Algebraic & Geometric Topology Articles (Project Euclid)
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The latest articles from Algebraic & Geometric 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 12:44 EDTThu, 19 Oct 2017 12:44 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Indecomposable nonorientable $\mathrm{PD}_3$–complexes
https://projecteuclid.org/euclid.agt/1508431438
<strong>Jonathan Hillman</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 2, 645--656.</p><p><strong>Abstract:</strong><br/>
We show that the orientable double covering space of an indecomposable, nonorientable [math] –complex has torsion-free fundamental group.
</p>projecteuclid.org/euclid.agt/1508431438_20171019124413Thu, 19 Oct 2017 12:44 EDTKlein-four connections and the Casson invariant for nontrivial admissible $U(2)$ bundleshttps://projecteuclid.org/euclid.agt/1510841485<strong>Christopher Scaduto</strong>, <strong>Matthew Stoffregen</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 2841--2861.</p><p><strong>Abstract:</strong><br/>
Given a rank-2 hermitian bundle over a [math] –manifold that is nontrivial admissible in the sense of Floer, one defines its Casson invariant as half the signed count of its projectively flat connections, suitably perturbed. We show that the [math] –divisibility of this integer invariant is controlled in part by a formula involving the mod 2 cohomology ring of the [math] –manifold. This formula counts flat connections on the induced adjoint bundle with Klein-four holonomy.
</p>projecteuclid.org/euclid.agt/1510841485_20171116091126Thu, 16 Nov 2017 09:11 ESTInfinite order corks via handle diagramshttps://projecteuclid.org/euclid.agt/1510841486<strong>Robert Gompf</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 2863--2891.</p><p><strong>Abstract:</strong><br/>
The author recently proved the existence of an infinite order cork: a compact, contractible submanifold [math] of a 4–manifold and an infinite order diffeomorphism [math] of [math] such that cutting out [math] and regluing it by distinct powers of [math] yields pairwise nondiffeomorphic manifolds. The present paper exhibits the first handle diagrams of this phenomenon, by translating the earlier proof into this language (for each of the infinitely many corks arising in the first paper). The cork twists in these papers are twists on incompressible tori. We give conditions guaranteeing that such twists do not change the diffeomorphism type of a 4–manifold, partially answering a question from the original paper. We also show that the “ [math] –moves” recently introduced by Akbulut are essentially equivalent to torus twists.
</p>projecteuclid.org/euclid.agt/1510841486_20171116091126Thu, 16 Nov 2017 09:11 ESTDetecting essential surfaces as intersections in the character varietyhttps://projecteuclid.org/euclid.agt/1510841487<strong>Michelle Chu</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 2893--2914.</p><p><strong>Abstract:</strong><br/>
We describe a family of hyperbolic knots whose character variety contain exactly two distinct components of characters of irreducible representations. The intersection points between the components carry rich topological information. In particular, these points are nonintegral and detect a Seifert surface.
</p>projecteuclid.org/euclid.agt/1510841487_20171116091126Thu, 16 Nov 2017 09:11 ESTThe surgery exact triangle in $\mathrm{Pin}(2)\mskip-1.5mu$–monopole Floer homologyhttps://projecteuclid.org/euclid.agt/1510841488<strong>Francesco Lin</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 2915--2960.</p><p><strong>Abstract:</strong><br/>
We prove the existence of an exact triangle for the [math] –monopole Floer homology groups of three-manifolds related by specific Dehn surgeries on a given knot. Unlike the counterpart in usual monopole Floer homology, only two of the three maps are those induced by the corresponding elementary cobordism. We use this triangle to describe the Manolescu correction terms of the manifolds obtained by [math] –surgery on alternating knots with Arf invariant [math] .
</p>projecteuclid.org/euclid.agt/1510841488_20171116091126Thu, 16 Nov 2017 09:11 ESTOdd knot invariants from quantum covering groupshttps://projecteuclid.org/euclid.agt/1510841489<strong>Sean Clark</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 2961--3005.</p><p><strong>Abstract:</strong><br/>
We show that the quantum covering group associated to [math] has an associated colored quantum knot invariant à la Reshetikhin–Turaev, which specializes to a quantum knot invariant for [math] , and to the usual quantum knot invariant for [math] . In particular, this furnishes an “odd” variant of [math] quantum invariants, even for knots labeled by spin representations. We then show that these knot invariants are essentially the same, up to a change of variables and a constant factor depending on the knot and weight.
</p>projecteuclid.org/euclid.agt/1510841489_20171116091126Thu, 16 Nov 2017 09:11 ESTLocalization of cofibration categories and groupoid $C^*$–algebrashttps://projecteuclid.org/euclid.agt/1510841490<strong>Markus Land</strong>, <strong>Thomas Nikolaus</strong>, <strong>Karol Szumiło</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 3007--3020.</p><p><strong>Abstract:</strong><br/>
We prove that relative functors out of a cofibration category are essentially the same as relative functors which are only defined on the subcategory of cofibrations. As an application we give a new construction of the functor that assigns to a groupoid its groupoid [math] –algebra and thereby its topological [math] –theory spectrum.
</p>projecteuclid.org/euclid.agt/1510841490_20171116091126Thu, 16 Nov 2017 09:11 ESTHOMFLY-PT homology for general link diagrams and braidlike isotopyhttps://projecteuclid.org/euclid.agt/1510841491<strong>Michael Abel</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 3021--3056.</p><p><strong>Abstract:</strong><br/>
Khovanov and Rozansky’s categorification of the homfly-pt polynomial is invariant under braidlike isotopies for any general link diagram and Markov moves for braid closures. To define homfly-pt homology, they required a link to be presented as a braid closure, because they did not prove invariance under the other oriented Reidemeister moves. In this text we prove that the Reidemeister IIb move fails in homfly-pt homology by using virtual crossing filtrations of the author and Rozansky. The decategorification of homfly-pt homology for general link diagrams gives a deformed version of the homfly-pt polynomial, [math] , which can be used to detect nonbraidlike isotopies. Finally, we will use [math] to prove that homfly-pt homology is not an invariant of virtual links, even when virtual links are presented as virtual braid closures.
</p>projecteuclid.org/euclid.agt/1510841491_20171116091126Thu, 16 Nov 2017 09:11 ESTThe topological sliceness of $3$–strand pretzel knotshttps://projecteuclid.org/euclid.agt/1510841492<strong>Allison Miller</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 3057--3079.</p><p><strong>Abstract:</strong><br/>
We give a complete characterization of the topological slice status of odd [math] –strand pretzel knots, proving that an odd [math] –strand pretzel knot is topologically slice if and only if it either is ribbon or has trivial Alexander polynomial. We also show that topologically slice even [math] –strand pretzel knots, except perhaps for members of Lecuona’s exceptional family, must be ribbon. These results follow from computations of the Casson–Gordon [math] –manifold signature invariants associated to the double branched covers of these knots.
</p>projecteuclid.org/euclid.agt/1510841492_20171116091126Thu, 16 Nov 2017 09:11 ESTAn index obstruction to positive scalar curvature on fiber bundles over aspherical manifoldshttps://projecteuclid.org/euclid.agt/1510841493<strong>Rudolf Zeidler</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 3081--3094.</p><p><strong>Abstract:</strong><br/>
We exhibit geometric situations where higher indices of the spinor Dirac operator on a spin manifold [math] are obstructions to positive scalar curvature on an ambient manifold [math] that contains [math] as a submanifold. In the main result of this note, we show that the Rosenberg index of [math] is an obstruction to positive scalar curvature on [math] if [math] is a fiber bundle of spin manifolds with [math] aspherical and [math] of finite asymptotic dimension. The proof is based on a new variant of the multipartitioned manifold index theorem which might be of independent interest. Moreover, we present an analogous statement for codimension-one submanifolds. We also discuss some elementary obstructions using the [math] -genus of certain submanifolds.
</p>projecteuclid.org/euclid.agt/1510841493_20171116091126Thu, 16 Nov 2017 09:11 ESTAn algebraic model for rational $\mathrm{SO}(3)$–spectrahttps://projecteuclid.org/euclid.agt/1510841494<strong>Magdalena Kędziorek</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 3095--3136.</p><p><strong>Abstract:</strong><br/>
Greenlees established an equivalence of categories between the homotopy category of rational [math] –spectra and the derived category [math] of a certain abelian category. In this paper we lift this equivalence of homotopy categories to the level of Quillen equivalences of model categories. Methods used in this paper provide the first step towards obtaining an algebraic model for the toral part of rational [math] –spectra, for any compact Lie group [math] .
</p>projecteuclid.org/euclid.agt/1510841494_20171116091126Thu, 16 Nov 2017 09:11 ESTBetti numbers and stability for configuration spaces via factorization homologyhttps://projecteuclid.org/euclid.agt/1510841495<strong>Ben Knudsen</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 3137--3187.</p><p><strong>Abstract:</strong><br/>
Using factorization homology, we realize the rational homology of the unordered configuration spaces of an arbitrary manifold [math] , possibly with boundary, as the homology of a Lie algebra constructed from the compactly supported cohomology of [math] . By locating the homology of each configuration space within the Chevalley–Eilenberg complex of this Lie algebra, we extend theorems of Bödigheimer, Cohen and Taylor and of Félix and Thomas, and give a new, combinatorial proof of the homological stability results of Church and Randal-Williams. Our method lends itself to explicit calculations, examples of which we include.
</p>projecteuclid.org/euclid.agt/1510841495_20171116091126Thu, 16 Nov 2017 09:11 ESTPresentably symmetric monoidal $\infty$–categories are represented by symmetric monoidal model categorieshttps://projecteuclid.org/euclid.agt/1510841496<strong>Thomas Nikolaus</strong>, <strong>Steffen Sagave</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 3189--3212.</p><p><strong>Abstract:</strong><br/>
We prove the theorem stated in the title. More precisely, we show the stronger statement that every symmetric monoidal left adjoint functor between presentably symmetric monoidal [math] -categories is represented by a strong symmetric monoidal left Quillen functor between simplicial, combinatorial and left proper symmetric monoidal model categories.
</p>projecteuclid.org/euclid.agt/1510841496_20171116091126Thu, 16 Nov 2017 09:11 EST$3$–manifolds built from injective handlebodieshttps://projecteuclid.org/euclid.agt/1510841506<strong>James Coffey</strong>, <strong>Hyam Rubinstein</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3213--3257.</p><p><strong>Abstract:</strong><br/>
This paper studies a class of closed orientable [math] –manifolds constructed from a gluing of three handlebodies, such that the inclusion of each handlebody is [math] –injective. This construction is the generalisation to handlebodies of the construction for gluing three solid tori to produce non-Haken Seifert fibred [math] –manifolds with infinite fundamental group. It is shown that there is an efficient algorithm to decide if a gluing of handlebodies satisfies the disk-condition. Also, an outline for the construction of the characteristic variety (JSJ decomposition) in such manifolds is given. Some non-Haken and atoroidal examples are given.
</p>projecteuclid.org/euclid.agt/1510841506_20171116091155Thu, 16 Nov 2017 09:11 ESTEquivariant iterated loop space theory and permutative $G$–categorieshttps://projecteuclid.org/euclid.agt/1510841507<strong>Bertrand Guillou</strong>, <strong>Peter May</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3259--3339.</p><p><strong>Abstract:</strong><br/>
We set up operadic foundations for equivariant iterated loop space theory. We start by building up from a discussion of the approximation theorem and recognition principle for [math] –fold loop [math] –spaces to several avatars of a recognition principle for infinite loop [math] –spaces. We then explain what genuine permutative [math] –categories are and, more generally, what [math] – [math] –categories are, giving examples showing how they arise. As an application, we prove the equivariant Barratt–Priddy–Quillen theorem as a statement about genuine [math] –spectra and use it to give a new, categorical proof of the tom Dieck splitting theorem for suspension [math] –spectra. Other examples are geared towards equivariant algebraic [math] –theory.
</p>projecteuclid.org/euclid.agt/1510841507_20171116091155Thu, 16 Nov 2017 09:11 ESTThe localized skein algebra is Frobeniushttps://projecteuclid.org/euclid.agt/1510841508<strong>Nel Abdiel</strong>, <strong>Charles Frohman</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3341--3373.</p><p><strong>Abstract:</strong><br/>
When [math] in the Kauffman bracket skein relation is set equal to a primitive [math] root of unity [math] with [math] not divisible by [math] , the Kauffman bracket skein algebra [math] of a finite-type surface [math] is a ring extension of the [math] –character ring of the fundamental group of [math] . We localize by inverting the nonzero characters to get an algebra [math] over the function field of the corresponding character variety. We prove that if [math] is noncompact, the algebra [math] is a symmetric Frobenius algebra. Along the way we prove [math] is finitely generated, [math] is a finite-rank module over the coordinate ring of the corresponding character variety, and learn to compute the trace that makes the algebra Frobenius.
</p>projecteuclid.org/euclid.agt/1510841508_20171116091155Thu, 16 Nov 2017 09:11 ESTGeneralized augmented alternating links and hyperbolic volumeshttps://projecteuclid.org/euclid.agt/1510841509<strong>Colin Adams</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3375--3397.</p><p><strong>Abstract:</strong><br/>
Augmented alternating links are links obtained by adding trivial components that bound twice-punctured disks to nonsplit reduced non- [math] –braid prime alternating projections. These links are known to be hyperbolic. Here, we extend to show that generalized augmented alternating links, which allow for new trivial components that bound [math] –punctured disks, are also hyperbolic. As an application we consider generalized belted sums of links and compute their volumes.
</p>projecteuclid.org/euclid.agt/1510841509_20171116091155Thu, 16 Nov 2017 09:11 ESTRepresentations of the Kauffman bracket skein algebra, II: Punctured surfaceshttps://projecteuclid.org/euclid.agt/1510841510<strong>Francis Bonahon</strong>, <strong>Helen Wong</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3399--3434.</p><p><strong>Abstract:</strong><br/>
In part I, we constructed invariants of irreducible finite-dimensional representations of the Kauffman bracket skein algebra of a surface. We introduce here an inverse construction, which to a set of possible invariants associates an irreducible representation that realizes these invariants. The current article is restricted to surfaces with at least one puncture, a condition that is lifted in subsequent work relying on this one. A step in the proof is of independent interest, and describes the arithmetic structure of the Thurston intersection form on the space of integer weight systems for a train track.
</p>projecteuclid.org/euclid.agt/1510841510_20171116091155Thu, 16 Nov 2017 09:11 ESTThe unstabilized canonical Heegaard splitting of a mapping torushttps://projecteuclid.org/euclid.agt/1510841511<strong>Yanqing Zou</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3435--3448.</p><p><strong>Abstract:</strong><br/>
Let [math] be a closed orientable surface of genus at least [math] . The action of an automorphism [math] on the curve complex of [math] is an isometry. Via this isometric action on the curve complex, a translation length is defined on [math] . The geometry of the mapping torus [math] depends on [math] . As it turns out, the structure of the minimal-genus Heegaard splitting also depends on [math] : the canonical Heegaard splitting of [math] , constructed from two parallel copies of [math] , is sometimes stabilized and sometimes unstabilized. We give an example of an infinite family of automorphisms for which the canonical Heegaard splitting of the mapping torus is stabilized. Interestingly, complexity bounds on [math] provide insight into the stability of the canonical Heegaard splitting of [math] . Using combinatorial techniques developed on [math] –manifolds, we prove that if the translation length of [math] is at least [math] , then the canonical Heegaard splitting of [math] is unstabilized.
</p>projecteuclid.org/euclid.agt/1510841511_20171116091155Thu, 16 Nov 2017 09:11 ESTNine generators of the skein space of the $3$–torushttps://projecteuclid.org/euclid.agt/1510841512<strong>Alessio Carrega</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3449--3460.</p><p><strong>Abstract:</strong><br/>
We show that the skein vector space of the [math] –torus is finitely generated. We show that it is generated by nine elements: the empty set, some simple closed curves representing the nonzero elements of the first homology group with coefficients in [math] , and a link consisting of two parallel copies of one of the previous nonempty knots.
</p>projecteuclid.org/euclid.agt/1510841512_20171116091155Thu, 16 Nov 2017 09:11 ESTQuasistabilization and basepoint moving maps in link Floer homologyhttps://projecteuclid.org/euclid.agt/1510841513<strong>Ian Zemke</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3461--3518.</p><p><strong>Abstract:</strong><br/>
We analyze the effect of adding, removing, and moving basepoints on link Floer homology. We prove that adding or removing basepoints via a procedure called quasistabilization is a natural operation on a certain version of link Floer homology, which we call [math] . We consider the effect on the full link Floer complex of moving basepoints, and develop a simple calculus for moving basepoints on the link Floer complexes. We apply it to compute the effect of several diffeomorphisms corresponding to moving basepoints. Using these techniques we prove a conjecture of Sarkar about the map on the full link Floer complex induced by a finger move along a link component.
</p>projecteuclid.org/euclid.agt/1510841513_20171116091155Thu, 16 Nov 2017 09:11 ESTCosimplicial groups and spaces of homomorphismshttps://projecteuclid.org/euclid.agt/1510841514<strong>Bernardo Villarreal</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3519--3545.</p><p><strong>Abstract:</strong><br/> Let [math] be a real linear algebraic group and [math] a finitely generated cosimplicial group. We prove that the space of homomorphisms [math] has a homotopy stable decomposition for each [math] . When [math] is a compact Lie group, we show that the decomposition is [math] –equivariant with respect to the induced action of conjugation by elements of [math] . In particular, under these hypotheses on [math] , we obtain stable decompositions for [math] and [math] , respectively, where [math] are the finitely generated free nilpotent groups of nilpotency class [math] . The spaces [math] assemble into a simplicial space [math] . When [math] we show that its geometric realization [math] , has a nonunital [math] –ring space structure whenever [math] is path connected for all [math] . </p>projecteuclid.org/euclid.agt/1510841514_20171116091155Thu, 16 Nov 2017 09:11 ESTGorenstein duality for real spectrahttps://projecteuclid.org/euclid.agt/1510841515<strong>J P C Greenlees</strong>, <strong>Lennart Meier</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3547--3619.</p><p><strong>Abstract:</strong><br/>
Following Hu and Kriz, we study the [math] –spectra [math] and [math] that refine the usual truncated Brown–Peterson and the Johnson–Wilson spectra. In particular, we show that they satisfy Gorenstein duality with a representation grading shift and identify their Anderson duals. We also compute the associated local cohomology spectral sequence in the cases [math] and [math] .
</p>projecteuclid.org/euclid.agt/1510841515_20171116091155Thu, 16 Nov 2017 09:11 ESTSlice implies mutant ribbon for odd $5$–stranded pretzel knotshttps://projecteuclid.org/euclid.agt/1510841516<strong>Kathryn Bryant</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3621--3664.</p><p><strong>Abstract:</strong><br/>
A pretzel knot [math] is called odd if all its twist parameters are odd and mutant ribbon if it is mutant to a simple ribbon knot. We prove that the family of odd [math] –stranded pretzel knots satisfies a weaker version of the slice-ribbon conjecture: all slice odd [math] –stranded pretzel knots are mutant ribbon , meaning they are mutant to a ribbon knot. We do this in stages by first showing that [math] –stranded pretzel knots having twist parameters with all the same sign or with exactly one parameter of a different sign have infinite order in the topological knot concordance group and thus in the smooth knot concordance group as well. Next, we show that any odd [math] –stranded pretzel knot with zero pairs or with exactly one pair of canceling twist parameters is not slice.
</p>projecteuclid.org/euclid.agt/1510841516_20171116091155Thu, 16 Nov 2017 09:11 ESTAxioms for higher twisted torsion invariants of smooth bundleshttps://projecteuclid.org/euclid.agt/1510841517<strong>Christopher Ohrt</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3665--3701.</p><p><strong>Abstract:</strong><br/>
This paper attempts to investigate the space of various characteristic classes for smooth manifold bundles with local system on the total space inducing a finite holonomy covering. These classes are known as twisted higher torsion classes. We will give a system of axioms that we require these cohomology classes to satisfy. Higher Franz–Reidemeister torsion and twisted versions of the higher Miller–Morita–Mumford classes will satisfy these axioms. We will show that the space of twisted torsion invariants is two-dimensional or one-dimensional depending on the torsion degree and is spanned by these two classes. The proof will greatly depend on results about the equivariant Hatcher constructions developed in Goodwillie, Igusa and Ohrt (2015).
</p>projecteuclid.org/euclid.agt/1510841517_20171116091155Thu, 16 Nov 2017 09:11 ESTSuper $q$–Howe duality and web categorieshttps://projecteuclid.org/euclid.agt/1510841518<strong>Daniel Tubbenhauer</strong>, <strong>Pedro Vaz</strong>, <strong>Paul Wedrich</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3703--3749.</p><p><strong>Abstract:</strong><br/>
We use super [math] –Howe duality to provide diagrammatic presentations of an idempotented form of the Hecke algebra and of categories of [math] –modules (and, more generally, [math] –modules) whose objects are tensor generated by exterior and symmetric powers of the vector representations. As an application, we give a representation-theoretic explanation and a diagrammatic version of a known symmetry of colored HOMFLY–PT polynomials.
</p>projecteuclid.org/euclid.agt/1510841518_20171116091155Thu, 16 Nov 2017 09:11 ESTUniform fellow traveling between surgery paths in the sphere graphhttps://projecteuclid.org/euclid.agt/1510841519<strong>Matt Clay</strong>, <strong>Yulan Qing</strong>, <strong>Kasra Rafi</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3751--3778.</p><p><strong>Abstract:</strong><br/>
We show that the Hausdorff distance between any forward and any backward surgery paths in the sphere graph is at most [math] . From this it follows that the Hausdorff distance between any two surgery paths with the same initial sphere system and same target sphere system is at most [math] . Our proof relies on understanding how surgeries affect the Guirardel core associated to sphere systems. We show that applying a surgery is equivalent to performing a Rips move on the Guirardel core.
</p>projecteuclid.org/euclid.agt/1510841519_20171116091155Thu, 16 Nov 2017 09:11 ESTOn the integral cohomology ring of toric orbifolds and singular toric varietieshttps://projecteuclid.org/euclid.agt/1510841520<strong>Anthony Bahri</strong>, <strong>Soumen Sarkar</strong>, <strong>Jongbaek Song</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3779--3810.</p><p><strong>Abstract:</strong><br/>
We examine the integral cohomology rings of certain families of [math] –dimensional orbifolds [math] that are equipped with a well-behaved action of the [math] –dimensional real torus. These orbifolds arise from two distinct but closely related combinatorial sources, namely from characteristic pairs [math] , where [math] is a simple convex [math] –polytope and [math] a labeling of its facets, and from [math] –dimensional fans [math] . In the literature, they are referred as toric orbifolds and singular toric varieties, respectively. Our first main result provides combinatorial conditions on [math] or on [math] which ensure that the integral cohomology groups [math] of the associated orbifolds are concentrated in even degrees. Our second main result assumes these conditions to be true, and expresses the graded ring [math] as a quotient of an algebra of polynomials that satisfy an integrality condition arising from the underlying combinatorial data. Also, we compute several examples.
</p>projecteuclid.org/euclid.agt/1510841520_20171116091155Thu, 16 Nov 2017 09:11 ESTRemarks on coloured triply graded link invariantshttps://projecteuclid.org/euclid.agt/1510841521<strong>Sabin Cautis</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3811--3836.</p><p><strong>Abstract:</strong><br/>
We explain how existing results (such as categorical [math] actions, associated braid group actions and infinite twists) can be used to define a triply graded link invariant which categorifies the homfly polynomial of links coloured by arbitrary partitions. The construction uses a categorified homfly clasp defined via cabling and infinite twists. We briefly discuss differentials and speculate on related structures.
</p>projecteuclid.org/euclid.agt/1510841521_20171116091155Thu, 16 Nov 2017 09:11 ESTCorrection to the articles “Homotopy theory of nonsymmetric operads”, I–IIhttps://projecteuclid.org/euclid.agt/1510841522<strong>Fernando Muro</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 6, 3837--3852.</p><p><strong>Abstract:</strong><br/>
We correct a mistake in Algebr. Geom. Topol. 11 (2011) 1541–1599 on the construction of push-outs along free morphisms of algebras over a nonsymmetric operad, and we fix the affected results from there and a follow-up article (Algebr. Geom. Topol. 14 (2014) 229–281).
</p>projecteuclid.org/euclid.agt/1510841522_20171116091155Thu, 16 Nov 2017 09:11 ESTWidth of a satellite knot and its companionhttps://projecteuclid.org/euclid.agt/1517454210<strong>Qilong Guo</strong>, <strong>Zhenkun Li</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 1--13.</p><p><strong>Abstract:</strong><br/>
In this paper, we give a proof of a conjecture which says that [math] , where [math] is the width of a knot, [math] is a satellite knot with [math] as its companion, and [math] is the winding number of the pattern. We also show that equality holds if [math] is a satellite knot with braid pattern.
</p>projecteuclid.org/euclid.agt/1517454210_20180131220354Wed, 31 Jan 2018 22:03 ESTThe closed-open string map for $S^1$–invariant Lagrangianshttps://projecteuclid.org/euclid.agt/1517454211<strong>Dmitry Tonkonog</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 15--68.</p><p><strong>Abstract:</strong><br/> Given a monotone Lagrangian submanifold invariant under a loop of Hamiltonian diffeomorphisms, we compute a piece of the closed-open string map into the Hochschild cohomology of the Lagrangian which captures the homology class of the loop’s orbit. Our applications include split-generation and nonformality results for real Lagrangians in projective spaces and other toric varieties; a particularly basic example is that the equatorial circle on the [math] –sphere carries a nonformal Fukaya [math] algebra in characteristic [math] . </p>projecteuclid.org/euclid.agt/1517454211_20180131220354Wed, 31 Jan 2018 22:03 ESTHeegaard Floer homology and knots determined by their complementshttps://projecteuclid.org/euclid.agt/1517454212<strong>Fyodor Gainullin</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 69--109.</p><p><strong>Abstract:</strong><br/>
We investigate the question of when different surgeries on a knot can produce identical manifolds. We show that given a knot in a homology sphere, unless the knot is quite special, there is a bound on the number of slopes that can produce a fixed manifold that depends only on this fixed manifold and the homology sphere the knot is in. By finding a different bound on the number of slopes, we show that non-null-homologous knots in certain homology [math] are determined by their complements. We also prove the surgery characterisation of the unknot for null-homologous knots in [math] –spaces. This leads to showing that all knots in some lens spaces are determined by their complements. Finally, we establish that knots of genus greater than [math] in the Brieskorn sphere [math] are also determined by their complements.
</p>projecteuclid.org/euclid.agt/1517454212_20180131220354Wed, 31 Jan 2018 22:03 ESTClassification of tight contact structures on small Seifert fibered $L$–spaceshttps://projecteuclid.org/euclid.agt/1517454213<strong>Irena Matkovič</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 111--152.</p><p><strong>Abstract:</strong><br/>
We identify tight contact structures on small Seifert fibered [math] –spaces as exactly the structures having nonvanishing contact invariant, and classify them by their induced [math] structures. The result (in the new case of [math] ) is based on the translation between convex surface theory and the tightness criterion of Lisca and Stipsicz.
</p>projecteuclid.org/euclid.agt/1517454213_20180131220354Wed, 31 Jan 2018 22:03 ESTAn infinite family of links with critical bridge sphereshttps://projecteuclid.org/euclid.agt/1517454214<strong>Daniel Rodman</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 153--186.</p><p><strong>Abstract:</strong><br/>
A closed orientable splitting surface in an oriented 3–manifold is a topologically minimal surface of index [math] if its associated disk complex is [math] –connected but not [math] –connected. A critical surface is a topologically minimal surface of index 2. In this paper, we use an equivalent combinatorial definition of critical surfaces to construct the first known critical bridge spheres for nontrivial links.
</p>projecteuclid.org/euclid.agt/1517454214_20180131220354Wed, 31 Jan 2018 22:03 ESTA second cohomology class of the symplectomorphism group with the discrete topologyhttps://projecteuclid.org/euclid.agt/1517454215<strong>Ryoji Kasagawa</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 187--219.</p><p><strong>Abstract:</strong><br/>
A second cohomology class of the group cohomology of the symplectomorphism group is defined for a symplectic manifold with first Chern class proportional to the class of symplectic form and with trivial first real cohomology. Some properties of it are studied. In particular, it is characterized in terms of cohomology classes of the universal symplectic fiber bundle over the classifying space of the symplectomorphism group with the discrete topology.
</p>projecteuclid.org/euclid.agt/1517454215_20180131220354Wed, 31 Jan 2018 22:03 ESTOn the stability of asymptotic property C for products and some group extensionshttps://projecteuclid.org/euclid.agt/1517454216<strong>Gregory Copeland Bell</strong>, <strong>Andrzej Nagórko</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 221--245.</p><p><strong>Abstract:</strong><br/>
We show that Dranishnikov’s asymptotic property C is preserved by direct products and the free product of discrete metric spaces. In particular, if [math] and [math] are groups with asymptotic property C, then both [math] and [math] have asymptotic property C. We also prove that a group [math] has asymptotic property C if [math] is exact, [math] and [math] has asymptotic property C. The groups are assumed to have left-invariant proper metrics and need not be finitely generated. These results settle questions of Dydak and Virk (2016), of Bell and Moran (2015) and an open problem in topology.
</p>projecteuclid.org/euclid.agt/1517454216_20180131220354Wed, 31 Jan 2018 22:03 ESTHigher cohomology operations and $R$–completionhttps://projecteuclid.org/euclid.agt/1517454217<strong>David Blanc</strong>, <strong>Debasis Sen</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 247--312.</p><p><strong>Abstract:</strong><br/>
Let [math] be either [math] or a field of characteristic [math] . For each [math] –good topological space [math] , we define a collection of higher cohomology operations which, together with the cohomology algebra [math] , suffice to determine [math] up to [math] –completion. We also provide a similar collection of higher cohomology operations which determine when two maps [math] between [math] –good spaces (inducing the same algebraic homomorphism [math] ) are [math] –equivalent.
</p>projecteuclid.org/euclid.agt/1517454217_20180131220354Wed, 31 Jan 2018 22:03 ESTOn high-dimensional representations of knot groupshttps://projecteuclid.org/euclid.agt/1517454218<strong>Stefan Friedl</strong>, <strong>Michael Heusener</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 313--332.</p><p><strong>Abstract:</strong><br/> Given a hyperbolic knot [math] and any [math] the abelian representations and the holonomy representation each give rise to an [math] –dimensional component in the [math] –character variety. A component of the [math] –character variety of dimension [math] is called high-dimensional. It was proved by D Cooper and D Long that there exist hyperbolic knots with high-dimensional components in the [math] –character variety. We show that given any nontrivial knot [math] and sufficiently large [math] the [math] –character variety of [math] admits high-dimensional components. </p>projecteuclid.org/euclid.agt/1517454218_20180131220354Wed, 31 Jan 2018 22:03 ESTDAHA and plane curve singularitieshttps://projecteuclid.org/euclid.agt/1517454219<strong>Ivan Cherednik</strong>, <strong>Ian Philipp</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 333--385.</p><p><strong>Abstract:</strong><br/>
We suggest a relatively simple and totally geometric conjectural description of uncolored daha superpolynomials of arbitrary algebraic knots (conjecturally coinciding with the reduced stable Khovanov–Rozansky polynomials) via the flagged Jacobian factors (new objects) of the corresponding unibranch plane curve singularities. This generalizes the Cherednik–Danilenko conjecture on the Betti numbers of Jacobian factors, the Gorsky combinatorial conjectural interpretation of superpolynomials of torus knots and that by Gorsky and Mazin for their constant term. The paper mainly focuses on nontorus algebraic knots. A connection with the conjecture due to Oblomkov, Rasmussen and Shende is possible, but our approach is different. A motivic version of our conjecture is related to [math] –adic orbital [math] –type integrals for anisotropic centralizers.
</p>projecteuclid.org/euclid.agt/1517454219_20180131220354Wed, 31 Jan 2018 22:03 ESTInertia groups of high-dimensional complex projective spaceshttps://projecteuclid.org/euclid.agt/1517454220<strong>Samik Basu</strong>, <strong>Ramesh Kasilingam</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 387--408.</p><p><strong>Abstract:</strong><br/>
For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension [math] , these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is nontrivial in many cases. In complex dimension [math] , we deduce some results on geometric structures on homotopy complex projective spaces and complex hyperbolic manifolds.
</p>projecteuclid.org/euclid.agt/1517454220_20180131220354Wed, 31 Jan 2018 22:03 ESTNonexistence of boundary maps for some hierarchically hyperbolic spaceshttps://projecteuclid.org/euclid.agt/1517454221<strong>Sarah C Mousley</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 409--439.</p><p><strong>Abstract:</strong><br/>
We provide negative answers to questions posed by Durham, Hagen and Sisto on the existence of boundary maps for some hierarchically hyperbolic spaces, namely maps from right-angled Artin groups to mapping class groups. We also prove results on existence of boundary maps for free subgroups of mapping class groups.
</p>projecteuclid.org/euclid.agt/1517454221_20180131220354Wed, 31 Jan 2018 22:03 ESTFinite Dehn surgeries on knots in $S^3$https://projecteuclid.org/euclid.agt/1517454222<strong>Yi Ni</strong>, <strong>Xingru Zhang</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 441--492.</p><p><strong>Abstract:</strong><br/>
We show that on a hyperbolic knot [math] in [math] , the distance between any two finite surgery slopes is at most [math] , and consequently, there are at most three nontrivial finite surgeries. Moreover, in the case where [math] admits three nontrivial finite surgeries, [math] must be the pretzel knot [math] . In the case where [math] admits two noncyclic finite surgeries or two finite surgeries at distance [math] , the two surgery slopes must be one of ten or seventeen specific pairs, respectively. For [math] –type finite surgeries, we improve a finiteness theorem due to Doig by giving an explicit bound on the possible resulting prism manifolds, and also prove that [math] and [math] are characterizing slopes for the torus knot [math] for each [math] .
</p>projecteuclid.org/euclid.agt/1517454222_20180131220354Wed, 31 Jan 2018 22:03 ESTA characterization for asymptotic dimension growthhttps://projecteuclid.org/euclid.agt/1517454223<strong>Goulnara Arzhantseva</strong>, <strong>Graham A Niblo</strong>, <strong>Nick Wright</strong>, <strong>Jiawen Zhang</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 493--524.</p><p><strong>Abstract:</strong><br/>
We give a characterization for asymptotic dimension growth. We apply it to [math] cube complexes of finite dimension, giving an alternative proof of Wright’s result on their finite asymptotic dimension. We also apply our new characterization to geodesic coarse median spaces of finite rank and establish that they have subexponential asymptotic dimension growth. This strengthens a recent result of S̆pakula and Wright.
</p>projecteuclid.org/euclid.agt/1517454223_20180131220354Wed, 31 Jan 2018 22:03 ESTClassifying spaces for $1$–truncated compact Lie groupshttps://projecteuclid.org/euclid.agt/1517454224<strong>Charles Rezk</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 525--546.</p><p><strong>Abstract:</strong><br/>
A [math] –truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of [math] , [math] , and [math] for compact Lie groups [math] and [math] with [math] [math] –truncated, showing that they are computed entirely in terms of spaces of homomorphisms from [math] to [math] . These results generalize the well-known case when [math] is finite, and the case when [math] is compact abelian due to Lashof, May, and Segal.
</p>projecteuclid.org/euclid.agt/1517454224_20180131220354Wed, 31 Jan 2018 22:03 ESTSecond mod $2$ homology of Artin groupshttps://projecteuclid.org/euclid.agt/1517454225<strong>Toshiyuki Akita</strong>, <strong>Ye Liu</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 547--568.</p><p><strong>Abstract:</strong><br/>
In this paper, we compute the second mod [math] homology of an arbitrary Artin group, without assuming the [math] conjecture. The key ingredients are (A) Hopf’s formula for the second integral homology of a group and (B) Howlett’s result on the second integral homology of Coxeter groups.
</p>projecteuclid.org/euclid.agt/1517454225_20180131220354Wed, 31 Jan 2018 22:03 ESTOn the third homotopy group of Orr's spacehttps://projecteuclid.org/euclid.agt/1517454226<strong>Emmanuel Dror Farjoun</strong>, <strong>Roman Mikhailov</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 569--582.</p><p><strong>Abstract:</strong><br/>
K Orr defined a Milnor-type invariant of links that lies in the third homotopy group of a certain space [math] . The problem of nontriviality of this third homotopy group has been open. We show that it is an infinitely generated group. The question of realization of its elements as links remains open.
</p>projecteuclid.org/euclid.agt/1517454226_20180131220354Wed, 31 Jan 2018 22:03 ESTGroups of homotopy classes of phantom mapshttps://projecteuclid.org/euclid.agt/1517454227<strong>Hiroshi Kihara</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 583--612.</p><p><strong>Abstract:</strong><br/>
We introduce a new approach to phantom maps which largely extends the rational-ization-completion approach developed by Meier and Zabrodsky. Our approach enables us to deal with the set [math] of homotopy classes of phantom maps and the subset [math] of homotopy classes of special phantom maps simultaneously. We give a sufficient condition for [math] and [math] to have natural group structures, which is much weaker than the conditions obtained by Meier and McGibbon. Previous calculations of [math] have generally assumed that [math] is trivial, in which case generalizations of Miller’s theorem are directly applicable, and calculations of [math] have rarely been reported. Here, we calculate not only [math] but also [math] in many important cases of nontrivial [math] .
</p>projecteuclid.org/euclid.agt/1517454227_20180131220354Wed, 31 Jan 2018 22:03 ESTLoop homology of some global quotient orbifoldshttps://projecteuclid.org/euclid.agt/1517454228<strong>Yasuhiko Asao</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 1, 613--633.</p><p><strong>Abstract:</strong><br/>
We determine the ring structure of the loop homology of some global quotient orbifolds. We can compute by our theorem the loop homology ring with suitable coefficients of the global quotient orbifolds of the form [math] for [math] being some kinds of homogeneous manifolds, and [math] being a finite subgroup of a path-connected topological group [math] acting on [math] . It is shown that these homology rings split into the tensor product of the loop homology ring [math] of the manifold [math] and that of the classifying space of the finite group, which coincides with the center of the group ring [math] .
</p>projecteuclid.org/euclid.agt/1517454228_20180131220354Wed, 31 Jan 2018 22:03 ESTA motivic Grothendieck–Teichmüller grouphttps://projecteuclid.org/euclid.agt/1521684012<strong>Ismaël Soudères</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 635--685.</p><p><strong>Abstract:</strong><br/>
We prove the Beilinson–Soulé vanishing conjecture for motives attached to the moduli spaces [math] of curves of genus [math] with [math] marked points. As part of the proof, we also show that these motives are mixed Tate. As a consequence of Levine’s work, we thus obtain a well-defined category of mixed Tate motives over the moduli space of curves [math] . We furthermore show that the morphisms between the moduli spaces [math] obtained by forgetting marked points and by embedding boundary components induce functors between the associated categories of mixed Tate motives. Finally, we explain how tangential base points fit into these functorialities.
The categories we construct are Tannakian, and therefore have attached Tannakian fundamental groups, connected by morphisms induced by those between the categories. This system of groups and morphisms leads to the definition of a motivic Grothendieck–Teichmüller group.
The proofs of the above results rely on the geometry of the tower of the moduli spaces [math] . This allows us to treat the general case of motives over [math] with coefficients in [math] , working in Spitzweck’s category of motives. From there, passing to [math] coefficients, we deal with the classical Tannakian formalism and explain how working over [math] yields a more concrete description of the Tannakian groups.
</p>projecteuclid.org/euclid.agt/1521684012_20180321220044Wed, 21 Mar 2018 22:00 EDTStable presentation length of $3$–manifold groupshttps://projecteuclid.org/euclid.agt/1521684015<strong>Ken’ichi Yoshida</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 687--722.</p><p><strong>Abstract:</strong><br/>
We introduce the stable presentation length of a finitely presentable group. The stable presentation length of the fundamental group of a [math] –manifold can be considered as an analogue of the simplicial volume. We show that, like the simplicial volume, the stable presentation length has some additive properties, and the simplicial volume of a closed [math] –manifold is bounded from above and below by constant multiples of the stable presentation length of its fundamental group.
</p>projecteuclid.org/euclid.agt/1521684015_20180321220044Wed, 21 Mar 2018 22:00 EDTIncomplete Tambara functorshttps://projecteuclid.org/euclid.agt/1521684018<strong>Andrew J Blumberg</strong>, <strong>Michael A Hill</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 723--766.</p><p><strong>Abstract:</strong><br/>
For a “genuine” equivariant commutative ring spectrum [math] , [math] admits a rich algebraic structure known as a Tambara functor. This algebraic structure mirrors the structure on [math] arising from the existence of multiplicative norm maps. Motivated by the surprising fact that Bousfield localization can destroy some of the norm maps, in previous work we studied equivariant commutative ring structures parametrized by [math] operads. In a precise sense, these interpolate between “naive” and “genuine” equivariant ring structures.
In this paper, we describe the algebraic analogue of [math] ring structures. We introduce and study categories of incomplete Tambara functors, described in terms of certain categories of bispans. Incomplete Tambara functors arise as [math] of [math] algebras, and interpolate between Green functors and Tambara functors. We classify all incomplete Tambara functors in terms of a basic structural result about polynomial functors. This classification gives a conceptual justification for our prior description of [math] operads and also allows us to easily describe the properties of the category of incomplete Tambara functors.
</p>projecteuclid.org/euclid.agt/1521684018_20180321220044Wed, 21 Mar 2018 22:00 EDTIdentifying lens spaces in polynomial timehttps://projecteuclid.org/euclid.agt/1521684019<strong>Greg Kuperberg</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 767--778.</p><p><strong>Abstract:</strong><br/>
We show that if a closed, oriented 3–manifold [math] is promised to be homeomorphic to a lens space [math] with [math] and [math] unknown, then we can compute both [math] and [math] in polynomial time in the size of the triangulation of [math] . The tricky part is the parameter [math] . The idea of the algorithm is to calculate Reidemeister torsion using numerical analysis over the complex numbers, rather than working directly in a cyclotomic field.
</p>projecteuclid.org/euclid.agt/1521684019_20180321220044Wed, 21 Mar 2018 22:00 EDTA combinatorial description of topological complexity for finite spaceshttps://projecteuclid.org/euclid.agt/1521684020<strong>Kohei Tanaka</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 779--796.</p><p><strong>Abstract:</strong><br/>
This paper presents a discrete analog of topological complexity for finite spaces using purely combinatorial terms. We demonstrate that this coincides with the genuine topological complexity of the original finite space. Furthermore, we study the relationship with simplicial complexity for simplicial complexes by taking the barycentric subdivision into account.
</p>projecteuclid.org/euclid.agt/1521684020_20180321220044Wed, 21 Mar 2018 22:00 EDTModuli of formal $A$–modules under change of $A$https://projecteuclid.org/euclid.agt/1521684021<strong>Andrew Salch</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 797--826.</p><p><strong>Abstract:</strong><br/>
We develop methods for computing the restriction map from the cohomology of the automorphism group of a height [math] formal group law (ie the height [math] Morava stabilizer group) to the cohomology of the automorphism group of an [math] –height [math] formal [math] –module, where [math] is the ring of integers in a degree [math] field extension of [math] . We then compute this map for the quadratic extensions of [math] and the height [math] Morava stabilizer group at primes [math] . We show that the these automorphism groups of formal modules are closed subgroups of the Morava stabilizer groups, and we use local class field theory to identify the automorphism group of an [math] –height [math] –formal [math] –module with the ramified part of the abelianization of the absolute Galois group of [math] , yielding an action of [math] on the Lubin–Tate/Morava [math] –theory spectrum [math] for each quadratic extension [math] . Finally, we run the associated descent spectral sequence to compute the [math] –homotopy groups of the homotopy fixed-points of this action; one consequence is that, for each element in the [math] –local homotopy groups of [math] , either that element or an appropriate dual of it is detected in the Galois cohomology of the abelian closure of some quadratic extension of [math] .
</p>projecteuclid.org/euclid.agt/1521684021_20180321220044Wed, 21 Mar 2018 22:00 EDTTopologically slice knots that are not smoothly slice in any definite $4$–manifoldhttps://projecteuclid.org/euclid.agt/1521684022<strong>Kouki Sato</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 827--837.</p><p><strong>Abstract:</strong><br/>
We prove that there exist infinitely many topologically slice knots which cannot bound a smooth null-homologous disk in any definite [math] –manifold. Furthermore, we show that we can take such knots so that they are linearly independent in the knot concordance group.
</p>projecteuclid.org/euclid.agt/1521684022_20180321220044Wed, 21 Mar 2018 22:00 EDTTopological complexity of $n$ points on a treehttps://projecteuclid.org/euclid.agt/1521684023<strong>Steven Scheirer</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 839--876.</p><p><strong>Abstract:</strong><br/>
The topological complexity of a path-connected space [math] , denoted by [math] , can be thought of as the minimum number of continuous rules needed to describe how to move from one point in [math] to another. The space [math] is often interpreted as a configuration space in some real-life context. Here, we consider the case where [math] is the space of configurations of [math] points on a tree [math] . We will be interested in two such configuration spaces. In the first, denoted by [math] , the points are distinguishable, while in the second, [math] , the points are indistinguishable. We determine [math] for any tree [math] and many values of [math] , and consequently determine [math] for the same values of [math] (provided the configuration spaces are path-connected).
</p>projecteuclid.org/euclid.agt/1521684023_20180321220044Wed, 21 Mar 2018 22:00 EDT$\Gamma$–structures and symmetric spaceshttps://projecteuclid.org/euclid.agt/1521684024<strong>Bernhard Hanke</strong>, <strong>Peter Quast</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 877--895.</p><p><strong>Abstract:</strong><br/>
$\Gamma$–structures are weak forms of multiplications on closed oriented manifolds. As was shown by Hopf the rational cohomology algebras of manifolds admitting $\Gamma$–structures are free over odd-degree generators. We prove that this condition is also sufficient for the existence of $\Gamma$–structures on manifolds which are nilpotent in the sense of homotopy theory. This includes homogeneous spaces with connected isotropy groups.
Passing to a more geometric perspective we show that on compact oriented Riemannian symmetric spaces with connected isotropy groups and free rational cohomology algebras the canonical products given by geodesic symmetries define $\Gamma$–structures. This extends work of Albers, Frauenfelder and Solomon on $\Gamma$–structures on Lagrangian Grassmannians.
</p>projecteuclid.org/euclid.agt/1521684024_20180321220044Wed, 21 Mar 2018 22:00 EDTConformal nets IV: The $3$-categoryhttps://projecteuclid.org/euclid.agt/1521684025<strong>Arthur Bartels</strong>, <strong>Christopher L Douglas</strong>, <strong>André Henriques</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 897--956.</p><p><strong>Abstract:</strong><br/>
Conformal nets are a mathematical model for conformal field theory, and defects between conformal nets are a model for an interaction or phase transition between two conformal field theories. We previously introduced a notion of composition, called fusion , between defects. We also described a notion of sectors between defects, modeling an interaction among or transformation between phase transitions, and defined fusion composition operations for sectors. In this paper we prove that altogether the collection of conformal nets, defects, sectors, and intertwiners, equipped with the fusion of defects and fusion of sectors, forms a symmetric monoidal [math] -category. This [math] -category encodes the algebraic structure of the possible interactions among conformal field theories.
</p>projecteuclid.org/euclid.agt/1521684025_20180321220044Wed, 21 Mar 2018 22:00 EDTA characterization of quaternionic Kleinian groups in dimension $2$ with complex trace fieldshttps://projecteuclid.org/euclid.agt/1521684026<strong>Sungwoon Kim</strong>, <strong>Joonhyung Kim</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 957--974.</p><p><strong>Abstract:</strong><br/>
Let [math] be a nonelementary discrete subgroup of [math] . We show that if the sum of diagonal entries of each element of [math] is a complex number, then [math] is conjugate to a subgroup of [math] .
</p>projecteuclid.org/euclid.agt/1521684026_20180321220044Wed, 21 Mar 2018 22:00 EDTRelative $2$–Segal spaceshttps://projecteuclid.org/euclid.agt/1521684027<strong>Matthew B Young</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 975--1039.</p><p><strong>Abstract:</strong><br/>
We introduce a relative version of the [math] –Segal simplicial spaces defined by Dyckerhoff and Kapranov, and Gálvez-Carrillo, Kock and Tonks. Examples of relative [math] –Segal spaces include the categorified unoriented cyclic nerve, real pseudoholomorphic polygons in almost complex manifolds and the [math] –construction from Grothendieck–Witt theory. We show that a relative [math] –Segal space defines a categorical representation of the Hall algebra associated to the base [math] –Segal space. In this way, after decategorification we recover a number of known constructions of Hall algebra representations. We also describe some higher categorical interpretations of relative [math] –Segal spaces.
</p>projecteuclid.org/euclid.agt/1521684027_20180321220044Wed, 21 Mar 2018 22:00 EDTOuter actions of $\mathrm{Out}(F_n)$ on small right-angled Artin groupshttps://projecteuclid.org/euclid.agt/1521684028<strong>Dawid Kielak</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1041--1065.</p><p><strong>Abstract:</strong><br/>
We determine the precise conditions under which [math] , the unique index-two subgroup of [math] , can act nontrivially via outer automorphisms on a RAAG whose defining graph has fewer than [math] vertices.
We also show that the outer automorphism group of a RAAG cannot act faithfully via outer automorphisms on a RAAG with a strictly smaller (in number of vertices) defining graph.
Along the way we determine the minimal dimensions of nontrivial linear representations of congruence quotients of the integral special linear groups over algebraically closed fields of characteristic zero, and provide a new lower bound on the cardinality of a set on which [math] can act nontrivially.
</p>projecteuclid.org/euclid.agt/1521684028_20180321220044Wed, 21 Mar 2018 22:00 EDTQuasi-invariant measures for some amenable groups acting on the linehttps://projecteuclid.org/euclid.agt/1521684029<strong>Nancy Guelman</strong>, <strong>Cristóbal Rivas</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1067--1076.</p><p><strong>Abstract:</strong><br/>
We show that if [math] is a solvable group acting on the line and if there is [math] having no fixed points, then there is a Radon measure [math] on the line quasi-invariant under [math] . In fact, our method allows for the same conclusion for [math] inside a class of groups that is closed under extensions and contains all solvable groups and all groups of subexponential growth.
</p>projecteuclid.org/euclid.agt/1521684029_20180321220044Wed, 21 Mar 2018 22:00 EDTNonfillable Legendrian knots in the $3$–spherehttps://projecteuclid.org/euclid.agt/1521684030<strong>Tolga Etgü</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1077--1088.</p><p><strong>Abstract:</strong><br/>
If $\Lambda$ is a Legendrian knot in the standard contact [math] –sphere that bounds an orientable exact Lagrangian surface $\Sigma$ in the standard symplectic [math] –ball, then the genus of $\Sigma$ is equal to the slice genus of (the smooth knot underlying) $\Lambda$, the rotation number of $\Lambda$ is zero as well as the sum of the Thurston–Bennequin number of $\Lambda$ and the Euler characteristic of $\Sigma$, and moreover, the linearized contact homology of $\Lambda$ with respect to the augmentation induced by $\Sigma$ is isomorphic to the (singular) homology of $\Sigma$. It was asked by Ekholm, Honda and Kálmán (2016) whether the converse of this statement holds. We give a negative answer, providing a family of Legendrian knots with augmentations which are not induced by any exact Lagrangian filling although the associated linearized contact homology is isomorphic to the homology of the smooth surface of minimal genus in the [math] –ball bounding the knot.
</p>projecteuclid.org/euclid.agt/1521684030_20180321220044Wed, 21 Mar 2018 22:00 EDTTaut branched surfaces from veering triangulationshttps://projecteuclid.org/euclid.agt/1521684031<strong>Michael Landry</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1089--1114.</p><p><strong>Abstract:</strong><br/>
Let [math] be a closed hyperbolic [math] –manifold with a fibered face [math] of the unit ball of the Thurston norm on [math] . If [math] satisfies a certain condition related to Agol’s veering triangulations, we construct a taut branched surface in [math] spanning [math] . This partially answers a 1986 question of Oertel, and extends an earlier partial answer due to Mosher.
</p>projecteuclid.org/euclid.agt/1521684031_20180321220044Wed, 21 Mar 2018 22:00 EDTTopological equivalences of E-infinity differential graded algebrashttps://projecteuclid.org/euclid.agt/1521684032<strong>Haldun Özgür Bayındır</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1115--1146.</p><p><strong>Abstract:</strong><br/>
Two DGAs are said to be topologically equivalent when the corresponding Eilenberg–Mac Lane ring spectra are weakly equivalent as ring spectra. Quasi-isomorphic DGAs are topologically equivalent, but the converse is not necessarily true. As a counterexample, Dugger and Shipley showed that there are DGAs that are nontrivially topologically equivalent, ie topologically equivalent but not quasi-isomorphic.
In this work, we define [math] topological equivalences and utilize the obstruction theories developed by Goerss, Hopkins and Miller to construct first examples of nontrivially [math] topologically equivalent [math] DGAs. Also, we show using these obstruction theories that for coconnective [math] –DGAs, [math] topological equivalences and quasi-isomorphisms agree. For [math] –DGAs with trivial first homology, we show that an [math] topological equivalence induces an isomorphism in homology that preserves the Dyer–Lashof operations and therefore induces an [math] –equivalence.
</p>projecteuclid.org/euclid.agt/1521684032_20180321220044Wed, 21 Mar 2018 22:00 EDTA rank inequality for the annular Khovanov homology of $2$–periodic linkshttps://projecteuclid.org/euclid.agt/1521684033<strong>Melissa Zhang</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1147--1194.</p><p><strong>Abstract:</strong><br/>
For a [math] –periodic link [math] in the thickened annulus and its quotient link [math] , we exhibit a spectral sequence with
[math]
This spectral sequence splits along quantum and [math] weight-space gradings, proving a rank inequality [math] for every pair of quantum and [math] weight-space gradings [math] . We also present a few decategorified consequences and discuss partial results toward a similar statement for the Khovanov homology of [math] –periodic links, as well as some frameworks for obstructing [math] –periodicity in links.
</p>projecteuclid.org/euclid.agt/1521684033_20180321220044Wed, 21 Mar 2018 22:00 EDTEuler characteristics and actions of automorphism groups of free groupshttps://projecteuclid.org/euclid.agt/1521684034<strong>Shengkui Ye</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1195--1204.</p><p><strong>Abstract:</strong><br/>
Let [math] be a connected orientable manifold with the Euler characteristic [math] . Denote by [math] the unique subgroup of index two in the automorphism group of a free group. Then any group action of [math] (and thus the special linear group [math] ) with [math] on [math] by homeomorphisms is trivial. This confirms a conjecture related to Zimmer’s program for these manifolds.
</p>projecteuclid.org/euclid.agt/1521684034_20180321220044Wed, 21 Mar 2018 22:00 EDTThe spectrum for commutative complex $K$–theoryhttps://projecteuclid.org/euclid.agt/1521684035<strong>Simon Philipp Gritschacher</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1205--1249.</p><p><strong>Abstract:</strong><br/>
We study commutative complex [math] –theory, a generalised cohomology theory built from spaces of ordered commuting tuples in the unitary groups. We show that the spectrum for commutative complex [math] –theory is stably equivalent to the [math] –group ring of [math] and thus obtain a splitting of its representing space [math] as a product of all the terms in the Whitehead tower for [math] , [math] . As a consequence of the spectrum level identification we obtain the ring of coefficients for this theory. Using the rational Hopf ring for [math] we describe the relationship of our results with a previous computation of the rational cohomology algebra of [math] . This gives an essentially complete description of the space [math] introduced by A Adem and J Gómez.
</p>projecteuclid.org/euclid.agt/1521684035_20180321220044Wed, 21 Mar 2018 22:00 EDTCorrigendum to the article The simplicial boundary of a CAT(0) cube complexhttps://projecteuclid.org/euclid.agt/1521684036<strong>Mark F Hagen</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1251--1256.</p><p><strong>Abstract:</strong><br/>
We correct Theorem 3.10 of [Algebr. Geom. Topol. 13 (2013) 1299–1367] in the infinite-dimensional case. No correction is needed in the finite-dimensional case.
</p>projecteuclid.org/euclid.agt/1521684036_20180321220044Wed, 21 Mar 2018 22:00 EDTCorrection to the article Preorientations of the derived motivic multiplicative grouphttps://projecteuclid.org/euclid.agt/1521684037<strong>Jens Hornbostel</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1257--1258.</p><p><strong>Abstract:</strong><br/>
We correct a claim concerning motivic [math] –deloopings.
</p>projecteuclid.org/euclid.agt/1521684037_20180321220044Wed, 21 Mar 2018 22:00 EDT