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 EDTThe intersection graph of an orientable generic surfacehttps://projecteuclid.org/euclid.agt/1510841406<strong>Doron Ben Hadar</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 3, 1675--1700.</p><p><strong>Abstract:</strong><br/> The intersection graph [math] of a generic surface [math] is the set of values which are either singularities or intersections. It is a multigraph whose edges are transverse intersections of two surfaces and whose vertices are triple intersections and branch values. [math] has an enhanced graph structure which Gui-Song Li referred to as a “daisy graph”. If [math] is oriented, then the orientation further refines the structure of [math] into what Li called an “arrowed daisy graph”. Li left open the question “which arrowed daisy graphs can be realized as the intersection graph of an oriented generic surface?” The main theorem of this article will answer this. I will also provide some generalizations and extensions to this theorem in Sections 4 and 5. </p>projecteuclid.org/euclid.agt/1510841406_20171116091008Thu, 16 Nov 2017 09:10 ESTEmbedding calculus knot invariants are of finite typehttps://projecteuclid.org/euclid.agt/1510841407<strong>Ryan Budney</strong>, <strong>James Conant</strong>, <strong>Robin Koytcheff</strong>, <strong>Dev Sinha</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 3, 1701--1742.</p><p><strong>Abstract:</strong><br/>
We show that the map on components from the space of classical long knots to the [math] stage of its Goodwillie–Weiss embedding calculus tower is a map of monoids whose target is an abelian group and which is invariant under clasper surgery. We deduce that this map on components is a finite type- [math] knot invariant. We compute the [math] –page in total degree zero for the spectral sequence converging to the components of this tower: it consists of [math] –modules of primitive chord diagrams, providing evidence for the conjecture that the tower is a universal finite-type invariant over the integers. Key to these results is the development of a group structure on the tower compatible with connected sum of knots, which in contrast with the corresponding results for the (weaker) homology tower requires novel techniques involving operad actions, evaluation maps and cosimplicial and subcubical diagrams.
</p>projecteuclid.org/euclid.agt/1510841407_20171116091008Thu, 16 Nov 2017 09:10 ESTAffine Hirsch foliations on $3$–manifoldshttps://projecteuclid.org/euclid.agt/1510841408<strong>Bin Yu</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 3, 1743--1770.</p><p><strong>Abstract:</strong><br/> This paper is devoted to discussing affine Hirsch foliations on [math] –manifolds. First, we prove that up to isotopic leaf-conjugacy, every closed orientable [math] –manifold [math] admits zero, one or two affine Hirsch foliations. Furthermore, every case is possible. Then we analyze the [math] –manifolds admitting two affine Hirsch foliations (we call these Hirsch manifolds ). On the one hand, we construct Hirsch manifolds by using exchangeable braided links (we call such Hirsch manifolds DEBL Hirsch manifolds ); on the other hand, we show that every Hirsch manifold virtually is a DEBL Hirsch manifold. Finally, we show that for every [math] , there are only finitely many Hirsch manifolds with strand number [math] . Here the strand number of a Hirsch manifold [math] is a positive integer defined by using strand numbers of braids. </p>projecteuclid.org/euclid.agt/1510841408_20171116091008Thu, 16 Nov 2017 09:10 ESTEquivariant corkshttps://projecteuclid.org/euclid.agt/1510841409<strong>Dave Auckly</strong>, <strong>Hee Jung Kim</strong>, <strong>Paul Melvin</strong>, <strong>Daniel Ruberman</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 3, 1771--1783.</p><p><strong>Abstract:</strong><br/>
For any finite subgroup [math] of [math] , we construct a contractible [math] –manifold [math] , with an effective [math] –action on its boundary, that can be embedded in a closed [math] –manifold so that cutting [math] out and regluing using distinct elements of [math] will always yield distinct smooth [math] –manifolds.
</p>projecteuclid.org/euclid.agt/1510841409_20171116091008Thu, 16 Nov 2017 09:10 ESTA homology-valued invariant for trivalent fatgraph spineshttps://projecteuclid.org/euclid.agt/1510841410<strong>Yusuke Kuno</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 3, 1785--1811.</p><p><strong>Abstract:</strong><br/>
We introduce an invariant for trivalent fatgraph spines of a once-bordered surface, which takes values in the first homology of the surface. This invariant is a secondary object coming from two 1–cocycles on the dual fatgraph complex, one introduced by Morita and Penner in 2008, and the other by Penner, Turaev and the author in 2013. We present an explicit formula for this invariant and investigate its properties. We also show that the mod 2 reduction of the invariant is the difference of two naturally defined spin structures on the surface.
</p>projecteuclid.org/euclid.agt/1510841410_20171116091008Thu, 16 Nov 2017 09:10 ESTThe augmentation category map induced by exact Lagrangian cobordismshttps://projecteuclid.org/euclid.agt/1510841411<strong>Yu Pan</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 3, 1813--1870.</p><p><strong>Abstract:</strong><br/>
To a Legendrian knot, one can associate an [math] category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study this functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.
</p>projecteuclid.org/euclid.agt/1510841411_20171116091008Thu, 16 Nov 2017 09:10 ESTTethers and homology stability for surfaceshttps://projecteuclid.org/euclid.agt/1510841412<strong>Allen Hatcher</strong>, <strong>Karen Vogtmann</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 3, 1871--1916.</p><p><strong>Abstract:</strong><br/>
Homological stability for sequences [math] of groups is often proved by studying the spectral sequence associated to the action of [math] on a highly connected simplicial complex whose stabilizers are related to [math] for [math] . When [math] is the mapping class group of a manifold, suitable simplicial complexes can be made using isotopy classes of various geometric objects in the manifold. We focus on the case of surfaces and show that by using more refined geometric objects consisting of certain configurations of curves with arcs that tether these curves to the boundary, the stabilizers can be greatly simplified and consequently also the spectral sequence argument. We give a careful exposition of this program and its basic tools, then illustrate the method using braid groups before treating mapping class groups of orientable surfaces in full detail.
</p>projecteuclid.org/euclid.agt/1510841412_20171116091008Thu, 16 Nov 2017 09:10 ESTThe mapping cone formula in Heegaard Floer homology and Dehn surgery on knots in $S^3$https://projecteuclid.org/euclid.agt/1510841433<strong>Fyodor Gainullin</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 1917--1951.</p><p><strong>Abstract:</strong><br/>
We write down an explicit formula for the [math] version of the Heegaard Floer homology (as an absolutely graded vector space over an arbitrary field) of the results of Dehn surgery on a knot [math] in [math] in terms of homological data derived from [math] . This allows us to prove some results about Dehn surgery on knots in [math] . In particular, we show that for a fixed manifold there are only finitely many alternating knots that can produce it by surgery. This is an improvement on a recent result by Lackenby and Purcell. We also derive a lower bound on the genus of knots depending on the manifold they give by surgery. Some new restrictions on Seifert fibred surgery are also presented.
</p>projecteuclid.org/euclid.agt/1510841433_20171116091049Thu, 16 Nov 2017 09:10 ESTThe $C_2$–spectrum $\mathrm{Tmf}_1(3)$ and its invertible moduleshttps://projecteuclid.org/euclid.agt/1510841434<strong>Michael Hill</strong>, <strong>Lennart Meier</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 1953--2011.</p><p><strong>Abstract:</strong><br/>
We explore the [math] –equivariant spectra [math] and [math] . In particular, we compute their [math] –equivariant Picard groups and the [math] –equivariant Anderson dual of [math] . This implies corresponding results for the fixed-point spectra [math] and [math] . Furthermore, we prove a real Landweber exact functor theorem.
</p>projecteuclid.org/euclid.agt/1510841434_20171116091049Thu, 16 Nov 2017 09:10 ESTAn algebraic model for commutative $H\mskip-1mu\mathbb{Z}$–algebrashttps://projecteuclid.org/euclid.agt/1510841435<strong>Birgit Richter</strong>, <strong>Brooke Shipley</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2013--2038.</p><p><strong>Abstract:</strong><br/>
We show that the homotopy category of commutative algebra spectra over the Eilenberg–Mac Lane spectrum of an arbitrary commutative ring [math] is equivalent to the homotopy category of [math] –monoids in unbounded chain complexes over [math] . We do this by establishing a chain of Quillen equivalences between the corresponding model categories. We also provide a Quillen equivalence to commutative monoids in the category of functors from the category of finite sets and injections to unbounded chain complexes.
</p>projecteuclid.org/euclid.agt/1510841435_20171116091049Thu, 16 Nov 2017 09:10 ESTEigenvalue varieties of Brunnian linkshttps://projecteuclid.org/euclid.agt/1510841436<strong>François Malabre</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2039--2050.</p><p><strong>Abstract:</strong><br/>
In this article, it is proved that the eigenvalue variety of the exterior of a nontrivial, non-Hopf, Brunnian link in [math] contains a nontrivial component of maximal dimension. Eigenvalue varieties were first introduced to generalize the [math] –polynomial of knots in [math] to manifolds with nonconnected toric boundary. The result presented here generalizes, for Brunnian links, the nontriviality of the [math] –polynomial of nontrivial knots in [math] .
</p>projecteuclid.org/euclid.agt/1510841436_20171116091049Thu, 16 Nov 2017 09:10 ESTA refinement of Betti numbers and homology in the presence of a continuous function, Ihttps://projecteuclid.org/euclid.agt/1510841437<strong>Dan Burghelea</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2051--2080.</p><p><strong>Abstract:</strong><br/> We propose a refinement of the Betti numbers and the homology with coefficients in a field of a compact ANR [math] , in the presence of a continuous real-valued function on [math] . The refinement of Betti numbers consists of finite configurations of points with multiplicities in the complex plane whose total cardinalities are the Betti numbers, and the refinement of homology consists of configurations of vector spaces indexed by points in the complex plane, with the same support as the first, whose direct sum is isomorphic to the homology. When the homology is equipped with a scalar product, these vector spaces are canonically realized as mutually orthogonal subspaces of the homology. The assignments above are in analogy with the collections of eigenvalues and generalized eigenspaces of a linear map in a finite-dimensional complex vector space. A number of remarkable properties of the above configurations are discussed. </p>projecteuclid.org/euclid.agt/1510841437_20171116091049Thu, 16 Nov 2017 09:10 ESTA categorification of the Alexander polynomial in embedded contact homologyhttps://projecteuclid.org/euclid.agt/1510841438<strong>Gilberto Spano</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2081--2124.</p><p><strong>Abstract:</strong><br/> Given a transverse knot [math] in a three-dimensional contact manifold [math] , Colin, Ghiggini, Honda and Hutchings defined a hat version [math] of embedded contact homology for [math] and conjectured that it is isomorphic to the knot Floer homology [math] . We define here a full version [math] and generalize the definitions to the case of links. We prove then that if [math] , then [math] and [math] categorify the (multivariable) Alexander polynomial of knots and links, obtaining expressions analogous to that for knot and link Floer homologies in the minus and, respectively, hat versions. </p>projecteuclid.org/euclid.agt/1510841438_20171116091049Thu, 16 Nov 2017 09:10 ESTOn mod $p$ $A_p$–spaceshttps://projecteuclid.org/euclid.agt/1510841439<strong>Ruizhi Huang</strong>, <strong>Jie Wu</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2125--2144.</p><p><strong>Abstract:</strong><br/>
We prove a necessary condition for the existence of an [math] –structure on [math] spaces, and also derive a simple proof for the finiteness of the number of [math] [math] –spaces of given rank. As a direct application, we compute a list of possible types of rank [math] [math] homotopy associative [math] –spaces.
</p>projecteuclid.org/euclid.agt/1510841439_20171116091049Thu, 16 Nov 2017 09:10 ESTAcylindrical group actions on quasi-treeshttps://projecteuclid.org/euclid.agt/1510841440<strong>Sahana Balasubramanya</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2145--2176.</p><p><strong>Abstract:</strong><br/>
A group [math] is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that every acylindrically hyperbolic group [math] has a generating set [math] such that the corresponding Cayley graph [math] is a (non-elementary) quasi-tree and the action of [math] on [math] is acylindrical. Our proof utilizes the notions of hyperbolically embedded subgroups and projection complexes. As an application, we obtain some new results about hyperbolically embedded subgroups and quasi-convex subgroups of acylindrically hyperbolic groups.
</p>projecteuclid.org/euclid.agt/1510841440_20171116091049Thu, 16 Nov 2017 09:10 ESTTranslation surfaces and the curve graph in genus twohttps://projecteuclid.org/euclid.agt/1510841441<strong>Duc-Manh Nguyen</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2177--2237.</p><p><strong>Abstract:</strong><br/> Let [math] be a (topological) compact closed surface of genus two. We associate to each translation surface [math] a subgraph [math] of the curve graph of [math] . The vertices of this subgraph are free homotopy classes of curves which can be represented either by a simple closed geodesic or by a concatenation of two parallel saddle connections (satisfying some additional properties) on [math] . The subgraph [math] is by definition [math] –invariant. Hence it may be seen as the image of the corresponding Teichmüller disk in the curve graph. We will show that [math] is always connected and has infinite diameter. The group [math] of affine automorphisms of [math] preserves naturally [math] , we show that [math] is precisely the stabilizer of [math] in [math] . We also prove that [math] is Gromov-hyperbolic if [math] is completely periodic in the sense of Calta. It turns out that the quotient of [math] by [math] is closely related to McMullen’s prototypes in the case that [math] is a Veech surface in [math] . We finally show that this quotient graph has finitely many vertices if and only if [math] is a Veech surface for [math] in both strata [math] and [math] . </p>projecteuclid.org/euclid.agt/1510841441_20171116091049Thu, 16 Nov 2017 09:10 ESTThe diagonal slice of Schottky spacehttps://projecteuclid.org/euclid.agt/1510841442<strong>Caroline Series</strong>, <strong>Ser Tan</strong>, <strong>Yasushi Yamashita</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2239--2282.</p><p><strong>Abstract:</strong><br/>
An irreducible representation of the free group on two generators [math] into [math] is determined up to conjugation by the traces of [math] and [math] . If the representation is faithful and discrete, the resulting manifold is in general a genus- [math] handlebody. We study the diagonal slice of the representation variety in which [math] . Using the symmetry, we are able to compute the Keen–Series pleating rays and thus fully determine the locus of faithful discrete representations. We also computationally determine the “Bowditch set” consisting of those parameter values for which no primitive elements in [math] have traces in [math] , and at most finitely many primitive elements have traces with absolute value at most [math] . The graphics make clear that this set is both strictly larger than, and significantly different from, the discreteness locus.
</p>projecteuclid.org/euclid.agt/1510841442_20171116091049Thu, 16 Nov 2017 09:10 ESTUntwisting information from Heegaard Floer homologyhttps://projecteuclid.org/euclid.agt/1510841443<strong>Kenan Ince</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2283--2306.</p><p><strong>Abstract:</strong><br/>
The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. We work with a generalization of the unknotting number due to Mathieu–Domergue, which we call the untwisting number. The [math] –untwisting number is the minimum number (over all diagrams of a knot) of full twists on at most [math] strands of a knot, with half of the strands oriented in each direction, necessary to transform that knot into the unknot. In previous work, we showed that the unknotting and untwisting numbers can be arbitrarily different. In this paper, we show that a common route for obstructing low unknotting number, the Montesinos trick, does not generalize to the untwisting number. However, we use a different approach to get conditions on the Heegaard Floer correction terms of the branched double cover of a knot with untwisting number one. This allows us to obstruct several [math] – and [math] –crossing knots from being unknotted by a single positive or negative twist. We also use the Ozsváth–Szabó [math] invariant and the Rasmussen [math] invariant to differentiate between the [math] – and [math] –untwisting numbers for certain [math] .
</p>projecteuclid.org/euclid.agt/1510841443_20171116091049Thu, 16 Nov 2017 09:10 ESTCyclotomic structure in the topological Hochschild homology of $DX$https://projecteuclid.org/euclid.agt/1510841444<strong>Cary Malkiewich</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2307--2356.</p><p><strong>Abstract:</strong><br/>
Let [math] be a finite CW complex, and let [math] be its dual in the category of spectra. We demonstrate that the Poincaré/Koszul duality between [math] and the free loop space [math] is in fact a genuinely [math] –equivariant duality that preserves the [math] –fixed points. Our proof uses an elementary but surprisingly useful rigidity theorem for the geometric fixed point functor [math] of orthogonal [math] –spectra.
</p>projecteuclid.org/euclid.agt/1510841444_20171116091049Thu, 16 Nov 2017 09:10 ESTSpectral sequences in smooth generalized cohomologyhttps://projecteuclid.org/euclid.agt/1510841445<strong>Daniel Grady</strong>, <strong>Hisham Sati</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2357--2412.</p><p><strong>Abstract:</strong><br/>
We consider spectral sequences in smooth generalized cohomology theories, including differential generalized cohomology theories. The main differential spectral sequences will be of the Atiyah–Hirzebruch (AHSS) type, where we provide a filtration by the Čech resolution of smooth manifolds. This allows for systematic study of torsion in differential cohomology. We apply this in detail to smooth Deligne cohomology, differential topological complex K-theory and to a smooth extension of integral Morava K-theory that we introduce. In each case, we explicitly identify the differentials in the corresponding spectral sequences, which exhibit an interesting and systematic interplay between (refinements of) classical cohomology operations, operations involving differential forms and operations on cohomology with [math] coefficients.
</p>projecteuclid.org/euclid.agt/1510841445_20171116091049Thu, 16 Nov 2017 09:10 ESTEpimorphisms between $2$–bridge knot groups and their crossing numbershttps://projecteuclid.org/euclid.agt/1510841446<strong>Masaaki Suzuki</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2413--2428.</p><p><strong>Abstract:</strong><br/>
Suppose that there exists an epimorphism from the knot group of a 2–bridge knot [math] onto that of another knot [math] . We study the relationship between their crossing numbers [math] and [math] . More specifically, it is shown that [math] is greater than or equal to [math] , and we estimate how many knot groups a 2–bridge knot group maps onto. Moreover, we formulate the generating function which determines the number of 2–bridge knot groups admitting epimorphisms onto the knot group of a given 2–bridge knot.
</p>projecteuclid.org/euclid.agt/1510841446_20171116091049Thu, 16 Nov 2017 09:10 ESTHomotopy decompositions of gauge groups over real surfaceshttps://projecteuclid.org/euclid.agt/1510841447<strong>Michael West</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2429--2480.</p><p><strong>Abstract:</strong><br/>
We analyse the homotopy types of gauge groups of principal [math] –bundles associated to pseudoreal vector bundles in the sense of Atiyah. We provide satisfactory homotopy decompositions of these gauge groups into factors in which the homotopy groups are well known. Therefore, we substantially build upon the low-dimensional homotopy groups as provided by Biswas, Huisman and Hurtubise.
</p>projecteuclid.org/euclid.agt/1510841447_20171116091049Thu, 16 Nov 2017 09:10 ESTCoarse medians and Property Ahttps://projecteuclid.org/euclid.agt/1510841448<strong>Ján Špakula</strong>, <strong>Nick Wright</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2481--2498.</p><p><strong>Abstract:</strong><br/>
We prove that uniformly locally finite quasigeodesic coarse median spaces of finite rank and at most exponential growth have Property A. This offers an alternative proof of the fact that mapping class groups have Property A.
</p>projecteuclid.org/euclid.agt/1510841448_20171116091049Thu, 16 Nov 2017 09:10 ESTGeometric embedding properties of Bestvina–Brady subgroupshttps://projecteuclid.org/euclid.agt/1510841449<strong>Hung Tran</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2499--2510.</p><p><strong>Abstract:</strong><br/>
We compute the relative divergence of right-angled Artin groups with respect to their Bestvina–Brady subgroups and the subgroup distortion of Bestvina–Brady subgroups. We also show that for each integer [math] , there is a free subgroup of rank [math] of some right-angled Artin group whose inclusion is not a quasi-isometric embedding. The corollary answers the question of Carr about the minimum rank [math] such that some right-angled Artin group has a free subgroup of rank [math] whose inclusion is not a quasi-isometric embedding. It is well known that a right-angled Artin group [math] is the fundamental group of a graph manifold whenever the defining graph [math] is a tree with at least three vertices. We show that the Bestvina–Brady subgroup [math] in this case is a horizontal surface subgroup.
</p>projecteuclid.org/euclid.agt/1510841449_20171116091049Thu, 16 Nov 2017 09:10 ESTNon-L–space integral homology $3$–spheres with no nice orderingshttps://projecteuclid.org/euclid.agt/1510841450<strong>Xinghua Gao</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2511--2522.</p><p><strong>Abstract:</strong><br/>
We give infinitely many examples of non-L–space irreducible integer homology [math] –spheres whose fundamental groups do not have nontrivial [math] representations.
</p>projecteuclid.org/euclid.agt/1510841450_20171116091049Thu, 16 Nov 2017 09:10 ESTNoncommutative formality implies commutative and Lie formalityhttps://projecteuclid.org/euclid.agt/1510841451<strong>Bashar Saleh</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2523--2542.</p><p><strong>Abstract:</strong><br/>
Over a field of characteristic zero we prove two formality conditions. We prove that a dg Lie algebra is formal if and only if its universal enveloping algebra is formal. We also prove that a commutative dg algebra is formal as a dg associative algebra if and only if it is formal as a commutative dg algebra. We present some consequences of these theorems in rational homotopy theory.
</p>projecteuclid.org/euclid.agt/1510841451_20171116091049Thu, 16 Nov 2017 09:10 ESTA note on cobordisms of algebraic knotshttps://projecteuclid.org/euclid.agt/1510841452<strong>József Bodnár</strong>, <strong>Daniele Celoria</strong>, <strong>Marco Golla</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 4, 2543--2564.</p><p><strong>Abstract:</strong><br/>
We use Heegaard Floer homology to study smooth cobordisms of algebraic knots and complex deformations of cusp singularities of curves. The main tool will be the concordance invariant [math] : we study its behaviour with respect to connected sums, providing an explicit formula in the case of [math] –space knots and proving subadditivity in general.
</p>projecteuclid.org/euclid.agt/1510841452_20171116091049Thu, 16 Nov 2017 09:10 ESTCategorical models for equivariant classifying spaceshttps://projecteuclid.org/euclid.agt/1510841476<strong>Bertrand Guillou</strong>, <strong>Peter May</strong>, <strong>Mona Merling</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 2565--2602.</p><p><strong>Abstract:</strong><br/>
Starting categorically, we give simple and precise models for classifying spaces of equivariant principal bundles. We need these models for work in progress in equivariant infinite loop space theory and equivariant algebraic [math] –theory, but the models are of independent interest in equivariant bundle theory and especially equivariant covering space theory.
</p>projecteuclid.org/euclid.agt/1510841476_20171116091126Thu, 16 Nov 2017 09:11 ESTBounds on alternating surgery slopeshttps://projecteuclid.org/euclid.agt/1510841477<strong>Duncan McCoy</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 2603--2634.</p><p><strong>Abstract:</strong><br/>
We show that if [math] –surgery on a nontrivial knot [math] yields the branched double cover of an alternating knot, then [math] . This generalises a bound for lens space surgeries first established by Rasmussen. We also show that all surgery coefficients yielding the double branched covers of alternating knots must be contained in an interval of width two and this full range can be realised only if the knot is a cable knot. The work of Greene and Gibbons shows that if [math] bounds a sharp [math] –manifold [math] , then the intersection form of [math] takes the form of a changemaker lattice. We extend this to show that the intersection form is determined uniquely by the knot [math] , the slope [math] and the Betti number [math] .
</p>projecteuclid.org/euclid.agt/1510841477_20171116091126Thu, 16 Nov 2017 09:11 ESTLink homology and equivariant gauge theoryhttps://projecteuclid.org/euclid.agt/1510841478<strong>Prayat Poudel</strong>, <strong>Nikolai Saveliev</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 2635--2685.</p><p><strong>Abstract:</strong><br/>
Singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the singular instanton Floer homology of a knot is graded by the knot signature mod [math] , thereby providing a purely topological way of fixing the absolute grading in the theory. Our approach also results in explicit computations of the generators and gradings of the singular instanton Floer chain complex for several classes of knots with simple double branched covers, such as two-bridge knots, some torus knots, and Montesinos knots, as well as for several families of two-component links.
</p>projecteuclid.org/euclid.agt/1510841478_20171116091126Thu, 16 Nov 2017 09:11 ESTHigher Toda brackets and the Adams spectral sequence in triangulated categorieshttps://projecteuclid.org/euclid.agt/1510841479<strong>J Daniel Christensen</strong>, <strong>Martin Frankland</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 2687--2735.</p><p><strong>Abstract:</strong><br/>
The Adams spectral sequence is available in any triangulated category equipped with a projective or injective class. Higher Toda brackets can also be defined in a triangulated category, as observed by B Shipley based on J Cohen’s approach for spectra. We provide a family of definitions of higher Toda brackets, show that they are equivalent to Shipley’s and show that they are self-dual. Our main result is that the Adams differential [math] in any Adams spectral sequence can be expressed as an [math] –fold Toda bracket and as an [math] order cohomology operation. We also show how the result simplifies under a sparseness assumption, discuss several examples and give an elementary proof of a result of Heller, which implies that the [math] –fold Toda brackets in principle determine the higher Toda brackets.
</p>projecteuclid.org/euclid.agt/1510841479_20171116091126Thu, 16 Nov 2017 09:11 ESTA generalized axis theorem for cube complexeshttps://projecteuclid.org/euclid.agt/1510841480<strong>Daniel Woodhouse</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 2737--2751.</p><p><strong>Abstract:</strong><br/>
We consider a finitely generated virtually abelian group [math] acting properly and without inversions on a [math] cube complex [math] . We prove that [math] stabilizes a finite-dimensional [math] subcomplex [math] that is isometrically embedded in the combinatorial metric. Moreover, we show that [math] is a product of finitely many quasilines. The result represents a higher-dimensional generalization of Haglund’s axis theorem.
</p>projecteuclid.org/euclid.agt/1510841480_20171116091126Thu, 16 Nov 2017 09:11 ESTOn growth of systole along congruence coverings of Hilbert modular varietieshttps://projecteuclid.org/euclid.agt/1510841481<strong>Plinio Murillo</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 2753--2762.</p><p><strong>Abstract:</strong><br/> We study how the systole of principal congruence coverings of a Hilbert modular variety grows when the degree of the covering goes to infinity. We prove that, given a Hilbert modular variety [math] of real dimension [math] defined over a number field [math] , the sequence of principal congruence coverings [math] eventually satisfies [math] where [math] is a constant independent of [math] . </p>projecteuclid.org/euclid.agt/1510841481_20171116091126Thu, 16 Nov 2017 09:11 ESTStable Postnikov data of Picard $2$–categorieshttps://projecteuclid.org/euclid.agt/1510841482<strong>Nick Gurski</strong>, <strong>Niles Johnson</strong>, <strong>Angélica Osorno</strong>, <strong>Marc Stephan</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 2763--2806.</p><p><strong>Abstract:</strong><br/>
Picard [math] –categories are symmetric monoidal [math] –categories with invertible [math] –, [math] – and [math] –cells. The classifying space of a Picard [math] –category [math] is an infinite loop space, the zeroth space of the [math] –theory spectrum [math] . This spectrum has stable homotopy groups concentrated in levels [math] , [math] and [math] . We describe part of the Postnikov data of [math] in terms of categorical structure. We use this to show that there is no strict skeletal Picard [math] –category whose [math] –theory realizes the [math] –truncation of the sphere spectrum. As part of the proof, we construct a categorical suspension, producing a Picard [math] –category [math] from a Picard [math] –category [math] , and show that it commutes with [math] –theory, in that [math] is stably equivalent to [math] .
</p>projecteuclid.org/euclid.agt/1510841482_20171116091126Thu, 16 Nov 2017 09:11 ESTLinks with finite $n$–quandleshttps://projecteuclid.org/euclid.agt/1510841483<strong>Jim Hoste</strong>, <strong>Patrick Shanahan</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 2807--2823.</p><p><strong>Abstract:</strong><br/>
Associated to every oriented link [math] in the 3–sphere is its fundamental quandle and, for each [math] , there is a certain quotient of the fundamental quandle called the [math] –quandle of the link. We prove a conjecture of Przytycki which asserts that the [math] –quandle of an oriented link [math] in the 3–sphere is finite if and only if the fundamental group of the [math] –fold cyclic branched cover of the 3–sphere, branched over [math] , is finite. We do this by extending into the setting of [math] –quandles, Joyce’s result that the fundamental quandle of a knot is isomorphic to a quandle whose elements are the cosets of the peripheral subgroup of the knot group. In addition to proving the conjecture, this relationship allows us to use the well-known Todd–Coxeter process to both enumerate the elements and find a multiplication table of a finite [math] –quandle of a link. We conclude the paper by using Dunbar’s classification of spherical 3–orbifolds to determine all links in the 3–sphere with a finite [math] –quandle for some [math] .
</p>projecteuclid.org/euclid.agt/1510841483_20171116091126Thu, 16 Nov 2017 09:11 ESTVanishing of $L^2$–Betti numbers and failure of acylindrical hyperbolicity of matrix groups over ringshttps://projecteuclid.org/euclid.agt/1510841484<strong>Feng Ji</strong>, <strong>Shengkui Ye</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 5, 2825--2840.</p><p><strong>Abstract:</strong><br/>
Let [math] be an infinite commutative ring with identity and [math] an integer. We prove that for each integer [math] , the [math] –Betti number [math] vanishes when [math] is the general linear group [math] , the special linear group [math] or the group [math] generated by elementary matrices. When [math] is an infinite principal ideal domain, similar results are obtained when [math] is the symplectic group [math] , the elementary symplectic group [math] , the split orthogonal group [math] or the elementary orthogonal group [math] . Furthermore, we prove that [math] is not acylindrically hyperbolic if [math] . We also prove similar results for a class of noncommutative rings. The proofs are based on a notion of [math] –rigid rings.
</p>projecteuclid.org/euclid.agt/1510841484_20171116091126Thu, 16 Nov 2017 09:11 ESTKlein-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 EST