator categories to higher operadshttps://projecteuclid.org/euclid.gt/1523584815<strong>Clark Barwick</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 1893--1959.</p><p><strong>Abstract:</strong><br/>
We introduce the notion of an operator category and two different models for homotopy theory of [math] –operads over an operator category — one of which extends Lurie’s theory of [math] –operads, the other of which is completely new, even in the commutative setting. We define perfect operator categories, and we describe a category [math] attached to a perfect operator category [math] that provides Segal maps. We define a wreath product of operator categories and a form of the Boardman–Vogt tensor product that lies over it. We then give examples of operator categories that provide universal properties for the operads [math] and [math] ( [math] ) and also a collection of new examples.
</p>projecteuclid.org/euclid.gt/1523584815_20180412220031Thu, 12 Apr 2018 22:00 EDTQuantitative bi-Lipschitz embeddings of bounded-curvature manifolds and orbifoldshttps://projecteuclid.org/euclid.gt/1523584816<strong>Sylvester Eriksson-Bique</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 1961--2026.</p><p><strong>Abstract:</strong><br/>
We construct bi-Lipschitz embeddings into Euclidean space for bounded-diameter subsets of manifolds and orbifolds of bounded curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. We also construct global bi-Lipschitz embeddings for spaces of the form [math] , where [math] is a discrete group acting properly discontinuously and by isometries on [math] . This generalizes results of Naor and Khot. Our approach is based on analyzing the structure of a bounded-curvature manifold at various scales by specializing methods from collapsing theory to a certain class of model spaces. In the process, we develop tools to prove collapsing theory results using algebraic techniques.
</p>projecteuclid.org/euclid.gt/1523584816_20180412220031Thu, 12 Apr 2018 22:00 EDTUnfolded Seiberg–Witten Floer spectra, I: Definition and invariancehttps://projecteuclid.org/euclid.gt/1523584817<strong>Tirasan Khandhawit</strong>, <strong>Jianfeng Lin</strong>, <strong>Hirofumi Sasahira</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2027--2114.</p><p><strong>Abstract:</strong><br/>
Let [math] be a closed and oriented [math] –manifold. We define different versions of unfolded Seiberg–Witten Floer spectra for [math] . These invariants generalize Manolescu’s Seiberg–Witten Floer spectrum for rational homology [math] –spheres. We also compute some examples when [math] is a Seifert space.
</p>projecteuclid.org/euclid.gt/1523584817_20180412220031Thu, 12 Apr 2018 22:00 EDTA family of compact complex and symplectic Calabi–Yau manifolds that are non-Kählerhttps://projecteuclid.org/euclid.gt/1523584818<strong>Lizhen Qin</strong>, <strong>Botong Wang</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2115--2144.</p><p><strong>Abstract:</strong><br/>
We construct a family of [math] –dimensional compact manifolds [math] which are simultaneously diffeomorphic to complex Calabi–Yau manifolds and symplectic Calabi–Yau manifolds. They have fundamental groups [math] , their odd-degree Betti numbers are even, they satisfy the hard Lefschetz property, and their real homotopy types are formal. However, [math] is never homotopy equivalent to a compact Kähler manifold for any topological space [math] . The main ingredient to show the non-Kählerness is a structure theorem of cohomology jump loci due to the second author.
</p>projecteuclid.org/euclid.gt/1523584818_20180412220031Thu, 12 Apr 2018 22:00 EDTRotation intervals and entropy on attracting annular continuahttps://projecteuclid.org/euclid.gt/1523584819<strong>Alejandro Passeggi</strong>, <strong>Rafael Potrie</strong>, <strong>Martín Sambarino</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2145--2186.</p><p><strong>Abstract:</strong><br/>
We show that if [math] is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of [math] is positive. Further, the entropy is shown to be associated to a [math] –robust rotational horseshoe . On the other hand, we construct examples of annular homeomorphisms with such attractors for which the rotation interval is uniformly large but the entropy approaches zero as much as desired.
The developed techniques allow us to obtain similar results in the context of Birkhoff attractors .
</p>projecteuclid.org/euclid.gt/1523584819_20180412220031Thu, 12 Apr 2018 22:00 EDTPrimes and fields in stable motivic homotopy theoryhttps://projecteuclid.org/euclid.gt/1523584820<strong>Jeremiah Heller</strong>, <strong>Kyle M Ormsby</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2187--2218.</p><p><strong>Abstract:</strong><br/>
Let [math] be a field of characteristic different from [math] . We establish surjectivity of Balmer’s comparison map
ρ
∙
:
Spc
(
SH
A
1
(
F
)
c
)
→
Spec
h
(
K
∗
M
W
(
F
)
)
from the tensor triangular spectrum of the homotopy category of compact motivic spectra to the homogeneous Zariski spectrum of Milnor–Witt [math] –theory. We also comment on the tensor triangular geometry of compact cellular motivic spectra, producing in particular novel field spectra in this category. We conclude with a list of questions about the structure of the tensor triangular spectrum of the stable motivic homotopy category.
</p>projecteuclid.org/euclid.gt/1523584820_20180412220031Thu, 12 Apr 2018 22:00 EDTSurgery for partially hyperbolic dynamical systems, I: Blow-ups of invariant submanifoldshttps://projecteuclid.org/euclid.gt/1523584821<strong>Andrey Gogolev</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2219--2252.</p><p><strong>Abstract:</strong><br/>
We suggest a method to construct new examples of partially hyperbolic diffeomorphisms. We begin with a partially hyperbolic diffeomorphism [math] which leaves invariant a submanifold [math] . We assume that [math] is an Anosov submanifold for [math] , that is, the restriction [math] is an Anosov diffeomorphism and the center distribution is transverse to [math] . By replacing each point in [math] with the projective space (real or complex) of lines normal to [math] , we obtain the blow-up [math] . Replacing [math] with [math] amounts to a surgery on the neighborhood of [math] which alters the topology of the manifold. The diffeomorphism [math] induces a canonical diffeomorphism [math] . We prove that under certain assumptions on the local dynamics of [math] at [math] the diffeomorphism [math] is also partially hyperbolic. We also present some modifications, such as the connected sum construction, which allows to “paste together” two partially hyperbolic diffeomorphisms to obtain a new one. Finally, we present several examples to which our results apply.
</p>projecteuclid.org/euclid.gt/1523584821_20180412220031Thu, 12 Apr 2018 22:00 EDTEigenvalues of curvature, Lyapunov exponents and Harder–Narasimhan filtrationshttps://projecteuclid.org/euclid.gt/1523584822<strong>Fei Yu</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2253--2298.</p><p><strong>Abstract:</strong><br/>
Inspired by the Katz–Mazur theorem on crystalline cohomology and by the numerical experiments of Eskin, Kontsevich and Zorich, we conjecture that the polygon of the Lyapunov spectrum lies above (or on) the Harder–Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons and the integral of eigenvalues of the curvature of the Hodge bundle by using the works of Atiyah and Bott, Forni, and Möller. We obtain several applications to Teichmüller dynamics conditional on the conjecture.
</p>projecteuclid.org/euclid.gt/1523584822_20180412220031Thu, 12 Apr 2018 22:00 EDTLower bounds for Lyapunov exponents of flat bundles on curveshttps://projecteuclid.org/euclid.gt/1523584823<strong>Alex Eskin</strong>, <strong>Maxim Kontsevich</strong>, <strong>Martin Möller</strong>, <strong>Anton Zorich</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2299--2338.</p><p><strong>Abstract:</strong><br/>
Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top [math] Lyapunov exponents of the flat bundle is always greater than or equal to the degree of any rank- [math] holomorphic subbundle. We generalize the original context from Teichmüller curves to any local system over a curve with nonexpanding cusp monodromies. As an application we obtain the large-genus limits of individual Lyapunov exponents in hyperelliptic strata of abelian differentials, which Fei Yu proved conditionally on his conjecture.
Understanding the case of equality with the degrees of subbundle coming from the Hodge filtration seems challenging, eg for Calabi–Yau-type families. We conjecture that equality of the sum of Lyapunov exponents and the degree is related to the monodromy group being a thin subgroup of its Zariski closure.
</p>projecteuclid.org/euclid.gt/1523584823_20180412220031Thu, 12 Apr 2018 22:00 EDTHyperbolic jigsaws and families of pseudomodular groups, Ihttps://projecteuclid.org/euclid.gt/1523584824<strong>Beicheng Lou</strong>, <strong>Ser Peow Tan</strong>, <strong>Anh Duc Vo</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2339--2366.</p><p><strong>Abstract:</strong><br/>
We show that there are infinitely many commensurability classes of pseudomodular groups, thus answering a question raised by Long and Reid. These are Fuchsian groups whose cusp set is all of the rationals but which are not commensurable to the modular group. We do this by introducing a general construction for the fundamental domains of Fuchsian groups obtained by gluing together marked ideal triangular tiles, which we call hyperbolic jigsaw groups.
</p>projecteuclid.org/euclid.gt/1523584824_20180412220031Thu, 12 Apr 2018 22:00 EDTSubflexible symplectic manifoldshttps://projecteuclid.org/euclid.gt/1523584825<strong>Emmy Murphy</strong>, <strong>Kyler Siegel</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2367--2401.</p><p><strong>Abstract:</strong><br/>
We introduce a class of Weinstein domains which are sublevel sets of flexible Weinstein manifolds but are not themselves flexible. These manifolds exhibit rather subtle behavior with respect to both holomorphic curve invariants and symplectic flexibility. We construct a large class of examples and prove that every flexible Weinstein manifold can be Weinstein homotoped to have a nonflexible sublevel set.
</p>projecteuclid.org/euclid.gt/1523584825_20180412220031Thu, 12 Apr 2018 22:00 EDTNormalized entropy versus volume for pseudo-Anosovshttps://projecteuclid.org/euclid.gt/1523584826<strong>Sadayoshi Kojima</strong>, <strong>Greg McShane</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2403--2426.</p><p><strong>Abstract:</strong><br/>
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 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 by the square root of the area of a surface) and the volume of its mapping torus, which leads to a better bound.
</p>projecteuclid.org/euclid.gt/1523584826_20180412220031Thu, 12 Apr 2018 22:00 EDTTowers of regular self-covers and linear endomorphisms of torihttps://projecteuclid.org/euclid.gt/1523584827<strong>Wouter van Limbeek</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2427--2464.</p><p><strong>Abstract:</strong><br/>
Let [math] be a closed manifold that admits a self-cover [math] of degree [math] . We say [math] is strongly regular if all iterates [math] are regular covers. In this case, we establish an algebraic structure theorem for the fundamental group of [math] : We prove that [math] surjects onto a nontrivial free abelian group [math] , and the self-cover is induced by a linear endomorphism of [math] . Under further hypotheses we show that a finite cover of [math] admits the structure of a principal torus bundle. We show that this applies when [math] is Kähler and [math] is a strongly regular, holomorphic self-cover, and prove that a finite cover of [math] splits as a product with a torus factor.
</p>projecteuclid.org/euclid.gt/1523584827_20180412220031Thu, 12 Apr 2018 22:00 EDTClassification and arithmeticity of toroidal compactifications with $3\bar{c}_2 = \bar{c}_1^{2} = 3$https://projecteuclid.org/euclid.gt/1523584828<strong>Luca F Di Cerbo</strong>, <strong>Matthew Stover</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 4, 2465--2510.</p><p><strong>Abstract:</strong><br/>
We classify the minimum-volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces, and they are all arithmetic; ie they are associated with quotients of the ball by an arithmetic lattice. Moreover, the associated lattices are all commensurable. The first compactification, originally discovered by Hirzebruch, is the blowup of an abelian surface at one point. The others are bielliptic surfaces blown up at one point. The bielliptic examples are new and are the first known examples of smooth toroidal compactifications birational to bielliptic surfaces.
</p>projecteuclid.org/euclid.gt/1523584828_20180412220031Thu, 12 Apr 2018 22:00 EDTKähler–Ricci flow, Kähler–Einstein metric, and K–stabilityhttps://projecteuclid.org/euclid.gt/1538186735<strong>Xiuxiong Chen</strong>, <strong>Song Sun</strong>, <strong>Bing Wang</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3145--3173.</p><p><strong>Abstract:</strong><br/>
We prove the existence of a Kähler–Einstein metric on a K–stable Fano manifold using the recent compactness result on Kähler–Ricci flows. The key ingredient is an algebrogeometric description of the asymptotic behavior of Kähler–Ricci flow on Fano manifolds. This is in turn based on a general finite-dimensional discussion, which is interesting on its own and could potentially apply to other problems. As one application, we relate the asymptotics of the Calabi flow on a polarized Kähler manifold to K–stability, assuming bounds on geometry.
</p>projecteuclid.org/euclid.gt/1538186735_20180928220544Fri, 28 Sep 2018 22:05 EDTTropical refined curve counting via motivic integrationhttps://projecteuclid.org/euclid.gt/1538186736<strong>Johannes Nicaise</strong>, <strong>Sam Payne</strong>, <strong>Franziska Schroeter</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3175--3234.</p><p><strong>Abstract:</strong><br/>
We propose a geometric interpretation of Block and Göttsche’s refined tropical curve counting invariants in terms of virtual [math] specializations of motivic measures of semialgebraic sets in relative Hilbert schemes. We prove that this interpretation is correct for linear series of genus 1, and in arbitrary genus after specializing from [math] –genus to Euler characteristic.
</p>projecteuclid.org/euclid.gt/1538186736_20180928220544Fri, 28 Sep 2018 22:05 EDTAdditive invariants for knots, links and graphs in $3$–manifoldshttps://projecteuclid.org/euclid.gt/1538186737<strong>Scott A Taylor</strong>, <strong>Maggy Tomova</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3235--3286.</p><p><strong>Abstract:</strong><br/>
We define two new families of invariants for ( [math] –manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and ( [math] ) additive under trivalent vertex sum of pairs. The first of these families is closely related to both bridge number and tunnel number. The second of these families is a variation and generalization of Gabai’s width for knots in the [math] –sphere. We give applications to the tunnel number and higher-genus bridge number of connected sums of knots.
</p>projecteuclid.org/euclid.gt/1538186737_20180928220544Fri, 28 Sep 2018 22:05 EDTPhase tropical hypersurfaceshttps://projecteuclid.org/euclid.gt/1538186738<strong>Gabriel Kerr</strong>, <strong>Ilia Zharkov</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3287--3320.</p><p><strong>Abstract:</strong><br/>
We prove a conjecture of Viro (Tr. Mat. Inst. Steklova 273 (2011) 271–303) that a smooth complex hypersurface in [math] is homeomorphic to the corresponding phase tropical hypersurface.
</p>projecteuclid.org/euclid.gt/1538186738_20180928220544Fri, 28 Sep 2018 22:05 EDTOn the Farrell–Jones conjecture for Waldhausen's $A$–theoryhttps://projecteuclid.org/euclid.gt/1538186739<strong>Nils-Edvin Enkelmann</strong>, <strong>Wolfgang Lück</strong>, <strong>Malte Pieper</strong>, <strong>Mark Ullmann</strong>, <strong>Christoph Winges</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3321--3394.</p><p><strong>Abstract:</strong><br/>
We prove the Farrell–Jones conjecture for (nonconnective) [math] –theory with coefficients and finite wreath products for hyperbolic groups, [math] –groups, cocompact lattices in almost connected Lie groups and fundamental groups of manifolds of dimension less or equal to three. Moreover, we prove inheritance properties such as passing to subgroups, colimits of direct systems of groups, finite direct products and finite free products. These results hold also for Whitehead spectra and spectra of stable pseudoisotopies in the topological, piecewise linear and smooth categories.
</p>projecteuclid.org/euclid.gt/1538186739_20180928220544Fri, 28 Sep 2018 22:05 EDTFaithful actions from hyperplane arrangementshttps://projecteuclid.org/euclid.gt/1538186740<strong>Yuki Hirano</strong>, <strong>Michael Wemyss</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3395--3433.</p><p><strong>Abstract:</strong><br/>
We show that if [math] is a smooth quasiprojective [math] –fold admitting a flopping contraction, then the fundamental group of an associated simplicial hyperplane arrangement acts faithfully on the derived category of [math] . The main technical advance is to use torsion pairs as an efficient mechanism to track various objects under iterations of the flop functor (or mutation functor). This allows us to relate compositions of the flop functor (or mutation functor) to the theory of Deligne normal form, and to give a criterion for when a finite composition of [math] –fold flops can be understood as a tilt at a single torsion pair. We also use this technique to give a simplified proof of a result of Brav and Thomas (Math. Ann. 351 (2011) 1005–1017) for Kleinian singularities.
</p>projecteuclid.org/euclid.gt/1538186740_20180928220544Fri, 28 Sep 2018 22:05 EDTGenerators for a complex hyperbolic braid grouphttps://projecteuclid.org/euclid.gt/1538186741<strong>Daniel Allcock</strong>, <strong>Tathagata Basak</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3435--3500.</p><p><strong>Abstract:</strong><br/>
We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic [math] –space, take the quotient of the remaining space by a discrete group, and find generators for the orbifold fundamental group of the quotient space. These generators have the most natural form: loops corresponding to the hyperplanes which come nearest the basepoint. Our results support the conjecture that motivated this study, the “monstrous proposal”, which posits a relationship between this braid group and the monster finite simple group.
</p>projecteuclid.org/euclid.gt/1538186741_20180928220544Fri, 28 Sep 2018 22:05 EDTA formal Riemannian structure on conformal classes and uniqueness for the $\sigma_2$–Yamabe problemhttps://projecteuclid.org/euclid.gt/1538186742<strong>Matthew Gursky</strong>, <strong>Jeffrey Streets</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3501--3573.</p><p><strong>Abstract:</strong><br/>
We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the [math] –Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere.
</p>projecteuclid.org/euclid.gt/1538186742_20180928220544Fri, 28 Sep 2018 22:05 EDTChern–Schwartz–MacPherson classes of degeneracy locihttps://projecteuclid.org/euclid.gt/1538186743<strong>László M Fehér</strong>, <strong>Richárd Rimányi</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3575--3622.</p><p><strong>Abstract:</strong><br/>
The Chern–Schwartz–MacPherson class (CSM) and the Segre–Schwartz–MacPherson class (SSM) are deformations of the fundamental class of an algebraic variety. They encode finer enumerative invariants of the variety than its fundamental class. In this paper we offer three contributions to the theory of equivariant CSM/SSM classes. First, we prove an interpolation characterization for CSM classes of certain representations. This method — inspired by recent work of Maulik and Okounkov and of Gorbounov, Rimányi, Tarasov and Varchenko — does not require a resolution of singularities and often produces explicit (not sieve) formulas for CSM classes. Second, using the interpolation characterization we prove explicit formulas — including residue generating sequences — for the CSM and SSM classes of matrix Schubert varieties. Third, we suggest that a stable version of the SSM class of matrix Schubert varieties will serve as the building block of equivariant SSM theory, similarly to how the Schur functions are the building blocks of fundamental class theory. We illustrate these phenomena, and related stability and (two-step) positivity properties for some relevant representations.
</p>projecteuclid.org/euclid.gt/1538186743_20180928220544Fri, 28 Sep 2018 22:05 EDTComputational complexity and $3$–manifolds and zombieshttps://projecteuclid.org/euclid.gt/1538186744<strong>Greg Kuperberg</strong>, <strong>Eric Samperton</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3623--3670.</p><p><strong>Abstract:</strong><br/>
We show the problem of counting homomorphisms from the fundamental group of a homology [math] –sphere [math] to a finite, nonabelian simple group [math] is almost parsimoniously [math] –complete, when [math] is fixed and [math] is the computational input. In the reduction, we guarantee that every nontrivial homomorphism is a surjection. As a corollary, any nontrivial information about the number of nontrivial homomorphisms is computationally intractable assuming standard conjectures in computer science. In particular, deciding if there is a nontrivial homomorphism is [math] –complete. Another corollary is that for any fixed integer [math] , it is [math] –complete to decide whether [math] admits a connected [math] –sheeted covering.
Given a classical reversible circuit [math] , we construct [math] so that evaluations of [math] with certain initialization and finalization conditions correspond to homomorphisms [math] . An intermediate state of [math] likewise corresponds to homomorphism [math] , where [math] is a Heegaard surface of [math] of genus [math] . We analyze the action on these homomorphisms by the pointed mapping class group [math] and its Torelli subgroup [math] . Using refinements of results of Dunfield and Thurston, we show that the actions of these groups are as large as possible when [math] is large. Our results and our construction are inspired by universality results in topological quantum computation, even though the present work is nonquantum.
One tricky step in the construction is handling an inert “zombie” symbol in the computational alphabet, which corresponds to a trivial homomorphism from the fundamental group of a subsurface of the Heegaard surface.
</p>projecteuclid.org/euclid.gt/1538186744_20180928220544Fri, 28 Sep 2018 22:05 EDT$C^*$–algebraic higher signatures and an invariance theorem in codimension twohttps://projecteuclid.org/euclid.gt/1538186745<strong>Nigel Higson</strong>, <strong>Thomas Schick</strong>, <strong>Zhizhang Xie</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3671--3699.</p><p><strong>Abstract:</strong><br/>
We revisit the construction of signature classes in [math] –algebra [math] –theory, and develop a variation that allows us to prove equality of signature classes in some situations involving homotopy equivalences of noncompact manifolds that are only defined outside a compact set. As an application, we prove a counterpart for signature classes of a codimension-two vanishing theorem for the index of the Dirac operator on spin manifolds (the latter is due to Hanke, Pape and Schick, and is a development of well-known work of Gromov and Lawson).
</p>projecteuclid.org/euclid.gt/1538186745_20180928220544Fri, 28 Sep 2018 22:05 EDTContractible stability spaces and faithful braid group actionshttps://projecteuclid.org/euclid.gt/1538186746<strong>Yu Qiu</strong>, <strong>Jon Woolf</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 6, 3701--3760.</p><p><strong>Abstract:</strong><br/>
We prove that any “finite-type” component of a stability space of a triangulated category is contractible. The motivating example of such a component is the stability space of the Calabi–Yau– [math] category [math] associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group [math] acts freely upon it by spherical twists, in particular that the spherical twist group [math] is isomorphic to [math] . This generalises the result of Brav–Thomas for the [math] case. Other classes of triangulated categories with finite-type components in their stability spaces include locally finite triangulated categories with finite-rank Grothendieck group and discrete derived categories of finite global dimension.
</p>projecteuclid.org/euclid.gt/1538186746_20180928220544Fri, 28 Sep 2018 22:05 EDT