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 EDTHyperplanes of Squier's cube complexeshttps://projecteuclid.org/euclid.agt/1540605641<strong>Anthony Genevois</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3205--3256.</p><p><strong>Abstract:</strong><br/>
To any semigroup presentation [math] and base word [math] may be associated a nonpositively curved cube complex [math] , called a Squier complex , whose underlying graph consists of the words of [math] equal to [math] modulo [math] , where two such words are linked by an edge when one can be transformed into the other by applying a relation of [math] . A group is a diagram group if it is the fundamental group of a Squier complex. We describe hyperplanes in these cube complexes. As a first application, we determine exactly when [math] is a special cube complex, as defined by Haglund and Wise, so that the associated diagram group embeds into a right-angled Artin group. A particular feature of Squier complexes is that the intersections of hyperplanes are “ordered” by a relation [math] . As a strong consequence on the geometry of [math] , we deduce, in finite dimensions, that its universal cover isometrically embeds into a product of finitely many trees with respect to the combinatorial metrics; in particular, we notice that (often) this allows us to embed quasi-isometrically the associated diagram group into a product of finitely many trees, giving information on its asymptotic dimension and its uniform Hilbert space compression. Finally, we exhibit a class of hyperplanes inducing a decomposition of [math] as a graph of spaces, and a fortiori a decomposition of the associated diagram group as a graph of groups, giving a new method to compute presentations of diagram groups. As an application, we associate a semigroup presentation [math] to any finite interval graph [math] , and we prove that the diagram group associated to [math] (for a given base word) is isomorphic to the right-angled Artin group [math] . This result has many consequences on the study of subgroups of diagram groups. In particular, we deduce that, for all [math] , the right-angled Artin group [math] embeds into a diagram group, answering a question of Guba and Sapir.
</p>projecteuclid.org/euclid.agt/1540605641_20181026220100Fri, 26 Oct 2018 22:01 EDTAction dimension of lattices in Euclidean buildingshttps://projecteuclid.org/euclid.agt/1540605642<strong>Kevin Schreve</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3257--3277.</p><p><strong>Abstract:</strong><br/>
We show that if a discrete group [math] acts properly and cocompactly on an [math] –dimensional, thick, Euclidean building, then [math] cannot act properly on a contractible [math] –manifold. As an application, if [math] is a torsion-free [math] –arithmetic group over a number field, we compute the minimal dimension of contractible manifold that admits a proper [math] –action. This partially answers a question of Bestvina, Kapovich, and Kleiner.
</p>projecteuclid.org/euclid.agt/1540605642_20181026220100Fri, 26 Oct 2018 22:01 EDTScl in free productshttps://projecteuclid.org/euclid.agt/1540605643<strong>Lvzhou Chen</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3279--3313.</p><p><strong>Abstract:</strong><br/>
We study stable commutator length (scl) in free products via surface maps into a wedge of spaces. We prove that scl is piecewise rational linear if it vanishes on each factor of the free product, generalizing a theorem of Danny Calegari. We further prove that the property of isometric embedding with respect to scl is preserved under taking free products. The method of proof gives a way to compute scl in free products which lets us generalize and derive in a new way several well-known formulas. Finally we show independently and in a new approach that scl in free products of cyclic groups behaves in a piecewise quasirational way when the word is fixed but the orders of factors vary, previously proved by Timothy Susse, settling a conjecture of Alden Walker.
</p>projecteuclid.org/euclid.agt/1540605643_20181026220100Fri, 26 Oct 2018 22:01 EDTSpectral order for contact manifolds with convex boundaryhttps://projecteuclid.org/euclid.agt/1540605644<strong>András Juhász</strong>, <strong>Sungkyung Kang</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3315--3338.</p><p><strong>Abstract:</strong><br/>
We extend the Heegaard Floer homological definition of spectral order for closed contact [math] –manifolds due to Kutluhan, Matić, Van Horn-Morris, and Wand to contact [math] –manifolds with convex boundary. We show that the order of a codimension-zero contact submanifold bounds the order of the ambient manifold from above. As the neighborhood of an overtwisted disk has order zero, we obtain that overtwisted contact structures have order zero. We also prove that the order of a small perturbation of a Giroux [math] –torsion domain has order at most two, hence any contact structure with positive Giroux torsion has order at most two (and, in particular, a vanishing contact invariant).
</p>projecteuclid.org/euclid.agt/1540605644_20181026220100Fri, 26 Oct 2018 22:01 EDTAlternating links have representativity $2$https://projecteuclid.org/euclid.agt/1540605645<strong>Thomas Kindred</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3339--3362.</p><p><strong>Abstract:</strong><br/>
We prove that if [math] is a nontrivial alternating link embedded (without crossings) in a closed surface [math] , then [math] has a compressing disk whose boundary intersects [math] in no more than two points. Moreover, whenever the surface is incompressible and [math] –incompressible in the link exterior, it can be isotoped to have a standard tube at some crossing of any reduced alternating diagram.
</p>projecteuclid.org/euclid.agt/1540605645_20181026220100Fri, 26 Oct 2018 22:01 EDTThe universal quantum invariant and colored ideal triangulationshttps://projecteuclid.org/euclid.agt/1540605646<strong>Sakie Suzuki</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3363--3402.</p><p><strong>Abstract:</strong><br/>
The Drinfeld double of a finite-dimensional Hopf algebra is a quasitriangular Hopf algebra with the canonical element as the universal [math] –matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal [math] –matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang–Baxter equation of the universal [math] –matrix. On the other hand, the Heisenberg double of a finite-dimensional Hopf algebra has the canonical element (the [math] –tensor ) satisfying the pentagon relation. In this paper we reconstruct the universal quantum invariant using the Heisenberg double, and extend it to an invariant of equivalence classes of colored ideal triangulations of [math] –manifolds up to colored moves . In this construction, a copy of the [math] –tensor is attached to each tetrahedron, and invariance under the colored Pachner [math] moves is shown by the pentagon relation of the [math] –tensor.
</p>projecteuclid.org/euclid.agt/1540605646_20181026220100Fri, 26 Oct 2018 22:01 EDT$A_{\infty}$–resolutions and the Golod property for monomial ringshttps://projecteuclid.org/euclid.agt/1540605647<strong>Robin Frankhuizen</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3403--3424.</p><p><strong>Abstract:</strong><br/>
Let [math] be a monomial ring whose minimal free resolution [math] is rooted. We describe an [math] –algebra structure on [math] . Using this structure, we show that [math] is Golod if and only if the product on [math] vanishes. Furthermore, we give a necessary and sufficient combinatorial condition for [math] to be Golod.
</p>projecteuclid.org/euclid.agt/1540605647_20181026220100Fri, 26 Oct 2018 22:01 EDTSymmetric chain complexes, twisted Blanchfield pairings and knot concordancehttps://projecteuclid.org/euclid.agt/1540605648<strong>Allison N Miller</strong>, <strong>Mark Powell</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3425--3476.</p><p><strong>Abstract:</strong><br/>
We give a formula for the duality structure of the [math] –manifold obtained by doing zero-framed surgery along a knot in the [math] –sphere, starting from a diagram of the knot. We then use this to give a combinatorial algorithm for computing the twisted Blanchfield pairing of such [math] –manifolds. With the twisting defined by Casson–Gordon-style representations, we use our computation of the twisted Blanchfield pairing to show that some subtle satellites of genus two ribbon knots yield nonslice knots. The construction is subtle in the sense that, once based, the infection curve lies in the second derived subgroup of the knot group.
</p>projecteuclid.org/euclid.agt/1540605648_20181026220100Fri, 26 Oct 2018 22:01 EDTDynamic characterizations of quasi-isometry and applications to cohomologyhttps://projecteuclid.org/euclid.agt/1540605649<strong>Xin Li</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3477--3535.</p><p><strong>Abstract:</strong><br/>
We build a bridge between geometric group theory and topological dynamical systems by establishing a dictionary between coarse equivalence and continuous orbit equivalence. As an application, we show that group homology and cohomology in a class of coefficients, including all induced and coinduced modules, are coarse invariants. We deduce that being of type [math] (over arbitrary rings) is a coarse invariant, and that being a (Poincaré) duality group over a ring is a coarse invariant among all groups which have finite cohomological dimension over that ring. Our results also imply that every coarse self-embedding of a Poincaré duality group must be a coarse equivalence. These results were only known under suitable finiteness assumptions, and our work shows that they hold in full generality.
</p>projecteuclid.org/euclid.agt/1540605649_20181026220100Fri, 26 Oct 2018 22:01 EDTSymplectic capacities from positive $S^1$–equivariant symplectic homologyhttps://projecteuclid.org/euclid.agt/1540605650<strong>Jean Gutt</strong>, <strong>Michael Hutchings</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3537--3600.</p><p><strong>Abstract:</strong><br/>
We use positive [math] –equivariant symplectic homology to define a sequence of symplectic capacities [math] for star-shaped domains in [math] . These capacities are conjecturally equal to the Ekeland–Hofer capacities, but they satisfy axioms which allow them to be computed in many more examples. In particular, we give combinatorial formulas for the capacities [math] of any “convex toric domain” or “concave toric domain”. As an application, we determine optimal symplectic embeddings of a cube into any convex or concave toric domain. We also extend the capacities [math] to functions of Liouville domains which are almost but not quite symplectic capacities.
</p>projecteuclid.org/euclid.agt/1540605650_20181026220100Fri, 26 Oct 2018 22:01 EDTHomotopy (pre)derivators of cofibration categories and quasicategorieshttps://projecteuclid.org/euclid.agt/1540605651<strong>Tobias Lenz</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3601--3646.</p><p><strong>Abstract:</strong><br/>
We prove that the homotopy prederivator of a cofibration category is equivalent to the homotopy prederivator of its associated quasicategory of frames , as introduced by Szumiło. We use this comparison result to deduce various abstract properties of the obtained prederivators.
</p>projecteuclid.org/euclid.agt/1540605651_20181026220100Fri, 26 Oct 2018 22:01 EDTThe distribution of knots in the Petaluma modelhttps://projecteuclid.org/euclid.agt/1540605652<strong>Chaim Even-Zohar</strong>, <strong>Joel Hass</strong>, <strong>Nathan Linial</strong>, <strong>Tahl Nowik</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3647--3667.</p><p><strong>Abstract:</strong><br/>
The representation of knots by petal diagrams (Adams et al 2012) naturally defines a sequence of distributions on the set of knots. We establish some basic properties of this randomized knot model. We prove that in the random [math] –petal model the probability of obtaining every specific knot type decays to zero as [math] , the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the [math] –petal model represents at least exponentially many distinct knots.
Past approaches to showing, in some random models, that individual knot types occur with vanishing probability rely on the prevalence of localized connect summands as the complexity of the knot increases. However, this phenomenon is not clear in other models, including petal diagrams, random grid diagrams and uniform random polygons. Thus we provide a new approach to investigate this question.
</p>projecteuclid.org/euclid.agt/1540605652_20181026220100Fri, 26 Oct 2018 22:01 EDTA note on the knot Floer homology of fibered knotshttps://projecteuclid.org/euclid.agt/1540605653<strong>John A Baldwin</strong>, <strong>David Shea Vela-Vick</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3669--3690.</p><p><strong>Abstract:</strong><br/>
We prove that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. Immediate applications include new proofs of Krcatovich’s result that knots with [math] –space surgeries are prime and Hedden and Watson’s result that the rank of knot Floer homology detects the trefoil among knots in the [math] –sphere. We also generalize the latter result, proving a similar theorem for nullhomologous knots in any [math] –manifold. We note that our method of proof inspired Baldwin and Sivek’s recent proof that Khovanov homology detects the trefoil. As part of this work, we also introduce a numerical refinement of the Ozsváth–Szabó contact invariant. This refinement was the inspiration for Hubbard and Saltz’s annular refinement of Plamenevskaya’s transverse link invariant in Khovanov homology.
</p>projecteuclid.org/euclid.agt/1540605653_20181026220100Fri, 26 Oct 2018 22:01 EDTBraid monodromy, orderings and transverse invariantshttps://projecteuclid.org/euclid.agt/1540605654<strong>Olga Plamenevskaya</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3691--3718.</p><p><strong>Abstract:</strong><br/>
A closed braid [math] naturally gives rise to a transverse link [math] in the standard contact [math] –space. We study the effect of the dynamical properties of the monodromy of [math] , such as right-veering, on the contact-topological properties of [math] and the values of transverse invariants in Heegaard Floer and Khovanov homologies. Using grid diagrams and the structure of Dehornoy’s braid ordering, we show that [math] is nonzero whenever [math] has fractional Dehn twist coefficient [math] . (For a [math] –braid, we get a sharp result: [math] if and only if the braid is right-veering.)
</p>projecteuclid.org/euclid.agt/1540605654_20181026220100Fri, 26 Oct 2018 22:01 EDTA signature invariant for knotted Klein graphshttps://projecteuclid.org/euclid.agt/1540605655<strong>Catherine Gille</strong>, <strong>Louis-Hadrien Robert</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 6, 3719--3747.</p><p><strong>Abstract:</strong><br/>
We define some signature invariants for a class of knotted trivalent graphs using branched covers. We relate them to classical signatures of knots and links. Finally, we explain how to compute these invariants through the example of Kinoshita’s knotted theta graph.
</p>projecteuclid.org/euclid.agt/1540605655_20181026220100Fri, 26 Oct 2018 22:01 EDTPoincaré duality complexes with highly connected universal coverhttps://projecteuclid.org/euclid.agt/1545102053<strong>Beatrice Bleile</strong>, <strong>Imre Bokor</strong>, <strong>Jonathan A Hillman</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 3749--3788.</p><p><strong>Abstract:</strong><br/>
Turaev conjectured that the classification, realization and splitting results for Poincaré duality complexes of dimension [math] ( [math] –complexes ) generalize to [math] –complexes with [math] –connected universal cover for [math] . Baues and Bleile showed that such complexes are classified, up to oriented homotopy equivalence, by the triple consisting of their fundamental group, orientation class and the image of their fundamental class in the homology of the fundamental group, verifying Turaev’s conjecture on classification.
We prove Turaev’s conjectures on realization and splitting. We show that a triple [math] , comprising a group [math] , a cohomology class [math] and a homology class [math] , can be realized by a [math] –complex with [math] –connected universal cover if and only if the Turaev map applied to [math] yields an equivalence. We show that a [math] –complex with [math] –connected universal cover is a nontrivial connected sum of two such complexes if and only if its fundamental group is a nontrivial free product of groups.
We then consider the indecomposable [math] –complexes of this type. When [math] is odd the results are similar to those for the case [math] . The indecomposables are either aspherical or have virtually free fundamental group. When [math] is even the indecomposables include manifolds which are neither aspherical nor have virtually free fundamental group, but if the group is virtually free and has no dihedral subgroup of order [math] then it has two ends.
</p>projecteuclid.org/euclid.agt/1545102053_20181217220124Mon, 17 Dec 2018 22:01 ESTCubical rigidification, the cobar construction and the based loop spacehttps://projecteuclid.org/euclid.agt/1545102054<strong>Manuel Rivera</strong>, <strong>Mahmoud Zeinalian</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 3789--3820.</p><p><strong>Abstract:</strong><br/>
We prove the following generalization of a classical result of Adams: for any pointed path-connected topological space [math] , that is not necessarily simply connected, the cobar construction of the differential graded (dg) coalgebra of normalized singular chains in [math] with vertices at [math] is weakly equivalent as a differential graded associative algebra (dga) to the singular chains on the Moore based loop space of [math] at [math] . We deduce this statement from several more general categorical results of independent interest. We construct a functor [math] from simplicial sets to categories enriched over cubical sets with connections, which, after triangulation of their mapping spaces, coincides with Lurie’s rigidification functor [math] from simplicial sets to simplicial categories. Taking normalized chains of the mapping spaces of [math] yields a functor [math] from simplicial sets to dg categories which is the left adjoint to the dg nerve functor. For any simplicial set [math] with [math] , [math] is a dga isomorphic to [math] , the cobar construction on the dg coalgebra [math] of normalized chains on [math] . We use these facts to show that [math] sends categorical equivalences between simplicial sets to maps of connected dg coalgebras which induce quasi-isomorphisms of dgas under the cobar functor, which is strictly stronger than saying the resulting dg coalgebras are quasi-isomorphic.
</p>projecteuclid.org/euclid.agt/1545102054_20181217220124Mon, 17 Dec 2018 22:01 ESTNoncrossing partitions and Milnor fibershttps://projecteuclid.org/euclid.agt/1545102055<strong>Thomas Brady</strong>, <strong>Michael J Falk</strong>, <strong>Colum Watt</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 3821--3838.</p><p><strong>Abstract:</strong><br/>
For a finite real reflection group [math] we use noncrossing partitions of type [math] to construct finite cell complexes with the homotopy type of the Milnor fiber of the associated [math] –discriminant [math] and that of the Milnor fiber of the defining polynomial of the associated reflection arrangement. These complexes support natural cyclic group actions realizing the geometric monodromy. Using the shellability of the noncrossing partition lattice, this cell complex yields a chain complex of homology groups computing the integral homology of the Milnor fiber of [math] .
</p>projecteuclid.org/euclid.agt/1545102055_20181217220124Mon, 17 Dec 2018 22:01 ESTKnot Floer homology and Khovanov–Rozansky homology for singular linkshttps://projecteuclid.org/euclid.agt/1545102056<strong>Nathan Dowlin</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 3839--3885.</p><p><strong>Abstract:</strong><br/>
The (untwisted) oriented cube of resolutions for knot Floer homology assigns a complex [math] to a singular resolution [math] of a knot [math] . Manolescu conjectured that when [math] is in braid position, the homology [math] is isomorphic to the HOMFLY-PT homology of [math] . Together with a naturality condition on the induced edge maps, this conjecture would prove the existence of a spectral sequence from HOMFLY-PT homology to knot Floer homology. Using a basepoint filtration on [math] , a recursion formula for HOMFLY-PT homology and additional [math] –like differentials on [math] , we prove Manolescu’s conjecture. The naturality condition remains open.
</p>projecteuclid.org/euclid.agt/1545102056_20181217220124Mon, 17 Dec 2018 22:01 ESTSome extensions in the Adams spectral sequence and the $51$–stemhttps://projecteuclid.org/euclid.agt/1545102057<strong>Guozhen Wang</strong>, <strong>Zhouli Xu</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 3887--3906.</p><p><strong>Abstract:</strong><br/>
We show a few nontrivial extensions in the classical Adams spectral sequence. In particular, we compute that the [math] –primary part of [math] is [math] . This was the last unsolved [math] –extension problem left by the recent work of Isaksen and the authors through the [math] –stem.
The proof of this result uses the [math] technique, which was introduced by the authors to prove [math] . This paper advertises this technique through examples that have simpler proofs than in our previous work.
</p>projecteuclid.org/euclid.agt/1545102057_20181217220124Mon, 17 Dec 2018 22:01 ESTDimension functions for spherical fibrationshttps://projecteuclid.org/euclid.agt/1545102058<strong>Cihan Okay</strong>, <strong>Ergün Yalçin</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 3907--3941.</p><p><strong>Abstract:</strong><br/>
Given a spherical fibration [math] over the classifying space [math] of a finite group [math] we define a dimension function for the [math] –fold fiber join of [math] , where [math] is some large positive integer. We show that the dimension functions satisfy the Borel–Smith conditions when [math] is large enough. As an application we prove that there exists no spherical fibration over the classifying space of [math] with [math] –effective Euler class, generalizing a result of Ünlü (2004) about group actions on finite complexes homotopy equivalent to a sphere. We have been informed that this result will also appear in upcoming work of Alejandro Adem and Jesper Grodal as a corollary of a previously announced program on homotopy group actions due to Grodal.
</p>projecteuclid.org/euclid.agt/1545102058_20181217220124Mon, 17 Dec 2018 22:01 ESTWeighted sheaves and homology of Artin groupshttps://projecteuclid.org/euclid.agt/1545102059<strong>Giovanni Paolini</strong>, <strong>Mario Salvetti</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 3943--4000.</p><p><strong>Abstract:</strong><br/>
We expand the theory of weighted sheaves over posets, and use it to study the local homology of Artin groups. First, we use such theory to relate the homology of classical braid groups with the homology of certain independence complexes of graphs. Then, in the context of discrete Morse theory on weighted sheaves, we introduce a particular class of acyclic matchings. Explicit formulas for the homology of the corresponding Morse complexes are given, in terms of the ranks of the associated incidence matrices. We use such method to perform explicit computations for the new affine case [math] , as well as for the cases [math] , [math] and [math] (which were already done before by different methods).
</p>projecteuclid.org/euclid.agt/1545102059_20181217220124Mon, 17 Dec 2018 22:01 ESTEquivariant complex bundles, fixed points and equivariant unitary bordismhttps://projecteuclid.org/euclid.agt/1545102060<strong>Andrés Ángel</strong>, <strong>José Manuel Gómez</strong>, <strong>Bernardo Uribe</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4001--4035.</p><p><strong>Abstract:</strong><br/>
We study the fixed points of the universal [math] –equivariant complex vector bundle of rank [math] and obtain a decomposition formula in terms of twisted equivariant universal complex vector bundles of smaller rank. We use this decomposition to describe the fixed points of the complex equivariant K–theory spectrum and the equivariant unitary bordism groups for adjacent families of subgroups.
</p>projecteuclid.org/euclid.agt/1545102060_20181217220124Mon, 17 Dec 2018 22:01 ESTDetecting a subclass of torsion-generated groupshttps://projecteuclid.org/euclid.agt/1545102061<strong>Emily Stark</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4037--4068.</p><p><strong>Abstract:</strong><br/>
We classify the groups quasi-isometric to a group generated by finite-order elements within the class of one-ended hyperbolic groups which are not Fuchsian and whose JSJ decomposition over two-ended subgroups does not contain rigid vertex groups. To do this, we characterize which JSJ trees of a group in this class admit a cocompact group action with quotient a tree. The conditions are stated in terms of two graphs we associate to the degree refinement of a group in this class. We prove there is a group in this class which is quasi-isometric to a Coxeter group but is not abstractly commensurable to a group generated by finite-order elements. Consequently, the subclass of groups in this class generated by finite-order elements is not quasi-isometrically rigid. We provide necessary conditions for two groups in this class to be abstractly commensurable. We use these conditions to prove there are infinitely many abstract commensurability classes within each quasi-isometry class of this class that contains a group generated by finite-order elements.
</p>projecteuclid.org/euclid.agt/1545102061_20181217220124Mon, 17 Dec 2018 22:01 ESTCohomology of symplectic groups and Meyer's signature theoremhttps://projecteuclid.org/euclid.agt/1545102064<strong>Dave Benson</strong>, <strong>Caterina Campagnolo</strong>, <strong>Andrew Ranicki</strong>, <strong>Carmen Rovi</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4069--4091.</p><p><strong>Abstract:</strong><br/>
Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of [math] , and can be computed using an element of [math] . If we denote by [math] the pullback of the universal cover of [math] , then by a theorem of Deligne, every finite index subgroup of [math] contains [math] . As a consequence, a class in the second cohomology of any finite quotient of [math] can at most enable us to compute the signature of a surface bundle modulo [math] . We show that this is in fact possible and investigate the smallest quotient of [math] that contains this information. This quotient [math] is a nonsplit extension of [math] by an elementary abelian group of order [math] . There is a central extension [math] , and [math] appears as a quotient of the metaplectic double cover [math] . It is an extension of [math] by an almost extraspecial group of order [math] , and has a faithful irreducible complex representation of dimension [math] . Provided [math] , the extension [math] is the universal central extension of [math] . Putting all this together, in Section 4 we provide a recipe for computing the signature modulo [math] , and indicate some consequences.
</p>projecteuclid.org/euclid.agt/1545102064_20181217220124Mon, 17 Dec 2018 22:01 ESTOn periodic groups of homeomorphisms of the $2$–dimensional spherehttps://projecteuclid.org/euclid.agt/1545102065<strong>Jonathan Conejeros</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4093--4107.</p><p><strong>Abstract:</strong><br/>
We prove that every finitely generated group of homeomorphisms of the [math] –dimensional sphere all of whose elements have a finite order which is a power of [math] and is such that there exists a uniform bound for the orders of the group elements is finite. We prove a similar result for groups of area-preserving homeomorphisms without the hypothesis that the orders of group elements are powers of [math] provided there is an element of even order.
</p>projecteuclid.org/euclid.agt/1545102065_20181217220124Mon, 17 Dec 2018 22:01 ESTAlgebraic and topological properties of big mapping class groupshttps://projecteuclid.org/euclid.agt/1545102066<strong>Priyam Patel</strong>, <strong>Nicholas G Vlamis</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4109--4142.</p><p><strong>Abstract:</strong><br/>
Let [math] be an orientable, connected topological surface of infinite type (that is, with infinitely generated fundamental group). The main theorem states that if the genus of [math] is finite and at least [math] , then the isomorphism type of the pure mapping class group associated to [math] , denoted by [math] , detects the homeomorphism type of [math] . As a corollary, every automorphism of [math] is induced by a homeomorphism, which extends a theorem of Ivanov from the finite-type setting. In the process of proving these results, we show that [math] is residually finite if and only if [math] has finite genus, demonstrating that the algebraic structure of [math] can distinguish finite- and infinite-genus surfaces. As an independent result, we also show that [math] fails to be residually finite for any infinite-type surface [math] . In addition, we give a topological generating set for [math] equipped with the compact-open topology. In particular, if [math] has at most one end accumulated by genus, then [math] is topologically generated by Dehn twists, otherwise it is topologically generated by Dehn twists along with handle shifts.
</p>projecteuclid.org/euclid.agt/1545102066_20181217220124Mon, 17 Dec 2018 22:01 ESTEquivariant cohomology Chern numbers determine equivariant unitary bordism for torus groupshttps://projecteuclid.org/euclid.agt/1545102068<strong>Zhi Lü</strong>, <strong>Wei Wang</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4143--4160.</p><p><strong>Abstract:</strong><br/>
We show that the integral equivariant cohomology Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary [math] –manifolds, which gives an affirmative answer to a conjecture posed by Guillemin, Ginzburg and Karshon ( Moment maps, cobordisms, and Hamiltonian group actions , Remark H.5 in Appendix H.3), where [math] is a torus. As a further application, we also obtain a satisfactory solution of their Question (A) (Appendix H.1.1) on unitary Hamiltonian [math] –manifolds. Our key ingredients in the proof are the universal toric genus defined by Buchstaber, Panov and Ray and the Kronecker pairing of bordism and cobordism. Our approach heavily exploits Quillen’s geometric interpretation of homotopic unitary cobordism theory. Moreover, this method can also be applied to the study of [math] –equivariant unoriented bordism and can still derive the classical result of tom Dieck.
</p>projecteuclid.org/euclid.agt/1545102068_20181217220124Mon, 17 Dec 2018 22:01 ESTSpaces of orders of some one-relator groupshttps://projecteuclid.org/euclid.agt/1545102069<strong>Juan Alonso</strong>, <strong>Joaquín Brum</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4161--4185.</p><p><strong>Abstract:</strong><br/>
We show that certain left-orderable groups admit no isolated left orders. The groups we consider are cyclic amalgamations of a free group with a general left-orderable group, the HNN extensions of free groups over cyclic subgroups, and a particular class of one-relator groups. In order to prove the results about orders, we develop perturbation techniques for actions of these groups on the line.
</p>projecteuclid.org/euclid.agt/1545102069_20181217220124Mon, 17 Dec 2018 22:01 ESTOn the asymptotic expansion of the quantum $\mathrm{SU}(2)$ invariant at $q = \exp(4\pi\sqrt{-1}/N)$ for closed hyperbolic $3$–manifolds obtained by integral surgery along the figure-eight knothttps://projecteuclid.org/euclid.agt/1545102070<strong>Tomotada Ohtsuki</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4187--4274.</p><p><strong>Abstract:</strong><br/>
It is known that the quantum [math] invariant of a closed [math] –manifold at [math] is of polynomial order as [math] . Recently, Chen and Yang conjectured that the quantum [math] invariant of a closed hyperbolic [math] –manifold at [math] is of order [math] , where [math] is a normalized complex volume of [math] . We can regard this conjecture as a kind of “volume conjecture”, which is an important topic from the viewpoint that it relates quantum topology and hyperbolic geometry.
In this paper, we give a concrete presentation of the asymptotic expansion of the quantum [math] invariant at [math] for closed hyperbolic [math] –manifolds obtained from the [math] –sphere by integral surgery along the figure-eight knot. In particular, the leading term of the expansion is [math] , which gives a proof of the Chen–Yang conjecture for such [math] –manifolds. Further, the semiclassical part of the expansion is a constant multiple of the square root of the Reidemeister torsion for such [math] –manifolds. We expect that the higher-order coefficients of the expansion would be “new” invariants, which are related to “quantization” of the hyperbolic structure of a closed hyperbolic [math] –manifold.
</p>projecteuclid.org/euclid.agt/1545102070_20181217220124Mon, 17 Dec 2018 22:01 ESTNotes on open book decompositions for Engel structureshttps://projecteuclid.org/euclid.agt/1545102071<strong>Vincent Colin</strong>, <strong>Francisco Presas</strong>, <strong>Thomas Vogel</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4275--4303.</p><p><strong>Abstract:</strong><br/>
We relate open book decompositions of a [math] –manifold [math] with its Engel structures. Our main result is, given an open book decomposition of [math] whose binding is a collection of [math] –tori and whose monodromy preserves a framing of a page, the construction of an Engel structure whose isotropic foliation is transverse to the interior of the pages and tangent to the binding.
In particular, the pages are contact manifolds and the monodromy is a compactly supported contactomorphism. As a consequence, on a parallelizable closed [math] –manifold, every open book with toric binding carries in the previous sense an Engel structure. Moreover, we show that among the supported Engel structures we construct, there are loose Engel structures.
</p>projecteuclid.org/euclid.agt/1545102071_20181217220124Mon, 17 Dec 2018 22:01 ESTAnick spaces and Kac–Moody groupshttps://projecteuclid.org/euclid.agt/1545102072<strong>Stephen Theriault</strong>, <strong>Jie Wu</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4305--4328.</p><p><strong>Abstract:</strong><br/>
For primes [math] we prove an approximation to Cohen, Moore and Neisendorfer’s conjecture that the loops on an Anick space retracts off the double loops on a mod- [math] Moore space. The approximation is then used to answer a question posed by Kitchloo regarding the topology of Kac–Moody groups. We show that, for certain rank- [math] Kac–Moody groups [math] , the based loops on [math] is [math] –locally homotopy equivalent to the product of the loops on a [math] –sphere and the loops on an Anick space.
</p>projecteuclid.org/euclid.agt/1545102072_20181217220124Mon, 17 Dec 2018 22:01 ESTLogarithmic Hennings invariants for restricted quantum ${\mathfrak{sl}}(2)$https://projecteuclid.org/euclid.agt/1545102073<strong>Anna Beliakova</strong>, <strong>Christian Blanchet</strong>, <strong>Nathan Geer</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4329--4358.</p><p><strong>Abstract:</strong><br/>
We construct a Hennings-type logarithmic invariant for restricted quantum [math] at a [math] root of unity. This quantum group [math] is not quasitriangular and hence not ribbon, but factorizable. The invariant is defined for a pair: a [math] –manifold [math] and a colored link [math] inside [math] . The link [math] is split into two parts colored by central elements and by trace classes, or elements in the [math] Hochschild homology of [math] , respectively. The two main ingredients of our construction are the universal invariant of a string link with values in tensor powers of [math] , and the modified trace introduced by the third author with his collaborators and computed on tensor powers of the regular representation. Our invariant is a colored extension of the logarithmic invariant constructed by Jun Murakami.
</p>projecteuclid.org/euclid.agt/1545102073_20181217220124Mon, 17 Dec 2018 22:01 ESTNonarithmetic hyperbolic manifolds and trace ringshttps://projecteuclid.org/euclid.agt/1545102074<strong>Olivier Mila</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4359--4373.</p><p><strong>Abstract:</strong><br/>
We give a sufficient condition on the hyperplanes used in the Belolipetsky–Thomson inbreeding construction to obtain nonarithmetic manifolds. We explicitly construct infinitely many examples of such manifolds that are pairwise noncommensurable and estimate their volume.
</p>projecteuclid.org/euclid.agt/1545102074_20181217220124Mon, 17 Dec 2018 22:01 ESTPretty rational models for Poincaré duality pairshttps://projecteuclid.org/euclid.agt/1549940428<strong>Hector Cordova Bulens</strong>, <strong>Pascal Lambrechts</strong>, <strong>Don Stanley</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 1--30.</p><p><strong>Abstract:</strong><br/>
We prove that a large class of Poincaré duality pairs of spaces admit rational models (in the sense of Sullivan) of a convenient form associated to some Poincaré duality CDGA.
</p>projecteuclid.org/euclid.agt/1549940428_20190211220040Mon, 11 Feb 2019 22:00 ESTTopological Hochschild homology of maximal orders in simple $\mathbb{Q}$–algebrashttps://projecteuclid.org/euclid.agt/1549940429<strong>Henry Yi-Wei Chan</strong>, <strong>Ayelet Lindenstrauss</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 31--75.</p><p><strong>Abstract:</strong><br/>
We calculate the topological Hochschild homology groups of a maximal order in a simple algebra over the rationals. Since the positive-dimensional [math] groups consist only of torsion, we do this one prime ideal at a time for all the nonzero prime ideals in the center of the maximal order. This allows us to reduce the problem to studying the topological Hochschild homology groups of maximal orders [math] in simple [math] –algebras. We show that the topological Hochschild homology of [math] splits as the tensor product of its Hochschild homology with [math] . We use this result in Brun’s spectral sequence to calculate [math] , and then we analyze the torsion to get [math] .
</p>projecteuclid.org/euclid.agt/1549940429_20190211220040Mon, 11 Feb 2019 22:00 ESTA simplicial James–Hopf map and decompositions of the unstable Adams spectral sequence for suspensionshttps://projecteuclid.org/euclid.agt/1549940430<strong>Fedor Pavutnitskiy</strong>, <strong>Jie Wu</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 77--108.</p><p><strong>Abstract:</strong><br/>
We use combinatorial group theory methods to extend the definition of the classical James–Hopf invariant to a simplicial group setting. This allows us to realize certain coalgebra idempotents at an [math] level and obtain a functorial decomposition of the spectral sequence, associated with the lower [math] –central series filtration on a free simplicial group.
</p>projecteuclid.org/euclid.agt/1549940430_20190211220040Mon, 11 Feb 2019 22:00 ESTDimensional reduction and the equivariant Chern characterhttps://projecteuclid.org/euclid.agt/1549940431<strong>Augusto Stoffel</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 109--150.</p><p><strong>Abstract:</strong><br/>
We propose a dimensional reduction procedure for [math] –dimensional supersymmetric euclidean field theories (EFTs) in the sense of Stolz and Teichner. Our construction is well suited in the presence of a finite gauge group or, more generally, for field theories over an orbifold. As an illustration, we give a geometric interpretation of the Chern character for manifolds with an action by a finite group.
</p>projecteuclid.org/euclid.agt/1549940431_20190211220040Mon, 11 Feb 2019 22:00 ESTConstructing the virtual fundamental class of a Kuranishi atlashttps://projecteuclid.org/euclid.agt/1549940432<strong>Dusa McDuff</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 151--238.</p><p><strong>Abstract:</strong><br/>
Consider a space [math] , such as a compact space of [math] –holomorphic stable maps, that is the zero set of a Kuranishi atlas. This note explains how to define the virtual fundamental class of [math] by representing [math] via the zero set of a map [math] , where [math] is a finite-dimensional vector space and the domain [math] is an oriented, weighted branched topological manifold. Moreover, [math] is equivariant under the action of the global isotropy group [math] on [math] and [math] . This tuple [math] together with a homeomorphism from [math] to [math] forms a single finite-dimensional model (or chart) for [math] . The construction assumes only that the atlas satisfies a topological version of the index condition that can be obtained from a standard, rather than a smooth, gluing theorem. However, if [math] is presented as the zero set of an [math] –Fredholm operator on a strong polyfold bundle, we outline a much more direct construction of the branched manifold [math] that uses an [math] –smooth partition of unity.
</p>projecteuclid.org/euclid.agt/1549940432_20190211220040Mon, 11 Feb 2019 22:00 EST$E_2$ structures and derived Koszul duality in string topologyhttps://projecteuclid.org/euclid.agt/1549940433<strong>Andrew J Blumberg</strong>, <strong>Michael A Mandell</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 239--279.</p><p><strong>Abstract:</strong><br/>
We construct an equivalence of [math] algebras between two models for the Thom spectrum of the free loop space that are related by derived Koszul duality. To do this, we describe the functoriality and invariance properties of topological Hochschild cohomology.
</p>projecteuclid.org/euclid.agt/1549940433_20190211220040Mon, 11 Feb 2019 22:00 ESTVanishing theorems for representation homology and the derived cotangent complexhttps://projecteuclid.org/euclid.agt/1549940434<strong>Yuri Berest</strong>, <strong>Ajay C Ramadoss</strong>, <strong>Wai-kit Yeung</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 281--339.</p><p><strong>Abstract:</strong><br/>
Let [math] be a reductive affine algebraic group defined over a field [math] of characteristic zero. We study the cotangent complex of the derived [math] –representation scheme [math] of a pointed connected topological space [math] . We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of [math] to the representation homology [math] to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in [math] and generalized lens spaces. In particular, for any finitely generated virtually free group [math] , we show that [math] for all [math] . For a closed Riemann surface [math] of genus [math] , we have [math] for all [math] . The sharp vanishing bounds for [math] actually depend on the genus: we conjecture that if [math] , then [math] for [math] , and if [math] , then [math] for [math] , where [math] is the center of [math] . We prove these bounds locally on the smooth locus of the representation scheme [math] in the case of complex connected reductive groups. One important consequence of our results is the existence of a well-defined [math] –theoretic virtual fundamental class for [math] in the sense of Ciocan-Fontanine and Kapranov (Geom. Topol. 13 (2009) 1779–1804). We give a new “Tor formula” for this class in terms of functor homology.
</p>projecteuclid.org/euclid.agt/1549940434_20190211220040Mon, 11 Feb 2019 22:00 ESTRelative phantom mapshttps://projecteuclid.org/euclid.agt/1549940435<strong>Kouyemon Iriye</strong>, <strong>Daisuke Kishimoto</strong>, <strong>Takahiro Matsushita</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 341--362.</p><p><strong>Abstract:</strong><br/>
The de Bruijn–Erdős theorem states that the chromatic number of an infinite graph equals the maximum of the chromatic numbers of its finite subgraphs. Such determination by finite subobjects appears in the definition of a phantom map, which is classical in algebraic topology. The topological method in combinatorics connects these two, which leads us to define the relative version of a phantom map: a map [math] is called a relative phantom map to a map [math] if the restriction of [math] to any finite subcomplex of [math] lifts to [math] through [math] , up to homotopy. There are two kinds of maps which are obviously relative phantom maps: (1) the composite of a map [math] with [math] ; (2) a usual phantom map [math] . A relative phantom map of type (1) is called trivial, and a relative phantom map out of a suspension which is a sum of (1) and (2) is called relatively trivial. We study the (relative) triviality of relative phantom maps and, in particular, we give rational homology conditions for the (relative) triviality.
</p>projecteuclid.org/euclid.agt/1549940435_20190211220040Mon, 11 Feb 2019 22:00 ESTDistortion of surfaces in graph manifoldshttps://projecteuclid.org/euclid.agt/1549940436<strong>G Christopher Hruska</strong>, <strong>Hoang Thanh Nguyen</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 363--395.</p><p><strong>Abstract:</strong><br/>
Let [math] be an immersed horizontal surface in a [math] –dimensional graph manifold. We show that the fundamental group of the surface [math] is quadratically distorted whenever the surface is virtually embedded (ie separable) and is exponentially distorted when the surface is not virtually embedded.
</p>projecteuclid.org/euclid.agt/1549940436_20190211220040Mon, 11 Feb 2019 22:00 ESTArrow calculus for welded and classical linkshttps://projecteuclid.org/euclid.agt/1549940437<strong>Jean-Baptiste Meilhan</strong>, <strong>Akira Yasuhara</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 397--456.</p><p><strong>Abstract:</strong><br/>
We develop a calculus for diagrams of knotted objects. We define arrow presentations, which encode the crossing information of a diagram into arrows in a way somewhat similar to Gauss diagrams, and more generally [math] –tree presentations, which can be seen as “higher-order Gauss diagrams”. This arrow calculus is used to develop an analogue of Habiro’s clasper theory for welded knotted objects, which contain classical link diagrams as a subset. This provides a “realization” of Polyak’s algebra of arrow diagrams at the welded level, and leads to a characterization of finite-type invariants of welded knots and long knots. As a corollary, we recover several topological results due to Habiro and Shima and to Watanabe on knotted surfaces in [math] –space. We also classify welded string links up to homotopy, thus recovering a result of the first author with Audoux, Bellingeri and Wagner.
</p>projecteuclid.org/euclid.agt/1549940437_20190211220040Mon, 11 Feb 2019 22:00 ESTTorsion homology and cellular approximationhttps://projecteuclid.org/euclid.agt/1549940438<strong>Ramón Flores</strong>, <strong>Fernando Muro</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 457--476.</p><p><strong>Abstract:</strong><br/>
We describe the role of the Schur multiplier in the structure of the [math] –torsion of discrete groups. More concretely, we show how the knowledge of [math] allows us to approximate many groups by colimits of copies of [math] –groups. Our examples include interesting families of noncommutative infinite groups, including Burnside groups, certain solvable groups and branch groups. We also provide a counterexample for a conjecture of Emmanuel Farjoun.
</p>projecteuclid.org/euclid.agt/1549940438_20190211220040Mon, 11 Feb 2019 22:00 ESTThe verbal width of acylindrically hyperbolic groupshttps://projecteuclid.org/euclid.agt/1549940439<strong>Mladen Bestvina</strong>, <strong>Kenneth Bromberg</strong>, <strong>Koji Fujiwara</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 477--489.</p><p><strong>Abstract:</strong><br/>
We show that the verbal width is infinite for acylindrically hyperbolic groups, which include hyperbolic groups, mapping class groups and [math] .
</p>projecteuclid.org/euclid.agt/1549940439_20190211220040Mon, 11 Feb 2019 22:00 ESTOn the homotopy types of $\mathrm{Sp}(n)$ gauge groupshttps://projecteuclid.org/euclid.agt/1549940440<strong>Daisuke Kishimoto</strong>, <strong>Akira Kono</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 491--502.</p><p><strong>Abstract:</strong><br/>
Let [math] be the gauge group of the principal [math] –bundle over [math] corresponding to [math] . We refine the result of Sutherland on the homotopy types of [math] and relate it to the order of a certain Samelson product in [math] . Then we classify the [math] –local homotopy types of [math] for [math] .
</p>projecteuclid.org/euclid.agt/1549940440_20190211220040Mon, 11 Feb 2019 22:00 ESTCohomology rings of compactifications of toric arrangementshttps://projecteuclid.org/euclid.agt/1549940441<strong>Corrado De Concini</strong>, <strong>Giovanni Gaiffi</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 503--532.</p><p><strong>Abstract:</strong><br/>
We previously (Adv. Math. 327 (2018) 390–409) constructed some projective wonderful models for the complement of a toric arrangement in an [math] –dimensional algebraic torus [math] . In this paper we describe their integer cohomology rings by generators and relations.
</p>projecteuclid.org/euclid.agt/1549940441_20190211220040Mon, 11 Feb 2019 22:00 ESTA note on the $(\infty,n)$–category of cobordismshttps://projecteuclid.org/euclid.agt/1552960819<strong>Damien Calaque</strong>, <strong>Claudia Scheimbauer</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 533--655.</p><p><strong>Abstract:</strong><br/>
In this extended note we give a precise definition of fully extended topological field theories à la Lurie. Using complete [math] –fold Segal spaces as a model, we construct an [math] –category of [math] –dimensional bordisms, possibly with tangential structure. We endow it with a symmetric monoidal structure and show that we can recover the usual category of bordisms.
</p>projecteuclid.org/euclid.agt/1552960819_20190318220045Mon, 18 Mar 2019 22:00 EDTParametrized homology via zigzag persistencehttps://projecteuclid.org/euclid.agt/1552960823<strong>Gunnar Carlsson</strong>, <strong>Vin de Silva</strong>, <strong>Sara Kališnik</strong>, <strong>Dmitriy Morozov</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 657--700.</p><p><strong>Abstract:</strong><br/>
This paper introduces parametrized homology, a continuous-parameter generalization of levelset zigzag persistent homology that captures the behavior of the homology of the fibers of a real-valued function on a topological space. This information is encoded as a “barcode” of real intervals, each corresponding to a homological feature supported over that interval; or, equivalently, as a persistence diagram. Points in the persistence diagram are classified algebraically into four classes; geometrically, the classes identify the distinct ways in which homological features perish at the boundaries of their interval of persistence. We study the conditions under which spaces fibered over the real line have a well-defined parametrized homology; we establish the stability of these invariants and we show how the four classes of persistence diagram correspond to the four diagrams that appear in the theory of extended persistence.
</p>projecteuclid.org/euclid.agt/1552960823_20190318220045Mon, 18 Mar 2019 22:00 EDTTopology of (small) Lagrangian cobordismshttps://projecteuclid.org/euclid.agt/1552960827<strong>Mads R Bisgaard</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 701--742.</p><p><strong>Abstract:</strong><br/>
We study the following quantitative phenomenon in symplectic topology: in many situations, if a Lagrangian cobordism is sufficiently small (in a sense we specify) then its topology is to a large extend determined by its boundary. This principle allows us to derive several homological uniqueness results for small Lagrangian cobordisms. In particular, under the smallness assumption, we prove homological uniqueness of the class of Lagrangian cobordisms, which, by Biran and Cornea’s Lagrangian cobordism theory, induces operations on a version of the derived Fukaya category. We also establish a link between our results and Vassilyev’s theory of Lagrange characteristic classes. Most currently known constructions of Lagrangian cobordisms yield small Lagrangian cobordisms in many examples.
</p>projecteuclid.org/euclid.agt/1552960827_20190318220045Mon, 18 Mar 2019 22:00 EDT$2$–associahedrahttps://projecteuclid.org/euclid.agt/1552960828<strong>Nathaniel Bottman</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 743--806.</p><p><strong>Abstract:</strong><br/>
For any [math] and [math] we construct a poset [math] called a [math] –associahedron . The [math] –associahedra arose in symplectic geometry, where they are expected to control maps between Fukaya categories of different symplectic manifolds. We prove that the completion [math] is an abstract polytope of dimension [math] . There are forgetful maps [math] , where [math] is the [math] –dimensional associahedron, and the [math] –associahedra specialize to the associahedra (in two ways) and to the multiplihedra. In an appendix, we work out the [math] – and [math] –dimensional [math] –associahedra in detail.
</p>projecteuclid.org/euclid.agt/1552960828_20190318220045Mon, 18 Mar 2019 22:00 EDTOn $\mathrm{BP}\langle 2\rangle$–cooperationshttps://projecteuclid.org/euclid.agt/1552960829<strong>Dominic Leon Culver</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 807--862.</p><p><strong>Abstract:</strong><br/>
We develop techniques to compute the cooperations algebra for the second truncated Brown–Peterson spectrum [math] . We prove that the cooperations algebra [math] decomposes as a direct sum of an [math] –vector space concentrated in Adams filtration [math] and an [math] –module which is concentrated in even degrees and is [math] –torsion-free. We also develop a recursive method which produces a basis for the [math] –torsion-free component.
</p>projecteuclid.org/euclid.agt/1552960829_20190318220045Mon, 18 Mar 2019 22:00 EDTHigher cyclic operadshttps://projecteuclid.org/euclid.agt/1552960830<strong>Philip Hackney</strong>, <strong>Marcy Robertson</strong>, <strong>Donald Yau</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 863--940.</p><p><strong>Abstract:</strong><br/>
We introduce a convenient definition for weak cyclic operads, which is based on unrooted trees and Segal conditions. More specifically, we introduce a category [math] of trees, which carries a tight relationship to the Moerdijk–Weiss category of rooted trees [math] . We prove a nerve theorem exhibiting colored cyclic operads as presheaves on [math] which satisfy a Segal condition. Finally, we produce a Quillen model category whose fibrant objects satisfy a weak Segal condition, and we consider these objects as an up-to-homotopy generalization of the concept of cyclic operad.
</p>projecteuclid.org/euclid.agt/1552960830_20190318220045Mon, 18 Mar 2019 22:00 EDTLeast dilatation of pure surface braidshttps://projecteuclid.org/euclid.agt/1552960831<strong>Marissa Loving</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 941--964.</p><p><strong>Abstract:</strong><br/>
We study the minimal dilatation of pseudo-Anosov pure surface braids and provide upper and lower bounds as a function of genus and the number of punctures. For a fixed number of punctures, these bounds tend to infinity as the genus does. We also bound the dilatation of pseudo-Anosov pure surface braids away from zero and give a constant upper bound in the case of a sufficient number of punctures.
</p>projecteuclid.org/euclid.agt/1552960831_20190318220045Mon, 18 Mar 2019 22:00 EDTTopological Hochschild homology and higher characteristicshttps://projecteuclid.org/euclid.agt/1552960832<strong>Jonathan A Campbell</strong>, <strong>Kate Ponto</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 965--1017.</p><p><strong>Abstract:</strong><br/>
We show that an important classical fixed-point invariant, the Reidemeister trace, arises as a topological Hochschild homology transfer. This generalizes a corresponding classical result for the Euler characteristic and is a first step in showing the Reidemeister trace is in the image of the cyclotomic trace. The main result follows from developing the relationship between shadows (see Astérisque 333, Soc. Math. France, Paris (2010)), topological Hochschild homology and Morita-invariance in bicategorical generality.
</p>projecteuclid.org/euclid.agt/1552960832_20190318220045Mon, 18 Mar 2019 22:00 EDTSpecies substitution, graph suspension, and graded Hopf algebras of painted tree polytopeshttps://projecteuclid.org/euclid.agt/1552960833<strong>Lisa Berry</strong>, <strong>Stefan Forcey</strong>, <strong>Maria Ronco</strong>, <strong>Patrick Showers</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 1019--1078.</p><p><strong>Abstract:</strong><br/>
Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here include ordered and unordered binary trees and ordered lists (labeled corollas). Some of the polytopes that constitute our main results are well known in other contexts. First we see the classical permutohedra, and then certain generalized permutohedra : specifically the graph associahedra of suspensions of certain simple graphs. As an aside we show that the stellohedra also appear as liftings of generalized permutohedra: graph composihedra for complete graphs. Thus our results give examples of Hopf algebras of tubings and marked tubings of graphs. We also show an alternative associative algebra structure on the graph tubings of star graphs.
</p>projecteuclid.org/euclid.agt/1552960833_20190318220045Mon, 18 Mar 2019 22:00 EDT