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 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 EDTHodge theory for intersection space cohomologyhttps://projecteuclid.org/euclid.gt/1571709623<strong>Markus Banagl</strong>, <strong>Eugénie Hunsicker</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 5, 2165--2225.</p><p><strong>Abstract:</strong><br/>
Given a perversity function in the sense of intersection homology theory, the method of intersection spaces assigns to certain oriented stratified spaces cell complexes whose ordinary reduced homology with real coefficients satisfies Poincaré duality across complementary perversities. The resulting homology theory is well known not to be isomorphic to intersection homology. For a two-strata pseudomanifold with product link bundle, we give a description of the cohomology of intersection spaces as a space of weighted [math] harmonic forms on the regular part, equipped with a fibred scattering metric. Some consequences of our methods for the signature are discussed as well.
</p>projecteuclid.org/euclid.gt/1571709623_20191021220047Mon, 21 Oct 2019 22:00 EDTOn the asymptotic dimension of the curve complexhttps://projecteuclid.org/euclid.gt/1571709624<strong>Mladen Bestvina</strong>, <strong>Ken Bromberg</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 5, 2227--2276.</p><p><strong>Abstract:</strong><br/>
We give a bound, linear in the complexity of the surface, to the asymptotic dimension of the curve complex as well as the capacity dimension of the ending lamination space.
</p>projecteuclid.org/euclid.gt/1571709624_20191021220047Mon, 21 Oct 2019 22:00 EDTSome finiteness results for groups of automorphisms of manifoldshttps://projecteuclid.org/euclid.gt/1571709625<strong>Alexander Kupers</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 5, 2277--2333.</p><p><strong>Abstract:</strong><br/>
We prove that in dimension [math] the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of [math] and various types of automorphisms of [math] –connected manifolds.
</p>projecteuclid.org/euclid.gt/1571709625_20191021220047Mon, 21 Oct 2019 22:00 EDTFourier–Mukai and autoduality for compactified Jacobians, IIhttps://projecteuclid.org/euclid.gt/1571709627<strong>Margarida Melo</strong>, <strong>Antonio Rapagnetta</strong>, <strong>Filippo Viviani</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 5, 2335--2395.</p><p><strong>Abstract:</strong><br/>
To every reduced (projective) curve [math] with planar singularities one can associate, following E Esteves, many fine compactified Jacobians, depending on the choice of a polarization on [math] , which are birational (possibly nonisomorphic) Calabi–Yau projective varieties with locally complete intersection singularities. We define a Poincaré sheaf on the product of any two (possibly equal) fine compactified Jacobians of [math] and show that the integral transform with kernel the Poincaré sheaf is an equivalence of their derived categories, hence it defines a Fourier–Mukai transform. As a corollary of this result, we prove that there is a natural equivariant open embedding of the connected component of the scheme parametrizing rank- [math] torsion-free sheaves on [math] into the connected component of the algebraic space parametrizing rank- [math] torsion-free sheaves on a given fine compactified Jacobian of [math] .
The main result can be interpreted in two ways. First of all, when the two fine compactified Jacobians are equal, the above Fourier–Mukai transform provides a natural autoequivalence of the derived category of any fine compactified Jacobian of [math] , which generalizes the classical result of S Mukai for Jacobians of smooth curves and the more recent result of D Arinkin for compactified Jacobians of integral curves with planar singularities. This provides further evidence for the classical limit of the geometric Langlands conjecture (as formulated by R Donagi and T Pantev). Second, when the two fine compactified Jacobians are different (and indeed possibly nonisomorphic), the above Fourier–Mukai transform provides a natural equivalence of their derived categories, thus it implies that any two fine compactified Jacobians of [math] are derived equivalent. This is in line with Kawamata’s conjecture that birational Calabi–Yau (smooth) varieties should be derived equivalent and it seems to suggest an extension of this conjecture to (mildly) singular Calabi–Yau varieties.
</p>projecteuclid.org/euclid.gt/1571709627_20191021220047Mon, 21 Oct 2019 22:00 EDTHomological stability of topological moduli spaceshttps://projecteuclid.org/euclid.gt/1571709628<strong>Manuel Krannich</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 5, 2397--2474.</p><p><strong>Abstract:</strong><br/>
Given a graded [math] –module over an [math] –algebra in spaces, we construct an augmented semi-simplicial space up to higher coherent homotopy over it, called its canonical resolution, whose graded connectivity yields homological stability for the graded pieces of the module with respect to constant and abelian coefficients. We furthermore introduce a notion of coefficient systems of finite degree in this context and show that, without further assumptions, the corresponding twisted homology groups stabilise as well. This generalises a framework of Randal-Williams and Wahl for families of discrete groups.
In many examples, the canonical resolution recovers geometric resolutions with known connectivity bounds. As a consequence, we derive new twisted homological stability results for various examples including moduli spaces of high-dimensional manifolds, unordered configuration spaces of manifolds with labels in a fibration, and moduli spaces of manifolds equipped with unordered embedded discs. This in turn implies representation stability for the ordered variants of the latter examples.
</p>projecteuclid.org/euclid.gt/1571709628_20191021220047Mon, 21 Oct 2019 22:00 EDTThe extended Bogomolny equations and generalized Nahm pole boundary conditionhttps://projecteuclid.org/euclid.gt/1571709629<strong>Siqi He</strong>, <strong>Rafe Mazzeo</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 5, 2475--2517.</p><p><strong>Abstract:</strong><br/>
We develop a Kobayashi–Hitchin-type correspondence between solutions of the extended Bogomolny equations on [math] with Nahm pole singularity at [math] and the Hitchin component of the stable [math] Higgs bundle; this verifies a conjecture of Gaiotto and Witten. We also develop a partial Kobayashi–Hitchin correspondence for solutions with a knot singularity in this program, corresponding to the non-Hitchin components in the moduli space of stable [math] Higgs bundles. We also prove existence and uniqueness of solutions with knot singularities on [math] .
</p>projecteuclid.org/euclid.gt/1571709629_20191021220047Mon, 21 Oct 2019 22:00 EDTHigher-order representation stability and ordered configuration spaces of manifoldshttps://projecteuclid.org/euclid.gt/1571709630<strong>Jeremy Miller</strong>, <strong>Jennifer C H Wilson</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 5, 2519--2591.</p><p><strong>Abstract:</strong><br/>
Using the language of twisted skew-commutative algebras, we define secondary representation stability , a stability pattern in the unstable homology of spaces that are representation stable in the sense of Church, Ellenberg and Farb (2015). We show that the rational homology of configuration spaces of ordered points in noncompact manifolds satisfies secondary representation stability. While representation stability for the homology of configuration spaces involves stabilizing by introducing a point “near infinity”, secondary representation stability involves stabilizing by introducing a pair of orbiting points — an operation that relates homology groups in different homological degrees. This result can be thought of as a representation-theoretic analogue of secondary homological stability in the sense of Galatius, Kupers and Randal-Williams (2018). In the course of the proof we establish some additional results: we give a new characterization of the homology of the complex of injective words, and we give a new proof of integral representation stability for configuration spaces of noncompact manifolds, extending previous results to nonorientable manifolds.
</p>projecteuclid.org/euclid.gt/1571709630_20191021220047Mon, 21 Oct 2019 22:00 EDTSpherical CR uniformization of Dehn surgeries of the Whitehead link complementhttps://projecteuclid.org/euclid.gt/1571709632<strong>Miguel Acosta</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 5, 2593--2664.</p><p><strong>Abstract:</strong><br/>
We apply a spherical CR Dehn surgery theorem in order to obtain infinitely many Dehn surgeries of the Whitehead link complement that carry spherical CR structures. We consider as a starting point the spherical CR uniformization of the Whitehead link complement constructed by Parker and Will, using a Ford domain in the complex hyperbolic plane [math] . We deform the Ford domain of Parker and Will in [math] in a one-parameter family. On one side, we obtain infinitely many spherical CR uniformizations on a particular Dehn surgery on one of the cusps of the Whitehead link complement. On the other side, we obtain spherical CR uniformizations for infinitely many Dehn surgeries on the same cusp of the Whitehead link complement. These manifolds are parametrized by an integer [math] , and the spherical CR structure obtained for [math] is the Deraux–Falbel spherical CR uniformization of the figure eight knot complement.
</p>projecteuclid.org/euclid.gt/1571709632_20191021220047Mon, 21 Oct 2019 22:00 EDTShake genus and slice genushttps://projecteuclid.org/euclid.gt/1571709633<strong>Lisa Piccirillo</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 5, 2665--2684.</p><p><strong>Abstract:</strong><br/>
An important difference between high-dimensional smooth manifolds and smooth [math] –manifolds that in a [math] –manifold it is not always possible to represent every middle-dimensional homology class with a smoothly embedded sphere. This is true even among the simplest [math] –manifolds: [math] obtained by attaching an [math] –framed [math] –handle to the [math] –ball along a knot [math] in [math] . The [math] –shake genus of [math] records the minimal genus among all smooth embedded surfaces representing a generator of the second homology of [math] and is clearly bounded above by the slice genus of [math] . We prove that slice genus is not an invariant of [math] , and thereby provide infinitely many examples of knots with [math] –shake genus strictly less than slice genus. This resolves Problem 1.41 of Kirby’s 1997 problem list. As corollaries we show that Rasmussen’s [math] invariant is not a [math] –trace invariant and we give examples, via the satellite operation, of bijective maps on the smooth concordance group which fix the identity but do not preserve slice genus. These corollaries resolve some questions from a conference at the Max Planck Institute, Bonn (2016).
</p>projecteuclid.org/euclid.gt/1571709633_20191021220047Mon, 21 Oct 2019 22:00 EDTGeometrically simply connected $4$–manifolds and stable cohomotopy Seiberg–Witten invariantshttps://projecteuclid.org/euclid.gt/1571709634<strong>Kouichi Yasui</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 5, 2685--2697.</p><p><strong>Abstract:</strong><br/>
We show that every positive definite closed [math] –manifold with [math] and without [math] –handles has a vanishing stable cohomotopy Seiberg–Witten invariant, and thus admits no symplectic structure. We also show that every closed oriented [math] –manifold with [math] and [math] and without [math] –handles admits no symplectic structure for at least one orientation of the manifold. In fact, relaxing the [math] –handle condition, we prove these results under more general conditions which are much easier to verify.
</p>projecteuclid.org/euclid.gt/1571709634_20191021220047Mon, 21 Oct 2019 22:00 EDTCorrection to the article An infinite-rank summand of topologically slice knotshttps://projecteuclid.org/euclid.gt/1571709635<strong>Jennifer Hom</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 5, 2699--2700.</p><p><strong>Abstract:</strong><br/>
We describe an error in the proof of a key proposition of our paper An infinite-rank summand of topologically slice knots (Geom. Topol. 19 (2015) 1063–1110), which was necessary for the proof of the main result. Alternative proofs of the main result are given by Ozsváth, Stipsicz and Szabó, and Dai, Hom, Stoffregen and Truong.
</p>projecteuclid.org/euclid.gt/1571709635_20191021220047Mon, 21 Oct 2019 22:00 EDTOn the symplectic cohomology of log Calabi–Yau surfaceshttps://projecteuclid.org/euclid.gt/1575687765<strong>James Pascaleff</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 6, 2701--2792.</p><p><strong>Abstract:</strong><br/>
We study the symplectic cohomology of affine algebraic surfaces that admit a compactification by a normal crossings anticanonical divisor. Using a toroidal structure near the compactification divisor, we describe the complex computing symplectic cohomology, and compute enough differentials to identify a basis for the degree zero part of the symplectic cohomology. This basis is indexed by integral points in a certain integral affine manifold, providing a relationship to the theta functions of Gross, Hacking and Keel. Included is a discussion of wrapped Floer cohomology of Lagrangian submanifolds and a description of the product structure in a special case. We also show that, after enhancing the coefficient ring, the degree zero symplectic cohomology defines a family degenerating to a singular surface obtained by gluing together several affine planes.
</p>projecteuclid.org/euclid.gt/1575687765_20191206220256Fri, 06 Dec 2019 22:02 ESTInradius collapsed manifoldshttps://projecteuclid.org/euclid.gt/1575687766<strong>Takao Yamaguchi</strong>, <strong>Zhilang Zhang</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 6, 2793--2860.</p><p><strong>Abstract:</strong><br/>
We study collapsed manifolds with boundary, where we assume a lower sectional curvature bound, two side bounds on the second fundamental forms of boundaries and upper diameter bound. Our main concern is the case when inradii of manifolds converge to zero. This is a typical case of collapsing manifolds with boundary. We determine the limit spaces of inradius collapsed manifolds as Alexandrov spaces with curvature uniformly bounded below. When the limit space has codimension one, we completely determine the topology of inradius collapsed manifold in terms of singular [math] –bundles. General inradius collapse to almost regular spaces are also characterized. In the general case of unbounded diameters, we prove that the number of boundary components of inradius collapsed manifolds is at most two, where the disconnected boundary happens if and only if the manifold has a topological product structure.
</p>projecteuclid.org/euclid.gt/1575687766_20191206220256Fri, 06 Dec 2019 22:02 ESTRationality, universal generation and the integral Hodge conjecturehttps://projecteuclid.org/euclid.gt/1575687767<strong>Mingmin Shen</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 6, 2861--2898.</p><p><strong>Abstract:</strong><br/>
We use the universal generation of algebraic cycles to relate (stable) rationality to the integral Hodge conjecture. We show that the Chow group of [math] –cycles on a cubic hypersurface is universally generated by lines. Applications are mainly in cubic hypersurfaces of low dimensions. For example, we show that if a generic cubic fourfold is stably rational then the Beauville–Bogomolov form on its variety of lines, viewed as an integral Hodge class on the self product of its variety of lines, is algebraic. In dimensions [math] and [math] , we relate stable rationality with the geometry of the associated intermediate Jacobian.
</p>projecteuclid.org/euclid.gt/1575687767_20191206220256Fri, 06 Dec 2019 22:02 ESTLocal topological rigidity of nongeometric $3$–manifoldshttps://projecteuclid.org/euclid.gt/1575687768<strong>Filippo Cerocchi</strong>, <strong>Andrea Sambusetti</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 6, 2899--2927.</p><p><strong>Abstract:</strong><br/>
We study Riemannian metrics on compact, orientable, nongeometric [math] –manifolds (ie those whose interior does not support any of the eight model geometries) with torsionless fundamental group and (possibly empty) nonspherical boundary. We prove a lower bound “à la Margulis” for the systole and a volume estimate for these manifolds, only in terms of upper bounds on the entropy and diameter. We then deduce corresponding local topological rigidity results for manifolds in this class whose entropy and diameter are bounded respectively by [math] and [math] . For instance, this class locally contains only finitely many topological types; and closed, irreducible manifolds in this class which are close enough (with respect to [math] and [math] ) are diffeomorphic. Several examples and counterexamples are produced to stress the differences with the geometric case.
</p>projecteuclid.org/euclid.gt/1575687768_20191206220256Fri, 06 Dec 2019 22:02 ESTBoundaries of Dehn fillingshttps://projecteuclid.org/euclid.gt/1575687769<strong>Daniel Groves</strong>, <strong>Jason Fox Manning</strong>, <strong>Alessandro Sisto</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 6, 2929--3002.</p><p><strong>Abstract:</strong><br/>
We begin an investigation into the behavior of Bowditch and Gromov boundaries under the operation of Dehn filling. In particular, we show many Dehn fillings of a toral relatively hyperbolic group with [math] –sphere boundary are hyperbolic with [math] –sphere boundary. As an application, we show that the Cannon conjecture implies a relatively hyperbolic version of the Cannon conjecture.
</p>projecteuclid.org/euclid.gt/1575687769_20191206220256Fri, 06 Dec 2019 22:02 ESTHomotopy groups of the observer moduli space of Ricci positive metricshttps://projecteuclid.org/euclid.gt/1575687770<strong>Boris Botvinnik</strong>, <strong>Mark G Walsh</strong>, <strong>David J Wraith</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 6, 3003--3040.</p><p><strong>Abstract:</strong><br/>
The observer moduli space of Riemannian metrics is the quotient of the space [math] of all Riemannian metrics on a manifold [math] by the group of diffeomorphisms [math] which fix both a basepoint [math] and the tangent space at [math] . The group [math] acts freely on [math] provided that [math] is connected. This offers certain advantages over the classic moduli space, which is the quotient by the full diffeomorphism group. Results due to Botvinnik, Hanke, Schick and Walsh, and Hanke, Schick and Steimle have demonstrated that the higher homotopy groups of the observer moduli space [math] of positive scalar curvature metrics are, in many cases, nontrivial. The aim in the current paper is to establish similar results for the moduli space [math] of metrics with positive Ricci curvature. In particular we show that for a given [math] , there are infinite-order elements in the homotopy group [math] provided the dimension [math] is odd and sufficiently large. In establishing this we make use of a gluing result of Perelman. We provide full details of the proof of this gluing theorem, which we believe have not appeared before in the literature. We also extend this to a family gluing theorem for Ricci positive manifolds.
</p>projecteuclid.org/euclid.gt/1575687770_20191206220256Fri, 06 Dec 2019 22:02 ESTContact integral geometry and the Heisenberg algebrahttps://projecteuclid.org/euclid.gt/1575687771<strong>Dmitry Faifman</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 6, 3041--3110.</p><p><strong>Abstract:</strong><br/>
Generalizing Weyl’s tube formula and building on Chern’s work, Alesker reinterpreted the Lipschitz–Killing curvature integrals as a family of valuations (finitely additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings. We uncover a similar structure for contact manifolds. Namely, we show that a contact manifold admits a canonical family of generalized valuations, which are universal under contact embeddings. Those valuations assign numerical invariants to even-dimensional submanifolds, which in a certain sense measure the curvature at points of tangency to the contact structure. Moreover, these valuations generalize to the class of manifolds equipped with the structure of a Heisenberg algebra on their cotangent bundle. Pursuing the analogy with Euclidean integral geometry, we construct symplectic-invariant distributions on Grassmannians to produce Crofton formulas on the contact sphere. Using closely related distributions, we obtain Crofton formulas also in the linear symplectic space.
</p>projecteuclid.org/euclid.gt/1575687771_20191206220256Fri, 06 Dec 2019 22:02 ESTMetric-minimizing surfaces revisitedhttps://projecteuclid.org/euclid.gt/1575687772<strong>Anton Petrunin</strong>, <strong>Stephan Stadler</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 6, 3111--3139.</p><p><strong>Abstract:</strong><br/>
A surface that does not admit a length nonincreasing deformation is called metric-minimizing . We show that metric-minimizing surfaces in [math] spaces are locally [math] with respect to their length metrics.
</p>projecteuclid.org/euclid.gt/1575687772_20191206220256Fri, 06 Dec 2019 22:02 ESTPlato's cave and differential formshttps://projecteuclid.org/euclid.gt/1575687773<strong>Fedor Manin</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 6, 3141--3202.</p><p><strong>Abstract:</strong><br/>
In the 1970s and again in the 1990s, Gromov gave a number of theorems and conjectures motivated by the notion that the real homotopy theory of compact manifolds and simplicial complexes influences the geometry of maps between them. The main technical result of this paper supports this intuition: we show that maps of differential algebras are closely shadowed, in a technical sense, by maps between the corresponding spaces. As a concrete application, we prove the following conjecture of Gromov: if [math] and [math] are finite complexes with [math] simply connected, then there are constants [math] and [math] such that any two homotopic [math] –Lipschitz maps have a [math] –Lipschitz homotopy (and if one of the maps is constant, [math] can be taken to be [math] ). We hope that it will lead more generally to a better understanding of the space of maps from [math] to [math] in this setting.
</p>projecteuclid.org/euclid.gt/1575687773_20191206220256Fri, 06 Dec 2019 22:02 ESTNonnegative Ricci curvature, stability at infinity and finite generation of fundamental groupshttps://projecteuclid.org/euclid.gt/1575687774<strong>Jiayin Pan</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 6, 3203--3231.</p><p><strong>Abstract:</strong><br/>
We study the fundamental group of an open [math] –manifold [math] of nonnegative Ricci curvature. We show that if there is an integer [math] such that any tangent cone at infinity of the Riemannian universal cover of [math] is a metric cone whose maximal Euclidean factor has dimension [math] , then [math] is finitely generated. In particular, this confirms the Milnor conjecture for a manifold whose universal cover has Euclidean volume growth and a unique tangent cone at infinity.
</p>projecteuclid.org/euclid.gt/1575687774_20191206220256Fri, 06 Dec 2019 22:02 ESTThe fundamental group of compact Kähler threefoldshttps://projecteuclid.org/euclid.gt/1578366024<strong>Benoît Claudon</strong>, <strong>Andreas Höring</strong>, <strong>Hsueh-Yung Lin</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 7, 3233--3271.</p><p><strong>Abstract:</strong><br/>
Let [math] be a compact Kähler manifold of dimension three. We prove that there exists a projective manifold [math] such that [math] . We also prove the bimeromorphic existence of algebraic approximations for compact Kähler manifolds of algebraic dimension [math] . Together with the work of Graf and the third author, this settles in particular the bimeromorphic Kodaira problem for compact Kähler threefolds.
</p>projecteuclid.org/euclid.gt/1578366024_20200106220049Mon, 06 Jan 2020 22:00 ESTResolution of singularities and geometric proofs of the Łojasiewicz inequalitieshttps://projecteuclid.org/euclid.gt/1578366029<strong>Paul M N Feehan</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 7, 3273--3313.</p><p><strong>Abstract:</strong><br/>
The Łojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanisław Łojasiewicz (1959, 1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). Here we first give an elementary geometric, coordinate-based proof of the Łojasiewicz inequalities in the special case where the function is [math] with simple normal crossings. We then prove, partly following Bierstone and Milman (1997) and using resolution of singularities for (real or complex) analytic varieties, that the gradient inequality for an arbitrary analytic function follows from the special case where it has simple normal crossings. In addition, we prove the Łojasiewicz inequalities when a function is [math] and generalized Morse–Bott of order [math] ; we earlier gave an elementary proof of the Łojasiewicz inequalities when a function is [math] and Morse–Bott on a Banach space.
</p>projecteuclid.org/euclid.gt/1578366029_20200106220049Mon, 06 Jan 2020 22:00 ESTModuli of stable maps in genus one and logarithmic geometry, Ihttps://projecteuclid.org/euclid.gt/1578366030<strong>Dhruv Ranganathan</strong>, <strong>Keli Santos-Parker</strong>, <strong>Jonathan Wise</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 7, 3315--3366.</p><p><strong>Abstract:</strong><br/>
This is the first in a pair of papers developing a framework for the application of logarithmic structures in the study of singular curves of genus [math] . We construct a smooth and proper moduli space dominating the main component of Kontsevich’s space of stable genus [math] maps to projective space. A variation on this theme furnishes a modular interpretation for Vakil and Zinger’s famous desingularization of the Kontsevich space of maps in genus [math] . Our methods also lead to smooth and proper moduli spaces of pointed genus [math] quasimaps to projective space. Finally, we present an application to the log minimal model program for [math] . We construct explicit factorizations of the rational maps among Smyth’s modular compactifications of pointed elliptic curves.
</p>projecteuclid.org/euclid.gt/1578366030_20200106220049Mon, 06 Jan 2020 22:00 ESTThe classification of Lagrangians nearby the Whitney immersionhttps://projecteuclid.org/euclid.gt/1578366031<strong>Georgios Dimitroglou Rizell</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 7, 3367--3458.</p><p><strong>Abstract:</strong><br/>
The Whitney immersion is a Lagrangian sphere inside the four-dimensional symplectic vector space which has a single transverse double point of Whitney self-intersection number [math] . This Lagrangian also arises as the Weinstein skeleton of the complement of a binodal cubic curve inside the projective plane, and the latter Weinstein manifold is thus the “standard” neighbourhood of Lagrangian immersions of this type. We classify the Lagrangians inside such a neighbourhood which are homologically essential, and which are either embedded or immersed with a single double point; they are shown to be Hamiltonian isotopic to either product tori, Chekanov tori, or rescalings of the Whitney immersion.
</p>projecteuclid.org/euclid.gt/1578366031_20200106220049Mon, 06 Jan 2020 22:00 ESTToric geometry of $\mathrm{G}_2$–manifoldshttps://projecteuclid.org/euclid.gt/1578366032<strong>Thomas Bruun Madsen</strong>, <strong>Andrew Swann</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 7, 3459--3500.</p><p><strong>Abstract:</strong><br/>
We consider [math] –manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of [math] –actions is found to be distinguished. For such actions multi-Hamiltonian with respect to both the three- and four-form, we derive a Gibbons–Hawking type ansatz giving the geometry on an open dense set in terms a symmetric [math] matrix of functions. This leads to particularly simple examples of explicit metrics with holonomy equal to [math] . We prove that the multimoment maps exhibit the full orbit space topologically as a smooth four-manifold containing a trivalent graph as the image of the set of special orbits and describe these graphs in some complete examples.
</p>projecteuclid.org/euclid.gt/1578366032_20200106220049Mon, 06 Jan 2020 22:00 ESTAppearance of stable minimal spheres along the Ricci flow in positive scalar curvaturehttps://projecteuclid.org/euclid.gt/1578366033<strong>Antoine Song</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 7, 3501--3535.</p><p><strong>Abstract:</strong><br/>
We construct spherical space forms [math] with positive scalar curvature and containing no stable embedded minimal surfaces such that the following happens along the Ricci flow starting at [math] : a stable embedded minimal [math] –sphere appears and a nontrivial singularity occurs. We also give in dimension [math] a general construction of Type I neckpinching and clarify the relationship between stable spheres and nontrivial Type I singularities of the Ricci flow. Some symmetry assumptions prevent the appearance of stable spheres, and this has consequences on the types of singularities which can occur for metrics with these symmetries.
</p>projecteuclid.org/euclid.gt/1578366033_20200106220049Mon, 06 Jan 2020 22:00 ESTDR/DZ equivalence conjecture and tautological relationshttps://projecteuclid.org/euclid.gt/1578366034<strong>Alexandr Buryak</strong>, <strong>Jérémy Guéré</strong>, <strong>Paolo Rossi</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 7, 3537--3600.</p><p><strong>Abstract:</strong><br/>
We present a family of conjectural relations in the tautological ring of the moduli spaces of stable curves which implies the strong double ramification/Dubrovin–Zhang equivalence conjecture introduced by the authors with Dubrovin (Comm. Math. Phys. 363 (2018) 191–260). Our tautological relations have the form of an equality between two different families of tautological classes, only one of which involves the double ramification cycle. We prove that both families behave the same way upon pullback and pushforward with respect to forgetting a marked point. We also prove that our conjectural relations are true in genus [math] and [math] and also when first pushed forward from [math] to [math] and then restricted to [math] for any [math] . Finally we show that, for semisimple CohFTs, the DR/DZ equivalence only depends on a subset of our relations, finite in each genus, which we prove for [math] . As an application we find a new formula for the class [math] as a linear combination of dual trees intersected with kappa- and psi-classes, and we check it for [math] .
</p>projecteuclid.org/euclid.gt/1578366034_20200106220049Mon, 06 Jan 2020 22:00 ESTTorsion contact forms in three dimensions have two or infinitely many Reeb orbitshttps://projecteuclid.org/euclid.gt/1578366035<strong>Dan Cristofaro-Gardiner</strong>, <strong>Michael Hutchings</strong>, <strong>Daniel Pomerleano</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 7, 3601--3645.</p><p><strong>Abstract:</strong><br/>
We prove that every nondegenerate contact form on a closed connected three-manifold such that the associated contact structure has torsion first Chern class has either two or infinitely many simple Reeb orbits. By previous results it follows that under the above assumptions, there are infinitely many simple Reeb orbits if the three-manifold is not the three-sphere or a lens space. We also show that for nontorsion contact structures, every nondegenerate contact form has at least four simple Reeb orbits.
</p>projecteuclid.org/euclid.gt/1578366035_20200106220049Mon, 06 Jan 2020 22:00 ESTThe Morel–Voevodsky localization theorem in spectral algebraic geometryhttps://projecteuclid.org/euclid.gt/1578366036<strong>Adeel A Khan</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 7, 3647--3685.</p><p><strong>Abstract:</strong><br/>
We prove an analogue of the Morel–Voevodsky localization theorem over spectral algebraic spaces. As a corollary we deduce a “derived nilpotent-invariance” result which, informally speaking, says that [math] –homotopy-invariance kills all higher homotopy groups of a connective commutative ring spectrum.
</p>projecteuclid.org/euclid.gt/1578366036_20200106220049Mon, 06 Jan 2020 22:00 ESTReidemeister torsion, complex volume and the Zograf infinite product for hyperbolic $3$–manifoldshttps://projecteuclid.org/euclid.gt/1578366037<strong>Jinsung Park</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 7, 3687--3734.</p><p><strong>Abstract:</strong><br/>
We prove an equality which involves Reidemeister torsion, complex volume and the Zograf infinite product for hyperbolic [math] –manifolds.
</p>projecteuclid.org/euclid.gt/1578366037_20200106220049Mon, 06 Jan 2020 22:00 ESTOn the nonrealizability of braid groups by homeomorphismshttps://projecteuclid.org/euclid.gt/1578366038<strong>Lei Chen</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 23, Number 7, 3735--3749.</p><p><strong>Abstract:</strong><br/>
We show that the projection [math] does not have a section for [math] ; ie the braid group [math] cannot be geometrically realized as a group of homeomorphisms of a disk fixing the boundary pointwise and [math] marked points in the interior as a set. We also give a new proof of a result of Markovic (2007) that the mapping class group of a surface of genus [math] cannot be geometrically realized as a group of homeomorphisms when [math] .
</p>projecteuclid.org/euclid.gt/1578366038_20200106220049Mon, 06 Jan 2020 22:00 ESTCoalgebraic formal curve spectra and spectral jet spaceshttps://projecteuclid.org/euclid.gt/1585706427<strong>Eric Peterson</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 24, Number 1, 1--47.</p><p><strong>Abstract:</strong><br/>
We import into homotopy theory the algebrogeometric construction of the cotangent space of a geometric point on a scheme. Specializing to the category of spectra local to a Morava [math] –theory of height [math] , we show that this can be used to produce a choice-free model of the determinantal sphere as well as an efficient Picard-graded cellular decomposition of [math] . Coupling these ideas to work of Westerland, we give a “Snaith’s theorem” for the Iwasawa extension of the [math] –local sphere.
</p>projecteuclid.org/euclid.gt/1585706427_20200331220031Tue, 31 Mar 2020 22:00 EDTSutured manifolds and polynomial invariants from higher rank bundleshttps://projecteuclid.org/euclid.gt/1585706428<strong>Aliakbar Daemi</strong>, <strong>Yi Xie</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 24, Number 1, 49--178.</p><p><strong>Abstract:</strong><br/>
For each integer [math] , Mariño and Moore defined generalized Donaldson invariants by the methods of quantum field theory, and made predictions about the values of these invariants. Subsequently, Kronheimer gave a rigorous definition of generalized Donaldson invariants using the moduli spaces of anti-self-dual connections on hermitian vector bundles of rank [math] . We confirm the predictions of Mariño and Moore for simply connected elliptic surfaces without multiple fibers and certain surfaces of general type in the case that [math] . The primary motivation is to study [math] –manifold instanton Floer homologies which are defined by higher rank bundles. In particular, the computation of the generalized Donaldson invariants is exploited to define a Floer homology theory for sutured [math] –manifolds.
</p>projecteuclid.org/euclid.gt/1585706428_20200331220031Tue, 31 Mar 2020 22:00 EDTContact handles, duality, and sutured Floer homologyhttps://projecteuclid.org/euclid.gt/1585706429<strong>András Juhász</strong>, <strong>Ian Zemke</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 24, Number 1, 179--307.</p><p><strong>Abstract:</strong><br/>
We give an explicit construction of the Honda–Kazez–Matić gluing maps in terms of contact handles. We use this to prove a duality result for turning a sutured manifold cobordism around and to compute the trace in the sutured Floer TQFT. We also show that the decorated link cobordism maps on the hat version of link Floer homology defined by the first author via sutured manifold cobordisms and by the second author via elementary cobordisms agree.
</p>projecteuclid.org/euclid.gt/1585706429_20200331220031Tue, 31 Mar 2020 22:00 EDTConical metrics on Riemann surfaces, I: The compactified configuration space and regularityhttps://projecteuclid.org/euclid.gt/1585706430<strong>Rafe Mazzeo</strong>, <strong>Xuwen Zhu</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 24, Number 1, 309--372.</p><p><strong>Abstract:</strong><br/>
We introduce a compactification of the space of simple positive divisors on a Riemann surface, as well as a compactification of the universal family of punctured surfaces above this space. These are real manifolds with corners. We then study the space of constant curvature metrics on this Riemann surface with prescribed conical singularities at these divisors. Our interest here is in the local deformation for these metrics, and in particular the behavior of these families as conic points coalesce. We prove a sharp regularity theorem for this phenomenon in the regime where these metrics are known to exist. This setting will be used in a subsequent paper to study the space of spherical conic metrics with large cone angles, where the existence theory is still incomplete. Of independent interest is how setting up the analysis on these compactified configuration spaces provides a good framework for analyzing “confluent families” of regular singular, ie conic, elliptic differential operators.
</p>projecteuclid.org/euclid.gt/1585706430_20200331220031Tue, 31 Mar 2020 22:00 EDT$\mathrm{GL}_2 \mathbb{R}$–invariant measures in marked strata: generic marked points, Earle–Kra for strata and illuminationhttps://projecteuclid.org/euclid.gt/1585706431<strong>Paul Apisa</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 24, Number 1, 373--408.</p><p><strong>Abstract:</strong><br/>
We show that nontrivial [math] –invariant point markings for translation surfaces in strata of abelian differentials exist only when the translation surface belongs to a hyperelliptic component. As an application, we establish constraints on sections of the universal curve restricted to orbifold covers of subvarieties of the moduli space of Riemann surfaces that contain a Teichmüller disk. We also solve the finite blocking problem for generic translation surfaces.
</p>projecteuclid.org/euclid.gt/1585706431_20200331220031Tue, 31 Mar 2020 22:00 EDTRecognition of being fibered for compact $3$–manifoldshttps://projecteuclid.org/euclid.gt/1585706432<strong>Andrei Jaikin-Zapirain</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 24, Number 1, 409--420.</p><p><strong>Abstract:</strong><br/>
Let [math] be a compact orientable aspherical [math] –manifold. We show that if the profinite completion of [math] is isomorphic to the profinite completion of a free-by-cyclic group or to the profinite completion of a surface-by-cyclic group, then [math] fibers over the circle with compact fiber.
</p>projecteuclid.org/euclid.gt/1585706432_20200331220031Tue, 31 Mar 2020 22:00 EDTEdge stabilization in the homology of graph braid groupshttps://projecteuclid.org/euclid.gt/1585706433<strong>Byung Hee An</strong>, <strong>Gabriel Drummond-Cole</strong>, <strong>Ben Knudsen</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 24, Number 1, 421--469.</p><p><strong>Abstract:</strong><br/>
We introduce a novel type of stabilization map on the configuration spaces of a graph which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges, and we show that this homology module is finitely generated. An analogue of classical homological and representation stability for manifolds, this result implies eventual polynomial growth of Betti numbers. We calculate the exact degree of this polynomial, in particular verifying an upper bound conjectured by Ramos. Because the action arises from a family of continuous maps, it lifts to an action at the level of singular chains which contains strictly more information than the homology-level action. We show that the resulting differential graded module is almost never formal over the ring of edges.
</p>projecteuclid.org/euclid.gt/1585706433_20200331220031Tue, 31 Mar 2020 22:00 EDTMin-max minimal disks with free boundary in Riemannian manifoldshttps://projecteuclid.org/euclid.gt/1585706434<strong>Longzhi Lin</strong>, <strong>Ao Sun</strong>, <strong>Xin Zhou</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 24, Number 1, 471--532.</p><p><strong>Abstract:</strong><br/>
We establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory for the Plateau problem of minimal disks, which can be used to generalize the famous work by Morse–Tompkins and Shiffman on minimal surfaces in [math] to the Riemannian setting.
More precisely, we generalize, to the free boundary setting, the min-max construction of minimal surfaces using harmonic replacement introduced by Colding–Minicozzi. As a key ingredient to this construction, we show an energy convexity for weakly harmonic maps with mixed Dirichlet and free boundaries from the half unit [math] –disk in [math] into any closed Riemannian manifold, which in particular yields the uniqueness of such weakly harmonic maps. This is a free boundary analogue of the energy convexity and uniqueness for weakly harmonic maps with Dirichlet boundary on the unit [math] –disk proved by Colding and Minicozzi.
</p>projecteuclid.org/euclid.gt/1585706434_20200331220031Tue, 31 Mar 2020 22:00 EDT