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 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 EDTParametrized spectra, multiplicative Thom spectra and the twisted Umkehr maphttps://projecteuclid.org/euclid.gt/1544756689<strong>Matthew Ando</strong>, <strong>Andrew J Blumberg</strong>, <strong>David Gepner</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 3761--3825.</p><p><strong>Abstract:</strong><br/>
We introduce a general theory of parametrized objects in the setting of [math] –categories. Although parametrised spaces and spectra are the most familiar examples, we establish our theory in the generality of families of objects of a presentable [math] –category parametrized over objects of an [math] –topos. We obtain a coherent functor formalism describing the relationship of the various adjoint functors associated to base-change and symmetric monoidal structures.
Our main applications are to the study of generalized Thom spectra. We obtain fiberwise constructions of twisted Umkehr maps for twisted generalized cohomology theories using a geometric fiberwise construction of Atiyah duality. In order to characterize the algebraic structures on generalized Thom spectra and twisted (co)homology, we express the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.
</p>projecteuclid.org/euclid.gt/1544756689_20181213220504Thu, 13 Dec 2018 22:05 ESTA Morse lemma for quasigeodesics in symmetric spaces and euclidean buildingshttps://projecteuclid.org/euclid.gt/1544756690<strong>Michael Kapovich</strong>, <strong>Bernhard Leeb</strong>, <strong>Joan Porti</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 3827--3923.</p><p><strong>Abstract:</strong><br/>
We prove a Morse lemma for regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings. We apply it to give a new coarse geometric characterization of Anosov subgroups of the isometry groups of such spaces simply as undistorted subgroups which are uniformly regular.
</p>projecteuclid.org/euclid.gt/1544756690_20181213220504Thu, 13 Dec 2018 22:05 ESTRicci flow from spaces with isolated conical singularitieshttps://projecteuclid.org/euclid.gt/1544756691<strong>Panagiotis Gianniotis</strong>, <strong>Felix Schulze</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 3925--3977.</p><p><strong>Abstract:</strong><br/>
Let [math] be a compact [math] –dimensional Riemannian manifold with a finite number of singular points, where the metric is asymptotic to a nonnegatively curved cone over [math] . We show that there exists a smooth Ricci flow starting from such a metric with curvature decaying like [math] . The initial metric is attained in Gromov–Hausdorff distance and smoothly away from the singular points. In the case that the initial manifold has isolated singularities asymptotic to a nonnegatively curved cone over [math] , where [math] acts freely and properly discontinuously, we extend the above result by showing that starting from such an initial condition there exists a smooth Ricci flow with isolated orbifold singularities.
</p>projecteuclid.org/euclid.gt/1544756691_20181213220504Thu, 13 Dec 2018 22:05 ESTQuasi-projectivity of even Artin groupshttps://projecteuclid.org/euclid.gt/1544756692<strong>Rubén Blasco-García</strong>, <strong>José Ignacio Cogolludo-Agustín</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 3979--4011.</p><p><strong>Abstract:</strong><br/>
Even Artin groups generalize right-angled Artin groups by allowing the labels in the defining graph to be even. We give a complete characterization of quasi-projective even Artin groups in terms of their defining graphs. Also, we show that quasi-projective even Artin groups are realizable by [math] quasi-projective spaces.
</p>projecteuclid.org/euclid.gt/1544756692_20181213220504Thu, 13 Dec 2018 22:05 ESTHigher enveloping algebrashttps://projecteuclid.org/euclid.gt/1544756693<strong>Ben Knudsen</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4013--4066.</p><p><strong>Abstract:</strong><br/>
We provide spectral Lie algebras with enveloping algebras over the operad of little [math] –framed [math] –dimensional disks for any choice of dimension [math] and structure group [math] , and we describe these objects in two complementary ways. The first description is an abstract characterization by a universal mapping property, which witnesses the higher enveloping algebra as the value of a left adjoint in an adjunction. The second, a generalization of the Poincaré–Birkhoff–Witt theorem, provides a concrete formula in terms of Lie algebra homology. Our construction pairs the theories of Koszul duality and Day convolution in order to lift to the world of higher algebra the fundamental combinatorics of Beilinson–Drinfeld’s theory of chiral algebras. Like that theory, ours is intimately linked to the geometry of configuration spaces and has the study of these spaces among its applications. We use it here to show that the stable homotopy types of configuration spaces are proper homotopy invariants.
</p>projecteuclid.org/euclid.gt/1544756693_20181213220504Thu, 13 Dec 2018 22:05 ESTVolumes of $\mathrm{SL}_n(\mathbb{C})$–representations of hyperbolic $3$–manifoldshttps://projecteuclid.org/euclid.gt/1544756694<strong>Wolfgang Pitsch</strong>, <strong>Joan Porti</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4067--4112.</p><p><strong>Abstract:</strong><br/>
Let [math] be a compact oriented three-manifold whose interior is hyperbolic of finite volume. We prove a variation formula for the volume on the variety of representations of [math] in [math] . Our proof follows the strategy of Reznikov’s rigidity when [math] is closed; in particular, we use Fuks’s approach to variations by means of Lie algebra cohomology. When [math] , we get Hodgson’s formula for variation of volume on the space of hyperbolic Dehn fillings. Our formula also recovers the variation of volume on the space of decorated triangulations obtained by Bergeron, Falbel and Guilloux and Dimofte, Gabella and Goncharov.
</p>projecteuclid.org/euclid.gt/1544756694_20181213220504Thu, 13 Dec 2018 22:05 ESTThe normal closure of big Dehn twists and plate spinning with rotating familieshttps://projecteuclid.org/euclid.gt/1544756695<strong>François Dahmani</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4113--4144.</p><p><strong>Abstract:</strong><br/>
We study the normal closure of a big power of one or several Dehn twists in a mapping class group. We prove that it has a presentation whose relators consist only of commutators between twists of disjoint support, thus answering a question of Ivanov. Our method is to use the theory of projection complexes of Bestvina, Bromberg and Fujiwara, together with the theory of rotating families, simultaneously on several spaces.
</p>projecteuclid.org/euclid.gt/1544756695_20181213220504Thu, 13 Dec 2018 22:05 ESTEndotrivial representations of finite groups and equivariant line bundles on the Brown complexhttps://projecteuclid.org/euclid.gt/1544756696<strong>Paul Balmer</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4145--4161.</p><p><strong>Abstract:</strong><br/>
We relate endotrivial representations of a finite group in characteristic [math] to equivariant line bundles on the simplicial complex of nontrivial [math] –subgroups, by means of weak homomorphisms.
</p>projecteuclid.org/euclid.gt/1544756696_20181213220504Thu, 13 Dec 2018 22:05 ESTIndicability, residual finiteness, and simple subquotients of groups acting on treeshttps://projecteuclid.org/euclid.gt/1544756697<strong>Pierre-Emmanuel Caprace</strong>, <strong>Phillip Wesolek</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4163--4204.</p><p><strong>Abstract:</strong><br/>
We establish three independent results on groups acting on trees. The first implies that a compactly generated locally compact group which acts continuously on a locally finite tree with nilpotent local action and no global fixed point is virtually indicable; that is to say, it has a finite-index subgroup which surjects onto [math] . The second ensures that irreducible cocompact lattices in a product of nondiscrete locally compact groups such that one of the factors acts vertex-transitively on a tree with a nilpotent local action cannot be residually finite. This is derived from a general result, of independent interest, on irreducible lattices in product groups. The third implies that every nondiscrete Burger–Mozes universal group of automorphisms of a tree with an arbitrary prescribed local action admits a compactly generated closed subgroup with a nondiscrete simple quotient. As applications, we answer a question of D Wise by proving the nonresidual finiteness of a certain lattice in a product of two regular trees, and we obtain a negative answer to a question of C Reid, concerning the structure theory of locally compact groups.
</p>projecteuclid.org/euclid.gt/1544756697_20181213220504Thu, 13 Dec 2018 22:05 ESTAn application of the Duistermaat–Heckman theorem and its extensions in Sasaki geometryhttps://projecteuclid.org/euclid.gt/1544756698<strong>Charles P Boyer</strong>, <strong>Hongnian Huang</strong>, <strong>Eveline Legendre</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4205--4234.</p><p><strong>Abstract:</strong><br/>
Building on an idea laid out by Martelli, Sparks and Yau (2008), we use the Duistermaat–Heckman localization formula and an extension of it to give rational and explicit expressions of the volume, the total transversal scalar curvature and the Einstein–Hilbert functional, seen as functionals on the Sasaki cone (Reeb cone). Studying the leading terms, we prove they are all proper. Among consequences thereof we get that the Einstein–Hilbert functional attains its minimal value and each Sasaki cone possesses at least one Reeb vector field with vanishing transverse Futaki invariant.
</p>projecteuclid.org/euclid.gt/1544756698_20181213220504Thu, 13 Dec 2018 22:05 ESTThe resolution of paracanonical curves of odd genushttps://projecteuclid.org/euclid.gt/1544756699<strong>Gavril Farkas</strong>, <strong>Michael Kemeny</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4235--4257.</p><p><strong>Abstract:</strong><br/>
We prove the Prym–Green conjecture on minimal free resolutions of paracanonical curves of odd genus. The proof proceeds via curves lying on ruled surfaces over an elliptic curve.
</p>projecteuclid.org/euclid.gt/1544756699_20181213220504Thu, 13 Dec 2018 22:05 ESTRigidity of Teichmüller spacehttps://projecteuclid.org/euclid.gt/1544756700<strong>Alex Eskin</strong>, <strong>Howard Masur</strong>, <strong>Kasra Rafi</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4259--4306.</p><p><strong>Abstract:</strong><br/>
We prove that every quasi-isometry of Teichmüller space equipped with the Teichmüller metric is a bounded distance from an isometry of Teichmüller space. That is, Teichmüller space is quasi-isometrically rigid.
</p>projecteuclid.org/euclid.gt/1544756700_20181213220504Thu, 13 Dec 2018 22:05 ESTStein fillings and $\mathrm{SU}(2)$ representationshttps://projecteuclid.org/euclid.gt/1544756701<strong>John A Baldwin</strong>, <strong>Steven Sivek</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 7, 4307--4380.</p><p><strong>Abstract:</strong><br/>
We recently defined invariants of contact [math] –manifolds using a version of instanton Floer homology for sutured manifolds. In this paper, we prove that if several contact structures on a [math] –manifold are induced by Stein structures on a single [math] –manifold with distinct Chern classes modulo torsion then their contact invariants in sutured instanton homology are linearly independent. As a corollary, we show that if a [math] –manifold bounds a Stein domain that is not an integer homology ball then its fundamental group admits a nontrivial homomorphism to [math] . We give several new applications of these results, proving the existence of nontrivial and irreducible [math] representations for a variety of [math] –manifold groups.
</p>projecteuclid.org/euclid.gt/1544756701_20181213220504Thu, 13 Dec 2018 22:05 ESTClassifying matchbox manifoldshttps://projecteuclid.org/euclid.gt/1552356077<strong>Alex Clark</strong>, <strong>Steven Hurder</strong>, <strong>Olga Lukina</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 1--27.</p><p><strong>Abstract:</strong><br/>
Matchbox manifolds are foliated spaces with totally disconnected transversals. Two matchbox manifolds which are homeomorphic have return equivalent dynamics, so that invariants of return equivalence can be applied to distinguish nonhomeomorphic matchbox manifolds. In this work we study the problem of showing the converse implication: when does return equivalence imply homeomorphism? For the class of weak solenoidal matchbox manifolds, we show that if the base manifolds satisfy a strong form of the Borel conjecture, then return equivalence for the dynamics of their foliations implies the total spaces are homeomorphic. In particular, we show that two equicontinuous [math] –like matchbox manifolds of the same dimension are homeomorphic if and only if their corresponding restricted pseudogroups are return equivalent. At the same time, we show that these results cannot be extended to include the “ adic surfaces”, which are a class of weak solenoids fibering over a closed surface of genus [math] .
</p>projecteuclid.org/euclid.gt/1552356077_20190311220143Mon, 11 Mar 2019 22:01 EDTQuasi-asymptotically conical Calabi–Yau manifoldshttps://projecteuclid.org/euclid.gt/1552356078<strong>Ronan J Conlon</strong>, <strong>Anda Degeratu</strong>, <strong>Frédéric Rochon</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 29--100.</p><p><strong>Abstract:</strong><br/>
We construct new examples of quasi-asymptotically conical ( [math] ) Calabi–Yau manifolds that are not quasi-asymptotically locally Euclidean ( [math] ). We do so by first providing a natural compactification of [math] –spaces by manifolds with fibered corners and by giving a definition of [math] –metrics in terms of an associated Lie algebra of smooth vector fields on this compactification. Thanks to this compactification and the Fredholm theory for elliptic operators on [math] –spaces developed by the second author and Mazzeo, we can in many instances obtain Kähler [math] –metrics having Ricci potential decaying sufficiently fast at infinity. This allows us to obtain [math] Calabi–Yau metrics in the Kähler classes of these metrics by solving a corresponding complex Monge–Ampère equation.
</p>projecteuclid.org/euclid.gt/1552356078_20190311220143Mon, 11 Mar 2019 22:01 EDTThe homotopy groups of the algebraic $K$–theory of the sphere spectrumhttps://projecteuclid.org/euclid.gt/1552356079<strong>Andrew J Blumberg</strong>, <strong>Michael A Mandell</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 101--134.</p><p><strong>Abstract:</strong><br/>
We calculate [math] , the homotopy groups of [math] away from [math] , in terms of the homotopy groups of [math] , the homotopy groups of [math] and the homotopy groups of [math] . This builds on work of Waldhausen, who computed the rational homotopy groups (building on work of Quillen and Borel) and Rognes, who calculated the groups at odd regular primes in terms of the homotopy groups of [math] and the homotopy groups of [math] .
</p>projecteuclid.org/euclid.gt/1552356079_20190311220143Mon, 11 Mar 2019 22:01 EDTTopology of automorphism groups of parabolic geometrieshttps://projecteuclid.org/euclid.gt/1552356080<strong>Charles Frances</strong>, <strong>Karin Melnick</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 135--169.</p><p><strong>Abstract:</strong><br/>
We prove for the automorphism group of an arbitrary parabolic geometry that the [math] – and [math] –topologies coincide, and the group admits the structure of a Lie group in this topology. We further show that this automorphism group is closed in the homeomorphism group of the underlying manifold.
</p>projecteuclid.org/euclid.gt/1552356080_20190311220143Mon, 11 Mar 2019 22:01 EDTRigidity of convex divisible domains in flag manifoldshttps://projecteuclid.org/euclid.gt/1552356081<strong>Wouter Van Limbeek</strong>, <strong>Andrew Zimmer</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 171--240.</p><p><strong>Abstract:</strong><br/>
In contrast to the many examples of convex divisible domains in real projective space, we prove that up to projective isomorphism there is only one convex divisible domain in the Grassmannian of [math] –planes in [math] when [math] . Moreover, this convex divisible domain is a model of the symmetric space associated to the simple Lie group [math] .
</p>projecteuclid.org/euclid.gt/1552356081_20190311220143Mon, 11 Mar 2019 22:01 EDTUbiquitous quasi-Fuchsian surfaces in cusped hyperbolic $3$–manifoldshttps://projecteuclid.org/euclid.gt/1552356082<strong>Daryl Cooper</strong>, <strong>David Futer</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 241--298.</p><p><strong>Abstract:</strong><br/>
We prove that every finite-volume hyperbolic [math] –manifold [math] contains a ubiquitous collection of closed, immersed, quasi-Fuchsian surfaces. These surfaces are ubiquitous in the sense that their preimages in the universal cover separate any pair of disjoint, nonasymptotic geodesic planes. The proof relies in a crucial way on the corresponding theorem of Kahn and Markovic for closed [math] –manifolds. As a corollary of this result and a companion statement about surfaces with cusps, we recover Wise’s theorem that the fundamental group of [math] acts freely and cocompactly on a [math] cube complex.
</p>projecteuclid.org/euclid.gt/1552356082_20190311220143Mon, 11 Mar 2019 22:01 EDTOperads of genus zero curves and the Grothendieck–Teichmüller grouphttps://projecteuclid.org/euclid.gt/1552356083<strong>Pedro Boavida de Brito</strong>, <strong>Geoffroy Horel</strong>, <strong>Marcy Robertson</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 299--346.</p><p><strong>Abstract:</strong><br/>
We show that the group of homotopy automorphisms of the profinite completion of the genus zero surface operad is isomorphic to the (profinite) Grothendieck–Teichmüller group. Using a result of Drummond-Cole, we deduce that the Grothendieck–Teichmüller group acts nontrivially on [math] , the operad of stable curves of genus zero. As a second application, we give an alternative proof that the framed little [math] –disks operad is formal.
</p>projecteuclid.org/euclid.gt/1552356083_20190311220143Mon, 11 Mar 2019 22:01 EDTBirational models of moduli spaces of coherent sheaves on the projective planehttps://projecteuclid.org/euclid.gt/1552356084<strong>Chunyi Li</strong>, <strong>Xiaolei Zhao</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 347--426.</p><p><strong>Abstract:</strong><br/>
We study the birational geometry of moduli spaces of semistable sheaves on the projective plane via Bridgeland stability conditions. We show that the entire MMP of their moduli spaces can be run via wall-crossing. Via a description of the walls, we give a numerical description of their movable cones, along with its chamber decomposition corresponding to minimal models. As an application, we show that for primitive vectors, all birational models corresponding to open chambers in the movable cone are smooth and irreducible.
</p>projecteuclid.org/euclid.gt/1552356084_20190311220143Mon, 11 Mar 2019 22:01 EDTMotivic hyper-Kähler resolution conjecture, I: Generalized Kummer varietieshttps://projecteuclid.org/euclid.gt/1552356085<strong>Lie Fu</strong>, <strong>Zhiyu Tian</strong>, <strong>Charles Vial</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 427--492.</p><p><strong>Abstract:</strong><br/>
Given a smooth projective variety [math] endowed with a faithful action of a finite group [math] , following Jarvis–Kaufmann–Kimura (Invent. Math. 168 (2007) 23–81), and Fantechi–Göttsche (Duke Math. J. 117 (2003) 197–227), we define the orbifold motive (or Chen–Ruan motive) of the quotient stack [math] as an algebra object in the category of Chow motives. Inspired by Ruan (Contemp. Math. 312 (2002) 187–233), one can formulate a motivic version of his cohomological hyper-Kähler resolution conjecture (CHRC). We prove this motivic version, as well as its K–theoretic analogue conjectured by Jarvis–Kaufmann–Kimura in loc. cit., in two situations related to an abelian surface [math] and a positive integer [math] . Case (A) concerns Hilbert schemes of points of [math] : the Chow motive of [math] is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack [math] . Case (B) concerns generalized Kummer varieties: the Chow motive of the generalized Kummer variety [math] is isomorphic as algebra objects, up to a suitable sign change, to the orbifold motive of the quotient stack [math] , where [math] is the kernel abelian variety of the summation map [math] . As a by-product, we prove the original cohomological hyper-Kähler resolution conjecture for generalized Kummer varieties. As an application, we provide multiplicative Chow–Künneth decompositions for Hilbert schemes of abelian surfaces and for generalized Kummer varieties. In particular, we have a multiplicative direct sum decomposition of their Chow rings with rational coefficients, which is expected to be the splitting of the conjectural Bloch–Beilinson–Murre filtration. The existence of such a splitting for holomorphic symplectic varieties is conjectured by Beauville (London Math. Soc. Lecture Note Ser. 344 (2007) 38–53). Finally, as another application, we prove that over a nonempty Zariski open subset of the base, there exists a decomposition isomorphism [math] in [math] which is compatible with the cup products on both sides, where [math] is the relative generalized Kummer variety associated to a (smooth) family of abelian surfaces [math] .
</p>projecteuclid.org/euclid.gt/1552356085_20190311220143Mon, 11 Mar 2019 22:01 EDTTowards a quantum Lefschetz hyperplane theorem in all generahttps://projecteuclid.org/euclid.gt/1552356086<strong>Honglu Fan</strong>, <strong>Yuan-Pin Lee</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 493--512.</p><p><strong>Abstract:</strong><br/>
An effective algorithm of determining Gromov–Witten invariants of smooth hypersurfaces in any genus (subject to a degree bound) from Gromov–Witten invariants of the ambient space is proposed.
</p>projecteuclid.org/euclid.gt/1552356086_20190311220143Mon, 11 Mar 2019 22:01 EDT(Log-)epiperimetric inequality and regularity over smooth cones for almost area-minimizing currentshttps://projecteuclid.org/euclid.gt/1552356087<strong>Max Engelstein</strong>, <strong>Luca Spolaor</strong>, <strong>Bozhidar Velichkov</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 1, 513--540.</p><p><strong>Abstract:</strong><br/>
We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing any given trace in the radial direction along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (eg work of Reifenberg, Taylor and White), we need no a priori assumptions on the structure of the cone (eg integrability). If the cone is integrable (not only through rotations), we recover the classical epiperimetric inequality. As a consequence we deduce a new regularity result for almost area-minimizing currents at singular points where at least one blowup is a multiplicity-one cone with isolated singularity. This result is similar to the one for stationary varifolds of Leon Simon (1983), but independent from it since almost-minimizers do not satisfy any equation.
</p>projecteuclid.org/euclid.gt/1552356087_20190311220143Mon, 11 Mar 2019 22:01 EDTDerived induction and restriction theoryhttps://projecteuclid.org/euclid.gt/1555466424<strong>Akhil Mathew</strong>, <strong>Niko Naumann</strong>, <strong>Justin Noel</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 2, 541--636.</p><p><strong>Abstract:</strong><br/>
Let [math] be a finite group. To any family [math] of subgroups of [math] , we associate a thick [math] –ideal [math] of the category of [math] –spectra with the property that every [math] –spectrum in [math] (which we call [math] –nilpotent) can be reconstructed from its underlying [math] –spectra as [math] varies over [math] . A similar result holds for calculating [math] –equivariant homotopy classes of maps into such spectra via an appropriate homotopy limit spectral sequence. In general, the condition [math] implies strong collapse results for this spectral sequence as well as its dual homotopy colimit spectral sequence. As applications, we obtain Artin- and Brauer-type induction theorems for [math] –equivariant [math] –homology and cohomology, and generalizations of Quillen’s [math] –isomorphism theorem when [math] is a homotopy commutative [math] –ring spectrum.
We show that the subcategory [math] contains many [math] –spectra of interest for relatively small families [math] . These include [math] –equivariant real and complex [math] –theory as well as the Borel-equivariant cohomology theories associated to complex-oriented ring spectra, the [math] –local sphere, the classical bordism theories, connective real [math] –theory and any of the standard variants of topological modular forms. In each of these cases we identify the minimal family for which these results hold.
</p>projecteuclid.org/euclid.gt/1555466424_20190416220041Tue, 16 Apr 2019 22:00 EDTStrand algebras and contact categorieshttps://projecteuclid.org/euclid.gt/1555466428<strong>Daniel V Mathews</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 2, 637--683.</p><p><strong>Abstract:</strong><br/>
We demonstrate an isomorphism between the homology of the strand algebra of bordered Floer homology, and the category algebra of the contact category introduced by Honda. This isomorphism provides a direct correspondence between various notions of Floer homology and arc diagrams, on the one hand, and contact geometry and topology on the other. In particular, arc diagrams correspond to quadrangulated surfaces, idempotents correspond to certain basic dividing sets, strand diagrams correspond to contact structures, and multiplication of strand diagrams corresponds to stacking of contact structures. The contact structures considered are cubulated, and the cubes are shown to behave equivalently to local fragments of strand diagrams.
</p>projecteuclid.org/euclid.gt/1555466428_20190416220041Tue, 16 Apr 2019 22:00 EDTGauge theory on Aloff–Wallach spaceshttps://projecteuclid.org/euclid.gt/1555466429<strong>Gavin Ball</strong>, <strong>Goncalo Oliveira</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 2, 685--743.</p><p><strong>Abstract:</strong><br/>
For gauge groups [math] and [math] we classify invariant [math] –instantons for homogeneous coclosed [math] –structures on Aloff–Wallach spaces [math] . As a consequence, we give examples where [math] –instantons can be used to distinguish between different strictly nearly parallel [math] –structures on the same Aloff–Wallach space. In addition to this, we find that while certain [math] –instantons exist for the strictly nearly parallel [math] –structure on [math] , no such [math] –instantons exist for the [math] –Sasakian one. As a further consequence of the classification, we produce examples of some other interesting phenomena, such as irreducible [math] –instantons that, as the structure varies, merge into the same reducible and obstructed one and [math] –instantons on nearly parallel [math] –manifolds that are not locally energy-minimizing.
</p>projecteuclid.org/euclid.gt/1555466429_20190416220041Tue, 16 Apr 2019 22:00 EDTUpsilon-like concordance invariants from $\mathfrak{sl}_n$ knot cohomologyhttps://projecteuclid.org/euclid.gt/1555466430<strong>Lukas Lewark</strong>, <strong>Andrew Lobb</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 2, 745--780.</p><p><strong>Abstract:</strong><br/>
We construct smooth concordance invariants of knots [math] which take the form of piecewise linear maps [math] for [math] . These invariants arise from [math] knot cohomology. We verify some properties which are analogous to those of the invariant [math] (which arises from knot Floer homology), and some which differ. We make some explicit computations and give some topological applications.
Further to this, we define a concordance invariant from equivariant [math] knot cohomology which subsumes many known concordance invariants arising from quantum knot cohomologies.
</p>projecteuclid.org/euclid.gt/1555466430_20190416220041Tue, 16 Apr 2019 22:00 EDTOrbifolds of $n$–dimensional defect TQFTshttps://projecteuclid.org/euclid.gt/1555466431<strong>Nils Carqueville</strong>, <strong>Ingo Runkel</strong>, <strong>Gregor Schaumann</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 2, 781--864.</p><p><strong>Abstract:</strong><br/>
We introduce the notion of [math] –dimensional topological quantum field theory (TQFT) with defects as a symmetric monoidal functor on decorated stratified bordisms of dimension [math] . The familiar closed or open–closed TQFTs are special cases of defect TQFTs, and for [math] and [math] our general definition recovers what had previously been studied in the literature.
Our main construction is that of “generalised orbifolds” for any [math] –dimensional defect TQFT: Given a defect TQFT [math] , one obtains a new TQFT [math] by decorating the Poincaré duals of triangulated bordisms with certain algebraic data [math] and then evaluating with [math] . The orbifold datum [math] is constrained by demanding invariance under [math] –dimensional Pachner moves. This procedure generalises both state sum models and gauging of finite symmetry groups for any [math] . After developing the general theory, we focus on the case [math] .
</p>projecteuclid.org/euclid.gt/1555466431_20190416220041Tue, 16 Apr 2019 22:00 EDTOn homology cobordism and local equivalence between plumbed manifoldshttps://projecteuclid.org/euclid.gt/1555466432<strong>Irving Dai</strong>, <strong>Matthew Stoffregen</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 2, 865--924.</p><p><strong>Abstract:</strong><br/>
We establish a structural understanding of the involutive Heegaard Floer homology for all linear combinations of almost-rational (AR) plumbed three-manifolds. We use this to show that the Neumann–Siebenmann invariant is a homology cobordism invariant for all linear combinations of AR plumbed homology spheres. As a corollary, we prove that if [math] is a linear combination of AR plumbed homology spheres with [math] , then [math] is not torsion in the homology cobordism group. A general computation of the involutive Heegaard Floer correction terms for these spaces is also included.
</p>projecteuclid.org/euclid.gt/1555466432_20190416220041Tue, 16 Apr 2019 22:00 EDTEquivariant concentration in topological groupshttps://projecteuclid.org/euclid.gt/1555466433<strong>Friedrich Martin Schneider</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 2, 925--956.</p><p><strong>Abstract:</strong><br/>
We prove that, if [math] is a second-countable topological group with a compatible right-invariant metric [math] and [math] is a sequence of compactly supported Borel probability measures on [math] converging to invariance with respect to the mass transportation distance over [math] and such that [math] concentrates to a fully supported, compact [math] –space [math] , then [math] is homeomorphic to a [math] –invariant subspace of the Samuel compactification of [math] . In particular, this confirms a conjecture by Pestov and generalizes a well-known result by Gromov and Milman on the extreme amenability of topological groups. Furthermore, we exhibit a connection between the average orbit diameter of a metrizable flow of an arbitrary amenable topological group and the limit of Gromov’s observable diameters along any net of Borel probability measures UEB–converging to invariance over the group.
</p>projecteuclid.org/euclid.gt/1555466433_20190416220041Tue, 16 Apr 2019 22:00 EDTFloer cohomology, multiplicity and the log canonical thresholdhttps://projecteuclid.org/euclid.gt/1555466434<strong>Mark McLean</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 2, 957--1056.</p><p><strong>Abstract:</strong><br/>
Let [math] be a polynomial over the complex numbers with an isolated singularity at [math] . We show that the multiplicity and the log canonical threshold of [math] at [math] are invariants of the link of [math] viewed as a contact submanifold of the sphere.
This is done by first constructing a spectral sequence converging to the fixed-point Floer cohomology of any iterate of the Milnor monodromy map whose [math] page is explicitly described in terms of a log resolution of [math] . This spectral sequence is a generalization of a formula by A’Campo. By looking at this spectral sequence, we get a purely Floer-theoretic description of the multiplicity and log canonical threshold of [math] .
</p>projecteuclid.org/euclid.gt/1555466434_20190416220041Tue, 16 Apr 2019 22:00 EDTLagrangian mean curvature flow of Whitney sphereshttps://projecteuclid.org/euclid.gt/1555466435<strong>Andreas Savas-Halilaj</strong>, <strong>Knut Smoczyk</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 2, 1057--1084.</p><p><strong>Abstract:</strong><br/>
It is shown that an equivariant Lagrangian sphere with a positivity condition on its Ricci curvature develops a type-II singularity under the Lagrangian mean curvature flow that rescales to the product of a grim reaper with a flat Lagrangian subspace. In particular this result applies to the Whitney spheres.
</p>projecteuclid.org/euclid.gt/1555466435_20190416220041Tue, 16 Apr 2019 22:00 EDTCohomology classes of strata of differentialshttps://projecteuclid.org/euclid.gt/1559700271<strong>Adrien Sauvaget</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 3, 1085--1171.</p><p><strong>Abstract:</strong><br/>
We introduce a space of stable meromorphic differentials with poles of prescribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of multiplicities of zeros of the differential. The main goal of this paper is to compute the Poincaré-dual cohomology classes of all strata. We prove that all these classes are tautological and give an algorithm to compute them.
In the second part of the paper we study the Picard group of the strata. We use the tools introduced in the first part to deduce several relations in these Picard groups.
</p>projecteuclid.org/euclid.gt/1559700271_20190604220436Tue, 04 Jun 2019 22:04 EDTOn strongly quasiconvex subgroupshttps://projecteuclid.org/euclid.gt/1559700272<strong>Hung Cong Tran</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 3, 1173--1235.</p><p><strong>Abstract:</strong><br/>
We develop a theory of strongly quasiconvex subgroups of an arbitrary finitely generated group. Strong quasiconvexity generalizes quasiconvexity in hyperbolic groups and is preserved under quasi-isometry. We show that strongly quasiconvex subgroups are also more reflective of the ambient group’s geometry than the stable subgroups defined by Durham and Taylor, while still having many properties analogous to those of quasiconvex subgroups of hyperbolic groups. We characterize strongly quasiconvex subgroups in terms of the lower relative divergence of ambient groups with respect to them.
We also study strong quasiconvexity and stability in relatively hyperbolic groups, right-angled Coxeter groups, and right-angled Artin groups. We give complete descriptions of strong quasiconvexity and stability in relatively hyperbolic groups and we characterize strongly quasiconvex special subgroups and stable special subgroups of two-dimensional right-angled Coxeter groups. In the case of right-angled Artin groups, we prove that the two notions of strong quasiconvexity and stability are equivalent when the right-angled Artin group is one-ended and the subgroups have infinite index. We also characterize nontrivial strongly quasiconvex subgroups of infinite index (ie nontrivial stable subgroups) in right-angled Artin groups by quadratic lower relative divergence, expanding the work of Koberda, Mangahas, and Taylor on characterizing purely loxodromic subgroups of right-angled Artin groups.
</p>projecteuclid.org/euclid.gt/1559700272_20190604220436Tue, 04 Jun 2019 22:04 EDTA finite $\mathbb{Q}$–bad spacehttps://projecteuclid.org/euclid.gt/1559700273<strong>Sergei O Ivanov</strong>, <strong>Roman Mikhailov</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 3, 1237--1249.</p><p><strong>Abstract:</strong><br/>
We prove that, for a free noncyclic group [math] , the second homology group [math] is an uncountable [math] –vector space, where [math] denotes the [math] –completion of [math] . This solves a problem of A K Bousfield for the case of rational coefficients. As a direct consequence of this result, it follows that a wedge of two or more circles is [math] –bad in the sense of Bousfield–Kan. The same methods as used in the proof of the above result serve to show that [math] is not a divisible group, where [math] is the integral pronilpotent completion of [math] .
</p>projecteuclid.org/euclid.gt/1559700273_20190604220436Tue, 04 Jun 2019 22:04 EDTThe geometry of maximal components of the $\mathsf{PSp}(4,\mathbb R)$ character varietyhttps://projecteuclid.org/euclid.gt/1559700274<strong>Daniele Alessandrini</strong>, <strong>Brian Collier</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 3, 1251--1337.</p><p><strong>Abstract:</strong><br/>
We describe the space of maximal components of the character variety of surface group representations into [math] and [math] .
For every real rank [math] Lie group of Hermitian type, we construct a mapping class group invariant complex structure on the maximal components. For the groups [math] and [math] , we give a mapping class group invariant parametrization of each maximal component as an explicit holomorphic fiber bundle over Teichmüller space. Special attention is put on the connected components which are singular: we give a precise local description of the singularities and their geometric interpretation. We also describe the quotient of the maximal components of [math] and [math] by the action of the mapping class group as a holomorphic submersion over the moduli space of curves.
These results are proven in two steps: first we use Higgs bundles to give a nonmapping class group equivariant parametrization, then we prove an analog of Labourie’s conjecture for maximal [math] –representations.
</p>projecteuclid.org/euclid.gt/1559700274_20190604220436Tue, 04 Jun 2019 22:04 EDTSasaki–Einstein metrics and K–stabilityhttps://projecteuclid.org/euclid.gt/1559700275<strong>Tristan C Collins</strong>, <strong>Gábor Székelyhidi</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 3, 1339--1413.</p><p><strong>Abstract:</strong><br/>
We show that a polarized affine variety with an isolated singularity admits a Ricci flat Kähler cone metric if and only if it is K–stable. This generalizes the Chen–Donaldson–Sun solution of the Yau–Tian–Donaldson conjecture to Kähler cones, or equivalently, Sasakian manifolds. As an application we show that the five-sphere admits infinitely many families of Sasaki–Einstein metrics.
</p>projecteuclid.org/euclid.gt/1559700275_20190604220436Tue, 04 Jun 2019 22:04 EDTGromov–Witten theory of elliptic fibrations: Jacobi forms and holomorphic anomaly equationshttps://projecteuclid.org/euclid.gt/1559700276<strong>Georg Oberdieck</strong>, <strong>Aaron Pixton</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 3, 1415--1489.</p><p><strong>Abstract:</strong><br/>
We conjecture that the relative Gromov–Witten potentials of elliptic fibrations are (cycle-valued) lattice quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture for the rational elliptic surface in all genera and curve classes numerically. The generating series are quasi-Jacobi forms for the lattice [math] . We also show the compatibility of the conjecture with the degeneration formula. As a corollary we deduce that the Gromov–Witten potentials of the Schoen Calabi–Yau threefold (relative to [math] ) are [math] quasi-bi-Jacobi forms and satisfy a holomorphic anomaly equation. This yields a partial verification of the BCOV holomorphic anomaly equation for Calabi–Yau threefolds. For abelian surfaces the holomorphic anomaly equation is proven numerically in primitive classes. The theory of lattice quasi-Jacobi forms is reviewed.
In the appendix the conjectural holomorphic anomaly equation is expressed as a matrix action on the space of (generalized) cohomological field theories. The compatibility of the matrix action with the Jacobi Lie algebra is proven. Holomorphic anomaly equations for K3 fibrations are discussed in an example.
</p>projecteuclid.org/euclid.gt/1559700276_20190604220436Tue, 04 Jun 2019 22:04 EDTA deformation of instanton homology for webshttps://projecteuclid.org/euclid.gt/1559700277<strong>Peter B Kronheimer</strong>, <strong>Tomasz S Mrowka</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 3, 1491--1547.</p><p><strong>Abstract:</strong><br/>
A deformation of the authors’ instanton homology for webs is constructed by introducing a local system of coefficients. In the case that the web is planar, the rank of the deformed instanton homology is equal to the number of Tait colorings of the web.
</p>projecteuclid.org/euclid.gt/1559700277_20190604220436Tue, 04 Jun 2019 22:04 EDTInfinite loop spaces and positive scalar curvature in the presence of a fundamental grouphttps://projecteuclid.org/euclid.gt/1559700278<strong>Johannes Ebert</strong>, <strong>Oscar Randal-Williams</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 3, 1549--1610.</p><p><strong>Abstract:</strong><br/>
This is a continuation of our previous work with Botvinnik on the nontriviality of the secondary index invariant on spaces of metrics of positive scalar curvature, in which we take the fundamental group of the manifolds into account. We show that the secondary index invariant associated to the vanishing of the Rosenberg index can be highly nontrivial for positive scalar curvature Spin manifolds with torsionfree fundamental groups which satisfy the Baum–Connes conjecture. This gives the first example of the nontriviality of the group [math] –algebra-valued secondary index invariant on higher homotopy groups. As an application, we produce a compact Spin [math] –manifold whose space of positive scalar curvature metrics has each rational homotopy group infinite-dimensional.
At a more technical level, we introduce the notion of “stable metrics” and prove a basic existence theorem for them, which generalises the Gromov–Lawson surgery technique, and we also give a method for rounding corners of manifolds with positive scalar curvature metrics.
</p>projecteuclid.org/euclid.gt/1559700278_20190604220436Tue, 04 Jun 2019 22:04 EDTSharp entropy bounds for self-shrinkers in mean curvature flowhttps://projecteuclid.org/euclid.gt/1559700279<strong>Or Hershkovits</strong>, <strong>Brian White</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 3, 1611--1619.</p><p><strong>Abstract:</strong><br/>
Let [math] be a smooth, closed, codimension-one self-shrinker (for mean curvature flow) with nontrivial [math] homology. We show that the entropy of [math] is greater than or equal to the entropy of a round [math] -sphere, and that if equality holds, then [math] is a round [math] -sphere in [math] .
</p>projecteuclid.org/euclid.gt/1559700279_20190604220436Tue, 04 Jun 2019 22:04 EDTHolomorphic curves in exploded manifolds: regularityhttps://projecteuclid.org/euclid.gt/1563242518<strong>Brett Parker</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 4, 1621--1690.</p><p><strong>Abstract:</strong><br/>
The category of exploded manifolds is an extension of the category of smooth manifolds; for exploded manifolds, some adiabatic limits appear as smooth families. This paper studies the [math] equation on variations of a given family of curves in an exploded manifold. Roughly, we prove that the [math] equation on variations of an exploded family of curves behaves as nicely as the [math] equation on variations of a smooth family of smooth curves, even though exploded families of curves allow the development of normal-crossing or log-smooth singularities. The resulting regularity results are foundational to the author’s construction of Gromov–Witten invariants for exploded manifolds.
</p>projecteuclid.org/euclid.gt/1563242518_20190715220211Mon, 15 Jul 2019 22:02 EDTThe simplicial EHP sequence in $\mathbb{A}^{1}$–algebraic topologyhttps://projecteuclid.org/euclid.gt/1563242519<strong>Kirsten Wickelgren</strong>, <strong>Ben Williams</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 4, 1691--1777.</p><p><strong>Abstract:</strong><br/>
We give a tool for understanding simplicial desuspension in [math] –algebraic topology: we show that [math] is a fiber sequence up to homotopy in [math] –localized [math] algebraic topology for [math] with [math] . It follows that there is an EHP spectral sequence [math]
</p>projecteuclid.org/euclid.gt/1563242519_20190715220211Mon, 15 Jul 2019 22:02 EDTHausdorff dimension of boundaries of relatively hyperbolic groupshttps://projecteuclid.org/euclid.gt/1563242520<strong>Leonid Potyagailo</strong>, <strong>Wen-yuan Yang</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 4, 1779--1840.</p><p><strong>Abstract:</strong><br/>
We study the Hausdorff dimension of the Floyd and Bowditch boundaries of a relatively hyperbolic group, and show that, for the Floyd metric and shortcut metrics, they are both equal to a constant times the growth rate of the group.
In the proof, we study a special class of conical points called uniformly conical points and establish that, in both boundaries, there exists a sequence of Alhfors regular sets with dimension tending to the Hausdorff dimension and these sets consist of uniformly conical points.
</p>projecteuclid.org/euclid.gt/1563242520_20190715220211Mon, 15 Jul 2019 22:02 EDTHyperbolicity as an obstruction to smoothability for one-dimensional actionshttps://projecteuclid.org/euclid.gt/1563242521<strong>Christian Bonatti</strong>, <strong>Yash Lodha</strong>, <strong>Michele Triestino</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 4, 1841--1876.</p><p><strong>Abstract:</strong><br/>
Ghys and Sergiescu proved in the 1980s that Thompson’s group [math] , and hence [math] , admits actions by [math] diffeomorphisms of the circle. They proved that the standard actions of these groups are topologically conjugate to a group of [math] diffeomorphisms. Monod defined a family of groups of piecewise projective homeomorphisms, and Lodha and Moore defined finitely presentable groups of piecewise projective homeomorphisms. These groups are of particular interest because they are nonamenable and contain no free subgroup. In contrast to the result of Ghys and Sergiescu, we prove that the groups of Monod and Lodha and Moore are not topologically conjugate to a group of [math] diffeomorphisms.
Furthermore, we show that the group of Lodha and Moore has no nonabelian [math] action on the interval. We also show that many of Monod’s groups [math] , for instance when [math] is such that [math] contains a rational homothety [math] , do not admit a [math] action on the interval. The obstruction comes from the existence of hyperbolic fixed points for [math] actions. With slightly different techniques, we also show that some groups of piecewise affine homeomorphisms of the interval or the circle are not smoothable.
</p>projecteuclid.org/euclid.gt/1563242521_20190715220211Mon, 15 Jul 2019 22:02 EDTHolomorphic curves in exploded manifolds: virtual fundamental classhttps://projecteuclid.org/euclid.gt/1563242522<strong>Brett Parker</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 4, 1877--1960.</p><p><strong>Abstract:</strong><br/>
We define Gromov–Witten invariants of exploded manifolds. The technical heart of this paper is a construction of a virtual fundamental class [math] of any Kuranishi category [math] (which is a simplified, more general version of an embedded Kuranishi structure). We also show how to integrate differential forms over [math] to obtain numerical invariants, and push forward such differential forms over suitable maps. We show that such invariants are independent of any choices, and are compatible with pullbacks, products and tropical completion of Kuranishi categories.
In the case of a compact symplectic manifold, this gives an alternative construction of Gromov–Witten invariants, including gravitational descendants.
</p>projecteuclid.org/euclid.gt/1563242522_20190715220211Mon, 15 Jul 2019 22:02 EDTCentral limit theorem for spectral partial Bergman kernelshttps://projecteuclid.org/euclid.gt/1563242523<strong>Steve Zelditch</strong>, <strong>Peng Zhou</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 4, 1961--2004.</p><p><strong>Abstract:</strong><br/>
Partial Bergman kernels [math] are kernels of orthogonal projections onto subspaces [math] of holomorphic sections of the [math] power of an ample line bundle over a Kähler manifold [math] . The subspaces of this article are spectral subspaces [math] of the Toeplitz quantization [math] of a smooth Hamiltonian [math] . It is shown that the relative partial density of states satisfies [math] where [math] . Moreover it is shown that this partial density of states exhibits “Erf” asymptotics along the interface [math] ; that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values [math] and [math] of [math] . Such “Erf” asymptotics are a universal edge effect. The different types of scaling asymptotics are reminiscent of the law of large numbers and the central limit theorem.
</p>projecteuclid.org/euclid.gt/1563242523_20190715220211Mon, 15 Jul 2019 22:02 EDTFinite type invariants of knots in homology $3$–spheres with respect to null LP–surgerieshttps://projecteuclid.org/euclid.gt/1563242524<strong>Delphine Moussard</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 4, 2005--2050.</p><p><strong>Abstract:</strong><br/>
We study a theory of finite type invariants for nullhomologous knots in rational homology [math] –spheres with respect to null Lagrangian-preserving surgeries. It is an analogue in the setting of the rational homology of the Garoufalidis–Rozansky theory for knots in integral homology [math] –spheres. We give a partial combinatorial description of the graded space associated with our theory and determine some cases when this description is complete. For nullhomologous knots in rational homology [math] –spheres with a trivial Alexander polynomial, we show that the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant built from integrals in configuration spaces are universal finite type invariants for this theory; in particular, this implies that they are equivalent for such knots.
</p>projecteuclid.org/euclid.gt/1563242524_20190715220211Mon, 15 Jul 2019 22:02 EDTKlt varieties with trivial canonical class: holonomy, differential forms, and fundamental groupshttps://projecteuclid.org/euclid.gt/1563242525<strong>Daniel Greb</strong>, <strong>Henri Guenancia</strong>, <strong>Stefan Kebekus</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 4, 2051--2124.</p><p><strong>Abstract:</strong><br/>
We investigate the holonomy group of singular Kähler–Einstein metrics on klt varieties with numerically trivial canonical divisor. Finiteness of the number of connected components, a Bochner principle for holomorphic tensors, and a connection between irreducibility of holonomy representations and stability of the tangent sheaf are established. As a consequence, known decompositions for tangent sheaves of varieties with trivial canonical divisor are refined. In particular, we show that up to finite quasi-étale covers, varieties with strongly stable tangent sheaf are either Calabi–Yau or irreducible holomorphic symplectic. These results form one building block for Höring and Peternell’s recent proof of a singular version of the Beauville–Bogomolov decomposition theorem.
</p>projecteuclid.org/euclid.gt/1563242525_20190715220211Mon, 15 Jul 2019 22:02 EDTCubulable Kähler groupshttps://projecteuclid.org/euclid.gt/1563242526<strong>Thomas Delzant</strong>, <strong>Pierre Py</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 4, 2125--2164.</p><p><strong>Abstract:</strong><br/>
We prove that a Kähler group which is cubulable, i.e. which acts properly discontinuously and cocompactly on a [math] cubical complex, has a finite-index subgroup isomorphic to a direct product of surface groups, possibly with a free abelian factor. Similarly, we prove that a closed aspherical Kähler manifold with a cubulable fundamental group has a finite cover which is biholomorphic to a topologically trivial principal torus bundle over a product of Riemann surfaces. Along the way, we prove a factorization result for essential actions of Kähler groups on irreducible, locally finite [math] cubical complexes, under the assumption that there is no fixed point in the visual boundary.
</p>projecteuclid.org/euclid.gt/1563242526_20190715220211Mon, 15 Jul 2019 22:02 EDT