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 EDTTopologically slice knots that are not smoothly slice in any definite $4$–manifoldhttps://projecteuclid.org/euclid.agt/1521684022<strong>Kouki Sato</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 827--837.</p><p><strong>Abstract:</strong><br/>
We prove that there exist infinitely many topologically slice knots which cannot bound a smooth null-homologous disk in any definite [math] –manifold. Furthermore, we show that we can take such knots so that they are linearly independent in the knot concordance group.
</p>projecteuclid.org/euclid.agt/1521684022_20180321220044Wed, 21 Mar 2018 22:00 EDTTopological complexity of $n$ points on a treehttps://projecteuclid.org/euclid.agt/1521684023<strong>Steven Scheirer</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 839--876.</p><p><strong>Abstract:</strong><br/>
The topological complexity of a path-connected space [math] , denoted by [math] , can be thought of as the minimum number of continuous rules needed to describe how to move from one point in [math] to another. The space [math] is often interpreted as a configuration space in some real-life context. Here, we consider the case where [math] is the space of configurations of [math] points on a tree [math] . We will be interested in two such configuration spaces. In the first, denoted by [math] , the points are distinguishable, while in the second, [math] , the points are indistinguishable. We determine [math] for any tree [math] and many values of [math] , and consequently determine [math] for the same values of [math] (provided the configuration spaces are path-connected).
</p>projecteuclid.org/euclid.agt/1521684023_20180321220044Wed, 21 Mar 2018 22:00 EDT$\Gamma$–structures and symmetric spaceshttps://projecteuclid.org/euclid.agt/1521684024<strong>Bernhard Hanke</strong>, <strong>Peter Quast</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 877--895.</p><p><strong>Abstract:</strong><br/>
$\Gamma$–structures are weak forms of multiplications on closed oriented manifolds. As was shown by Hopf the rational cohomology algebras of manifolds admitting $\Gamma$–structures are free over odd-degree generators. We prove that this condition is also sufficient for the existence of $\Gamma$–structures on manifolds which are nilpotent in the sense of homotopy theory. This includes homogeneous spaces with connected isotropy groups.
Passing to a more geometric perspective we show that on compact oriented Riemannian symmetric spaces with connected isotropy groups and free rational cohomology algebras the canonical products given by geodesic symmetries define $\Gamma$–structures. This extends work of Albers, Frauenfelder and Solomon on $\Gamma$–structures on Lagrangian Grassmannians.
</p>projecteuclid.org/euclid.agt/1521684024_20180321220044Wed, 21 Mar 2018 22:00 EDTConformal nets IV: The $3$-categoryhttps://projecteuclid.org/euclid.agt/1521684025<strong>Arthur Bartels</strong>, <strong>Christopher L Douglas</strong>, <strong>André Henriques</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 897--956.</p><p><strong>Abstract:</strong><br/>
Conformal nets are a mathematical model for conformal field theory, and defects between conformal nets are a model for an interaction or phase transition between two conformal field theories. We previously introduced a notion of composition, called fusion , between defects. We also described a notion of sectors between defects, modeling an interaction among or transformation between phase transitions, and defined fusion composition operations for sectors. In this paper we prove that altogether the collection of conformal nets, defects, sectors, and intertwiners, equipped with the fusion of defects and fusion of sectors, forms a symmetric monoidal [math] -category. This [math] -category encodes the algebraic structure of the possible interactions among conformal field theories.
</p>projecteuclid.org/euclid.agt/1521684025_20180321220044Wed, 21 Mar 2018 22:00 EDTA characterization of quaternionic Kleinian groups in dimension $2$ with complex trace fieldshttps://projecteuclid.org/euclid.agt/1521684026<strong>Sungwoon Kim</strong>, <strong>Joonhyung Kim</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 957--974.</p><p><strong>Abstract:</strong><br/>
Let [math] be a nonelementary discrete subgroup of [math] . We show that if the sum of diagonal entries of each element of [math] is a complex number, then [math] is conjugate to a subgroup of [math] .
</p>projecteuclid.org/euclid.agt/1521684026_20180321220044Wed, 21 Mar 2018 22:00 EDTRelative $2$–Segal spaceshttps://projecteuclid.org/euclid.agt/1521684027<strong>Matthew B Young</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 975--1039.</p><p><strong>Abstract:</strong><br/>
We introduce a relative version of the [math] –Segal simplicial spaces defined by Dyckerhoff and Kapranov, and Gálvez-Carrillo, Kock and Tonks. Examples of relative [math] –Segal spaces include the categorified unoriented cyclic nerve, real pseudoholomorphic polygons in almost complex manifolds and the [math] –construction from Grothendieck–Witt theory. We show that a relative [math] –Segal space defines a categorical representation of the Hall algebra associated to the base [math] –Segal space. In this way, after decategorification we recover a number of known constructions of Hall algebra representations. We also describe some higher categorical interpretations of relative [math] –Segal spaces.
</p>projecteuclid.org/euclid.agt/1521684027_20180321220044Wed, 21 Mar 2018 22:00 EDTOuter actions of $\mathrm{Out}(F_n)$ on small right-angled Artin groupshttps://projecteuclid.org/euclid.agt/1521684028<strong>Dawid Kielak</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1041--1065.</p><p><strong>Abstract:</strong><br/>
We determine the precise conditions under which [math] , the unique index-two subgroup of [math] , can act nontrivially via outer automorphisms on a RAAG whose defining graph has fewer than [math] vertices.
We also show that the outer automorphism group of a RAAG cannot act faithfully via outer automorphisms on a RAAG with a strictly smaller (in number of vertices) defining graph.
Along the way we determine the minimal dimensions of nontrivial linear representations of congruence quotients of the integral special linear groups over algebraically closed fields of characteristic zero, and provide a new lower bound on the cardinality of a set on which [math] can act nontrivially.
</p>projecteuclid.org/euclid.agt/1521684028_20180321220044Wed, 21 Mar 2018 22:00 EDTQuasi-invariant measures for some amenable groups acting on the linehttps://projecteuclid.org/euclid.agt/1521684029<strong>Nancy Guelman</strong>, <strong>Cristóbal Rivas</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1067--1076.</p><p><strong>Abstract:</strong><br/>
We show that if [math] is a solvable group acting on the line and if there is [math] having no fixed points, then there is a Radon measure [math] on the line quasi-invariant under [math] . In fact, our method allows for the same conclusion for [math] inside a class of groups that is closed under extensions and contains all solvable groups and all groups of subexponential growth.
</p>projecteuclid.org/euclid.agt/1521684029_20180321220044Wed, 21 Mar 2018 22:00 EDTNonfillable Legendrian knots in the $3$–spherehttps://projecteuclid.org/euclid.agt/1521684030<strong>Tolga Etgü</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1077--1088.</p><p><strong>Abstract:</strong><br/>
If $\Lambda$ is a Legendrian knot in the standard contact [math] –sphere that bounds an orientable exact Lagrangian surface $\Sigma$ in the standard symplectic [math] –ball, then the genus of $\Sigma$ is equal to the slice genus of (the smooth knot underlying) $\Lambda$, the rotation number of $\Lambda$ is zero as well as the sum of the Thurston–Bennequin number of $\Lambda$ and the Euler characteristic of $\Sigma$, and moreover, the linearized contact homology of $\Lambda$ with respect to the augmentation induced by $\Sigma$ is isomorphic to the (singular) homology of $\Sigma$. It was asked by Ekholm, Honda and Kálmán (2016) whether the converse of this statement holds. We give a negative answer, providing a family of Legendrian knots with augmentations which are not induced by any exact Lagrangian filling although the associated linearized contact homology is isomorphic to the homology of the smooth surface of minimal genus in the [math] –ball bounding the knot.
</p>projecteuclid.org/euclid.agt/1521684030_20180321220044Wed, 21 Mar 2018 22:00 EDTTaut branched surfaces from veering triangulationshttps://projecteuclid.org/euclid.agt/1521684031<strong>Michael Landry</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1089--1114.</p><p><strong>Abstract:</strong><br/>
Let [math] be a closed hyperbolic [math] –manifold with a fibered face [math] of the unit ball of the Thurston norm on [math] . If [math] satisfies a certain condition related to Agol’s veering triangulations, we construct a taut branched surface in [math] spanning [math] . This partially answers a 1986 question of Oertel, and extends an earlier partial answer due to Mosher.
</p>projecteuclid.org/euclid.agt/1521684031_20180321220044Wed, 21 Mar 2018 22:00 EDTTopological equivalences of E-infinity differential graded algebrashttps://projecteuclid.org/euclid.agt/1521684032<strong>Haldun Özgür Bayındır</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1115--1146.</p><p><strong>Abstract:</strong><br/>
Two DGAs are said to be topologically equivalent when the corresponding Eilenberg–Mac Lane ring spectra are weakly equivalent as ring spectra. Quasi-isomorphic DGAs are topologically equivalent, but the converse is not necessarily true. As a counterexample, Dugger and Shipley showed that there are DGAs that are nontrivially topologically equivalent, ie topologically equivalent but not quasi-isomorphic.
In this work, we define [math] topological equivalences and utilize the obstruction theories developed by Goerss, Hopkins and Miller to construct first examples of nontrivially [math] topologically equivalent [math] DGAs. Also, we show using these obstruction theories that for coconnective [math] –DGAs, [math] topological equivalences and quasi-isomorphisms agree. For [math] –DGAs with trivial first homology, we show that an [math] topological equivalence induces an isomorphism in homology that preserves the Dyer–Lashof operations and therefore induces an [math] –equivalence.
</p>projecteuclid.org/euclid.agt/1521684032_20180321220044Wed, 21 Mar 2018 22:00 EDTA rank inequality for the annular Khovanov homology of $2$–periodic linkshttps://projecteuclid.org/euclid.agt/1521684033<strong>Melissa Zhang</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1147--1194.</p><p><strong>Abstract:</strong><br/>
For a [math] –periodic link [math] in the thickened annulus and its quotient link [math] , we exhibit a spectral sequence with
[math]
This spectral sequence splits along quantum and [math] weight-space gradings, proving a rank inequality [math] for every pair of quantum and [math] weight-space gradings [math] . We also present a few decategorified consequences and discuss partial results toward a similar statement for the Khovanov homology of [math] –periodic links, as well as some frameworks for obstructing [math] –periodicity in links.
</p>projecteuclid.org/euclid.agt/1521684033_20180321220044Wed, 21 Mar 2018 22:00 EDTEuler characteristics and actions of automorphism groups of free groupshttps://projecteuclid.org/euclid.agt/1521684034<strong>Shengkui Ye</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1195--1204.</p><p><strong>Abstract:</strong><br/>
Let [math] be a connected orientable manifold with the Euler characteristic [math] . Denote by [math] the unique subgroup of index two in the automorphism group of a free group. Then any group action of [math] (and thus the special linear group [math] ) with [math] on [math] by homeomorphisms is trivial. This confirms a conjecture related to Zimmer’s program for these manifolds.
</p>projecteuclid.org/euclid.agt/1521684034_20180321220044Wed, 21 Mar 2018 22:00 EDTThe spectrum for commutative complex $K$–theoryhttps://projecteuclid.org/euclid.agt/1521684035<strong>Simon Philipp Gritschacher</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1205--1249.</p><p><strong>Abstract:</strong><br/>
We study commutative complex [math] –theory, a generalised cohomology theory built from spaces of ordered commuting tuples in the unitary groups. We show that the spectrum for commutative complex [math] –theory is stably equivalent to the [math] –group ring of [math] and thus obtain a splitting of its representing space [math] as a product of all the terms in the Whitehead tower for [math] , [math] . As a consequence of the spectrum level identification we obtain the ring of coefficients for this theory. Using the rational Hopf ring for [math] we describe the relationship of our results with a previous computation of the rational cohomology algebra of [math] . This gives an essentially complete description of the space [math] introduced by A Adem and J Gómez.
</p>projecteuclid.org/euclid.agt/1521684035_20180321220044Wed, 21 Mar 2018 22:00 EDTCorrigendum to the article The simplicial boundary of a CAT(0) cube complexhttps://projecteuclid.org/euclid.agt/1521684036<strong>Mark F Hagen</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1251--1256.</p><p><strong>Abstract:</strong><br/>
We correct Theorem 3.10 of [Algebr. Geom. Topol. 13 (2013) 1299–1367] in the infinite-dimensional case. No correction is needed in the finite-dimensional case.
</p>projecteuclid.org/euclid.agt/1521684036_20180321220044Wed, 21 Mar 2018 22:00 EDTCorrection to the article Preorientations of the derived motivic multiplicative grouphttps://projecteuclid.org/euclid.agt/1521684037<strong>Jens Hornbostel</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 2, 1257--1258.</p><p><strong>Abstract:</strong><br/>
We correct a claim concerning motivic [math] –deloopings.
</p>projecteuclid.org/euclid.agt/1521684037_20180321220044Wed, 21 Mar 2018 22:00 EDTCombinatorial spin structures on triangulated manifoldshttps://projecteuclid.org/euclid.agt/1524708092<strong>Ryan Budney</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1259--1279.</p><p><strong>Abstract:</strong><br/>
We give a combinatorial description of spin and [math] –structures on triangulated manifolds of arbitrary dimension. These encodings of spin and [math] –structures are established primarily for the purpose of aiding in computations. The novelty of the approach is that we rely heavily on the naturality of binary symmetric groups to avoid lengthy explicit constructions of smoothings of PL manifolds.
</p>projecteuclid.org/euclid.agt/1524708092_20180425220139Wed, 25 Apr 2018 22:01 EDTThe nonmultiplicativity of the signature modulo $8$ of a fibre bundle is an Arf–Kervaire invarianthttps://projecteuclid.org/euclid.agt/1524708093<strong>Carmen Rovi</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1281--1322.</p><p><strong>Abstract:</strong><br/>
It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle [math] is multiplicative if the fundamental group [math] acts trivially on [math] , with [math] . Hambleton, Korzeniewski and Ranicki proved that in any case the signature is multiplicative modulo [math] , that is, [math] . We present two results concerning the multiplicativity modulo [math] : firstly we identify [math] with a [math] –valued Arf–Kervaire invariant of a Pontryagin squaring operation. Furthermore, we prove that if [math] is [math] –dimensional and the action of [math] is trivial on [math] , this Arf–Kervaire invariant takes value [math] and hence the signature is multiplicative modulo [math] , that is, [math] .
</p>projecteuclid.org/euclid.agt/1524708093_20180425220139Wed, 25 Apr 2018 22:01 EDTGerms of fibrations of spheres by great circles always extend to the whole spherehttps://projecteuclid.org/euclid.agt/1524708094<strong>Patricia Cahn</strong>, <strong>Herman Gluck</strong>, <strong>Haggai Nuchi</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1323--1360.</p><p><strong>Abstract:</strong><br/>
We prove that every germ of a smooth fibration of an odd-dimensional round sphere by great circles extends to such a fibration of the entire sphere, a result previously known only in dimension three.
</p>projecteuclid.org/euclid.agt/1524708094_20180425220139Wed, 25 Apr 2018 22:01 EDTThin position for knots, links, and graphs in $3$–manifoldshttps://projecteuclid.org/euclid.agt/1524708095<strong>Scott Taylor</strong>, <strong>Maggy Tomova</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1361--1409.</p><p><strong>Abstract:</strong><br/>
We define a new notion of thin position for a graph in a [math] –manifold which combines the ideas of thin position for manifolds first originated by Scharlemann and Thompson with the ideas of thin position for knots first originated by Gabai. This thin position has the property that connect-summing annuli and pairs-of-pants show up as thin levels. In a forthcoming paper, this new thin position allows us to define two new families of invariants of knots, links, and graphs in [math] –manifolds. The invariants in one family are similar to bridge number, and the invariants in the other family are similar to Gabai’s width for knots in the [math] –sphere. The invariants in both families detect the unknot and are additive under connected sum and trivalent vertex sum.
</p>projecteuclid.org/euclid.agt/1524708095_20180425220139Wed, 25 Apr 2018 22:01 EDTA colored Khovanov spectrum and its tail for $\mathit{B}$–adequate linkshttps://projecteuclid.org/euclid.agt/1524708096<strong>Michael Willis</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1411--1459.</p><p><strong>Abstract:</strong><br/>
We define a Khovanov spectrum for [math] –colored links and quantum spin networks and derive some of its basic properties. In the case of [math] –colored [math] –adequate links, we show a stabilization of the spectra as the coloring [math] , generalizing the tail behavior of the colored Jones polynomial. Finally, we also provide an alternative, simpler stabilization in the case of the colored unknot.
</p>projecteuclid.org/euclid.agt/1524708096_20180425220139Wed, 25 Apr 2018 22:01 EDTNoncharacterizing slopes for hyperbolic knotshttps://projecteuclid.org/euclid.agt/1524708097<strong>Kenneth Baker</strong>, <strong>Kimihiko Motegi</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1461--1480.</p><p><strong>Abstract:</strong><br/>
A nontrivial slope [math] on a knot [math] in [math] is called a characterizing slope if whenever the result of [math] –surgery on a knot [math] is orientation-preservingly homeomorphic to the result of [math] –surgery on [math] , then [math] is isotopic to [math] . Ni and Zhang ask: for any hyperbolic knot [math] , is a slope [math] with [math] sufficiently large a characterizing slope? In this article, we prove that if we can take an unknot [math] so that [math] –surgery on [math] results in [math] and [math] is not a meridian of [math] , then [math] has infinitely many noncharacterizing slopes. As the simplest known example, the hyperbolic, two-bridge knot [math] has no integral characterizing slopes. This answers the above question in the negative. We also prove that any L-space knot never admits such an unknot [math] .
</p>projecteuclid.org/euclid.agt/1524708097_20180425220139Wed, 25 Apr 2018 22:01 EDTA trivial tail homology for non-$A$–adequate linkshttps://projecteuclid.org/euclid.agt/1524708098<strong>Christine Ruey Shan Lee</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1481--1513.</p><p><strong>Abstract:</strong><br/>
We prove a conjecture of Rozansky’s concerning his categorification of the tail of the colored Jones polynomial for an [math] –adequate link. We show that the tail homology groups he constructs are trivial for non- [math] –adequate links.
</p>projecteuclid.org/euclid.agt/1524708098_20180425220139Wed, 25 Apr 2018 22:01 EDTTopology of holomorphic Lefschetz pencils on the four-torushttps://projecteuclid.org/euclid.agt/1524708099<strong>Noriyuki Hamada</strong>, <strong>Kenta Hayano</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1515--1572.</p><p><strong>Abstract:</strong><br/>
We discuss topological properties of holomorphic Lefschetz pencils on the four-torus. Relying on the theory of moduli spaces of polarized abelian surfaces, we first prove that, under some mild assumptions, the (smooth) isomorphism class of a holomorphic Lefschetz pencil on the four-torus is uniquely determined by its genus and divisibility. We then explicitly give a system of vanishing cycles of the genus- [math] holomorphic Lefschetz pencil on the four-torus due to Smith, and obtain those of holomorphic pencils with higher genera by taking finite unbranched coverings. One can also obtain the monodromy factorization associated with Smith’s pencil in a combinatorial way. This construction allows us to generalize Smith’s pencil to higher genera, which is a good source of pencils on the (topological) four-torus. As another application of the combinatorial construction, for any torus bundle over the torus with a section we construct a genus- [math] Lefschetz pencil whose total space is homeomorphic to that of the given bundle.
</p>projecteuclid.org/euclid.agt/1524708099_20180425220139Wed, 25 Apr 2018 22:01 EDTHyperbolic tangle surgeries and nested linkshttps://projecteuclid.org/euclid.agt/1524708100<strong>John Harnois</strong>, <strong>Hayley Olson</strong>, <strong>Rolland Trapp</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1573--1602.</p><p><strong>Abstract:</strong><br/>
Changes to gluing patterns for fully augmented links are shown to result in generalized fully augmented links. The first changes considered result in [math] –tangle surgeries on hyperbolic fully augmented links that produce hyperbolic generalized fully augmented links. These surgeries motivate the definition of nested links, and a characterization of hyperbolic nested links is given. Finally, the geometry of nested links is compared to that of fully augmented links.
</p>projecteuclid.org/euclid.agt/1524708100_20180425220139Wed, 25 Apr 2018 22:01 EDTMacfarlane hyperbolic $3$–manifoldshttps://projecteuclid.org/euclid.agt/1524708101<strong>Joseph A Quinn</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1603--1632.</p><p><strong>Abstract:</strong><br/>
We identify and study a class of hyperbolic [math] –manifolds (which we call Macfarlane manifolds) whose quaternion algebras admit a geometric interpretation analogous to Hamilton’s classical model for Euclidean rotations. We characterize these manifolds arithmetically, and show that infinitely many commensurability classes of them arise in diverse topological and arithmetic settings. We then use this perspective to introduce a new method for computing their Dirichlet domains. We give similar results for a class of hyperbolic surfaces and explore their occurrence as subsurfaces of Macfarlane manifolds.
</p>projecteuclid.org/euclid.agt/1524708101_20180425220139Wed, 25 Apr 2018 22:01 EDTDivergence of $\mathrm{CAT}(0)$ cube complexes and Coxeter groupshttps://projecteuclid.org/euclid.agt/1524708102<strong>Ivan Levcovitz</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1633--1673.</p><p><strong>Abstract:</strong><br/>
We provide geometric conditions on a pair of hyperplanes of a [math] cube complex that imply divergence bounds for the cube complex. As an application, we classify all right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cubic. This generalizes a theorem of Dani and Thomas that addressed the class of [math] –dimensional right-angled Coxeter groups. As another application, we provide an inductive graph-theoretic criterion on a right-angled Coxeter group’s defining graph which allows us to recognize arbitrary integer degree polynomial divergence for many infinite classes of right-angled Coxeter groups. We also provide similar divergence results for some classes of Coxeter groups that are not right-angled.
</p>projecteuclid.org/euclid.agt/1524708102_20180425220139Wed, 25 Apr 2018 22:01 EDTGenerating families and augmentations for Legendrian surfaceshttps://projecteuclid.org/euclid.agt/1524708103<strong>Dan Rutherford</strong>, <strong>Michael G Sullivan</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1675--1731.</p><p><strong>Abstract:</strong><br/>
We study augmentations of a Legendrian surface [math] in the [math] –jet space, [math] , of a surface [math] . We introduce two types of algebraic/combinatorial structures related to the front projection of [math] that we call chain homotopy diagrams (CHDs) and Morse complex [math] –families (MC2Fs), and show that the existence of a [math] –graded CHD or a [math] –graded MC2F is equivalent to the existence of a [math] –graded augmentation of the Legendrian contact homology DGA to [math] . A CHD is an assignment of chain complexes, chain maps, and homotopy operators to the [math] –, [math] –, and [math] –cells of a compatible polygonal decomposition of the base projection of [math] with restrictions arising from the front projection of [math] . An MC2F consists of a collection of formal handleslide sets and chain complexes, subject to axioms based on the behavior of Morse complexes in [math] –parameter families. We prove that if a Legendrian surface has a tame-at-infinity generating family, then it has a [math] –graded MC2F and hence a [math] –graded augmentation. In addition, continuation maps and a monodromy representation of [math] are associated to augmentations, and then used to provide more refined obstructions to the existence of generating families that (i) are linear at infinity or (ii) have trivial bundle domain. We apply our methods in several examples.
</p>projecteuclid.org/euclid.agt/1524708103_20180425220139Wed, 25 Apr 2018 22:01 EDTCompact Stein surfaces as branched covers with same branch setshttps://projecteuclid.org/euclid.agt/1524708104<strong>Takahiro Oba</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1733--1751.</p><p><strong>Abstract:</strong><br/>
For each integer [math] , we construct a braided surface [math] in [math] and simple branched covers of [math] branched along [math] such that the covers have the same degrees and are mutually diffeomorphic, but Stein structures associated to the covers are mutually not homotopic. As a corollary, for each integer [math] , we also construct a transverse link [math] in the standard contact [math] –sphere and simple branched covers of [math] branched along [math] such that the covers have the same degrees and are mutually diffeomorphic, but contact manifolds associated to the covers are mutually not contactomorphic.
</p>projecteuclid.org/euclid.agt/1524708104_20180425220139Wed, 25 Apr 2018 22:01 EDTThe relative lattice path operadhttps://projecteuclid.org/euclid.agt/1524708105<strong>Alexandre Quesney</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1753--1798.</p><p><strong>Abstract:</strong><br/>
We construct a set-theoretic coloured operad [math] that may be thought of as a combinatorial model for the Swiss cheese operad. This is the relative (or Swiss cheese) version of the lattice path operad constructed by Batanin and Berger. By adapting their condensation process we obtain a topological (resp. chain) operad that we show to be weakly equivalent to the topological (resp. chain) Swiss cheese operad.
</p>projecteuclid.org/euclid.agt/1524708105_20180425220139Wed, 25 Apr 2018 22:01 EDTComparing $4$–manifolds in the pants complex via trisectionshttps://projecteuclid.org/euclid.agt/1524708106<strong>Gabriel Islambouli</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1799--1822.</p><p><strong>Abstract:</strong><br/>
Given two smooth, oriented, closed [math] –manifolds, [math] and [math] , we construct two invariants, [math] and [math] , coming from distances in the pants complex and the dual curve complex, respectively. To do this, we adapt work of Johnson on Heegaard splittings of [math] –manifolds to the trisections of [math] –manifolds introduced by Gay and Kirby. Our main results are that the invariants are independent of the choices made throughout the process, as well as interpretations of “nearby” manifolds. This naturally leads to various graphs of [math] –manifolds coming from unbalanced trisections, and we briefly explore their properties.
</p>projecteuclid.org/euclid.agt/1524708106_20180425220139Wed, 25 Apr 2018 22:01 EDTThe nonorientable $4$–genus for knots with $8$ or $9$ crossingshttps://projecteuclid.org/euclid.agt/1524708107<strong>Stanislav Jabuka</strong>, <strong>Tynan Kelly</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1823--1856.</p><p><strong>Abstract:</strong><br/>
The nonorientable [math] –genus of a knot in the [math] –sphere is defined as the smallest first Betti number of any nonorientable surface smoothly and properly embedded in the [math] –ball with boundary the given knot. We compute the nonorientable [math] –genus for all knots with crossing number [math] or [math] . As applications we prove a conjecture of Murakami and Yasuhara and compute the clasp and slicing number of a knot.
</p>projecteuclid.org/euclid.agt/1524708107_20180425220139Wed, 25 Apr 2018 22:01 EDTThe eta-inverted sphere over the rationalshttps://projecteuclid.org/euclid.agt/1524708108<strong>Glen Matthew Wilson</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 3, 1857--1881.</p><p><strong>Abstract:</strong><br/>
We calculate the motivic stable homotopy groups of the two-complete sphere spectrum after inverting multiplication by the Hopf map [math] over fields of cohomological dimension at most [math] with characteristic different from [math] (this includes the [math] –adic fields [math] and the finite fields [math] of odd characteristic) and the field of rational numbers; the ring structure is also determined.
</p>projecteuclid.org/euclid.agt/1524708108_20180425220139Wed, 25 Apr 2018 22:01 EDTAlgebraic ending laminations and quasiconvexityhttps://projecteuclid.org/euclid.agt/1525312822<strong>Mahan Mj</strong>, <strong>Kasra Rafi</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 1883--1916.</p><p><strong>Abstract:</strong><br/>
We explicate a number of notions of algebraic laminations existing in the literature, particularly in the context of an exact sequence
1
→
H
→
G
→
Q
→
1
of hyperbolic groups. These laminations arise in different contexts: existence of Cannon–Thurston maps; closed geodesics exiting ends of manifolds; dual to actions on [math] –trees.
We use the relationship between these laminations to prove quasiconvexity results for finitely generated infinite-index subgroups of [math] , the normal subgroup in the exact sequence above.
</p>projecteuclid.org/euclid.agt/1525312822_20180502220029Wed, 02 May 2018 22:00 EDTThe concordance invariant tau in link grid homologyhttps://projecteuclid.org/euclid.agt/1525312823<strong>Alberto Cavallo</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 1917--1951.</p><p><strong>Abstract:</strong><br/>
We introduce a generalization of the Ozsváth–Szabó [math] –invariant to links by studying a filtered version of link grid homology. We prove that this invariant remains unchanged under strong concordance and we show that it produces a lower bound for the slice genus of a link. We show that this bound is sharp for torus links and we also give an application to Legendrian link invariants in the standard contact [math] –sphere.
</p>projecteuclid.org/euclid.agt/1525312823_20180502220029Wed, 02 May 2018 22:00 EDTSymplectic homology and the Eilenberg–Steenrod axiomshttps://projecteuclid.org/euclid.agt/1525312824<strong>Kai Cieliebak</strong>, <strong>Alexandru Oancea</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 1953--2130.</p><p><strong>Abstract:</strong><br/>
We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg–Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, the Legendrian duality sequence, and the exact sequence relating symplectic homology and Rabinowitz Floer homology. New consequences of this framework include a Mayer–Vietoris exact sequence for symplectic homology, invariance of Rabinowitz Floer homology under subcritical handle attachment, and a new product on Rabinowitz Floer homology unifying the pair-of-pants product on symplectic homology with a secondary coproduct on positive symplectic homology.
In the appendix, joint with Peter Albers, we discuss obstructions to the existence of certain Liouville cobordisms.
</p>projecteuclid.org/euclid.agt/1525312824_20180502220029Wed, 02 May 2018 22:00 EDTThe double $n$–space property for contractible $n$–manifoldshttps://projecteuclid.org/euclid.agt/1525312825<strong>Peter Sparks</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 2131--2149.</p><p><strong>Abstract:</strong><br/>
Motivated by a recent paper of Gabai (J. Topol. 4 (2011) 529–534) on the Whitehead contractible 3–manifold, we investigate contractible manifolds [math] which decompose or split as [math] with [math] or [math] . Of particular interest to us is the case [math] . Our main results exhibit large collections of 4–manifolds that split in this manner.
</p>projecteuclid.org/euclid.agt/1525312825_20180502220029Wed, 02 May 2018 22:00 EDTSymplectic embeddings of four-dimensional polydisks into ballshttps://projecteuclid.org/euclid.agt/1525312826<strong>Katherine Christianson</strong>, <strong>Jo Nelson</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 2151--2178.</p><p><strong>Abstract:</strong><br/>
We obtain new obstructions to symplectic embeddings of the four-dimensional polydisk [math] into the ball [math] for [math] , extending work done by Hind and Lisi and by Hutchings. Schlenk’s folding construction permits us to conclude our bound on [math] is optimal. Our proof makes use of the combinatorial criterion necessary for one “convex toric domain” to symplectically embed into another introduced by Hutchings (2016). We also observe that the computational complexity of this criterion can be reduced from [math] to [math] .
</p>projecteuclid.org/euclid.agt/1525312826_20180502220029Wed, 02 May 2018 22:00 EDTEquivariant dendroidal setshttps://projecteuclid.org/euclid.agt/1525312827<strong>Luís Alexandre Pereira</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 2179--2244.</p><p><strong>Abstract:</strong><br/>
We extend the Cisinski–Moerdijk–Weiss theory of [math] –operads to the equivariant setting to obtain a notion of [math] - [math] –operads that encode “equivariant operads with norm maps” up to homotopy. At the root of this work is the identification of a suitable category of [math] –trees together with a notion of [math] –inner horns capable of encoding the compositions of norm maps.
Additionally, we follow Blumberg and Hill by constructing suitable variants associated to each of the indexing systems featured in their work.
</p>projecteuclid.org/euclid.agt/1525312827_20180502220029Wed, 02 May 2018 22:00 EDTThe number of fiberings of a surface bundle over a surfacehttps://projecteuclid.org/euclid.agt/1525312828<strong>Lei Chen</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 2245--2263.</p><p><strong>Abstract:</strong><br/>
For a closed manifold [math] , let [math] be the number of ways that [math] can be realized as a surface bundle, up to [math] –fiberwise diffeomorphism. We consider the case when [math] . We give the first computation of [math] where [math] but [math] is not a product. In particular, we prove [math] for the Atiyah–Kodaira manifold and any finite cover of a trivial surface bundle. We also give an example where [math] .
</p>projecteuclid.org/euclid.agt/1525312828_20180502220029Wed, 02 May 2018 22:00 EDTRefinements of the holonomic approximation lemmahttps://projecteuclid.org/euclid.agt/1525312829<strong>Daniel Álvarez-Gavela</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 2265--2303.</p><p><strong>Abstract:</strong><br/>
The holonomic approximation lemma of Eliashberg and Mishachev is a powerful tool in the philosophy of the [math] –principle. By carefully keeping track of the quantitative geometry behind the holonomic approximation process, we establish several refinements of this lemma. Gromov’s idea from convex integration of working “one pure partial derivative at a time” is central to the discussion. We give applications of our results to flexible symplectic and contact topology.
</p>projecteuclid.org/euclid.agt/1525312829_20180502220029Wed, 02 May 2018 22:00 EDTStability phenomena in the homology of tree braid groupshttps://projecteuclid.org/euclid.agt/1525312830<strong>Eric Ramos</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 2305--2337.</p><p><strong>Abstract:</strong><br/>
For a tree [math] , we study the changing behaviors in the homology groups [math] as [math] varies, where [math] . We prove that the ranks of these homologies can be described by a single polynomial for all [math] , and construct this polynomial explicitly in terms of invariants of the tree [math] . To accomplish this we prove that the group [math] can be endowed with the structure of a finitely generated graded module over an integral polynomial ring, and further prove that it naturally decomposes as a direct sum of graded shifts of squarefree monomial ideals. Following this, we spend time considering how our methods might be generalized to braid groups of arbitrary graphs, and make various conjectures in this direction.
</p>projecteuclid.org/euclid.agt/1525312830_20180502220029Wed, 02 May 2018 22:00 EDTQuasiautomorphism groups of type $F_\infty$https://projecteuclid.org/euclid.agt/1525312831<strong>Samuel Audino</strong>, <strong>Delaney R Aydel</strong>, <strong>Daniel S Farley</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 2339--2369.</p><p><strong>Abstract:</strong><br/>
The groups [math] , [math] , [math] , [math] and [math] are groups of quasiautomorphisms of the infinite binary tree. Their names indicate a similarity with Thompson’s well-known groups [math] , [math] and [math] .
We will use the theory of diagram groups over semigroup presentations to prove that all of the above groups (and several generalizations) have type [math] . Our proof uses certain types of hybrid diagrams, which have properties in common with both planar diagrams and braided diagrams. The diagram groups defined by hybrid diagrams also act properly and isometrically on [math] cubical complexes.
</p>projecteuclid.org/euclid.agt/1525312831_20180502220029Wed, 02 May 2018 22:00 EDTOn hyperbolic knots in $S^3$ with exceptional surgeries at maximal distancehttps://projecteuclid.org/euclid.agt/1525312832<strong>Benjamin Audoux</strong>, <strong>Ana G Lecuona</strong>, <strong>Fionntan Roukema</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 2371--2417.</p><p><strong>Abstract:</strong><br/>
Baker showed that [math] of the [math] classes of Berge knots are obtained by surgery on the minimally twisted [math] –chain link. We enumerate all hyperbolic knots in [math] obtained by surgery on the minimally twisted [math] –chain link that realize the maximal known distances between slopes corresponding to exceptional (lens, lens), (lens, toroidal) and (lens, Seifert fibred) pairs. In light of Baker’s work, the classification in this paper conjecturally accounts for “most” hyperbolic knots in [math] realizing the maximal distance between these exceptional pairs. As a byproduct, we obtain that all examples that arise from the [math] –chain link actually arise from the magic manifold. The classification highlights additional examples not mentioned in Martelli and Petronio’s survey of the exceptional fillings on the magic manifold. Of particular interest is an example of a knot with two lens space surgeries that is not obtained by filling the Berge manifold (ie the exterior of the unique hyperbolic knot in a solid torus with two nontrivial surgeries producing solid tori).
</p>projecteuclid.org/euclid.agt/1525312832_20180502220029Wed, 02 May 2018 22:00 EDTOn some adjunctions in equivariant stable homotopy theoryhttps://projecteuclid.org/euclid.agt/1525312833<strong>Po Hu</strong>, <strong>Igor Kriz</strong>, <strong>Petr Somberg</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 2419--2442.</p><p><strong>Abstract:</strong><br/>
We investigate certain adjunctions in derived categories of equivariant spectra, including a right adjoint to fixed points, a right adjoint to pullback by an isometry of universes, and a chain of two right adjoints to geometric fixed points. This leads to a variety of interesting other adjunctions, including a chain of six (sometimes seven) adjoints involving the restriction functor to a subgroup of a finite group on equivariant spectra indexed over the trivial universe.
</p>projecteuclid.org/euclid.agt/1525312833_20180502220029Wed, 02 May 2018 22:00 EDTThe homology of configuration spaces of trees with loopshttps://projecteuclid.org/euclid.agt/1525312834<strong>Safia Chettih</strong>, <strong>Daniel Lütgehetmann</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 2443--2469.</p><p><strong>Abstract:</strong><br/>
We show that the homology of ordered configuration spaces of finite trees with loops is torsion-free. We introduce configuration spaces with sinks, which allow for taking quotients of the base space. Furthermore, we give a concrete generating set for all homology groups of configuration spaces of trees with loops and the first homology group of configuration spaces of general finite graphs. An important technique in the paper is the identification of the [math] –page and differentials of Mayer–Vietoris spectral sequences for configuration spaces.
</p>projecteuclid.org/euclid.agt/1525312834_20180502220029Wed, 02 May 2018 22:00 EDTOn the virtually cyclic dimension of mapping class groups of punctured sphereshttps://projecteuclid.org/euclid.agt/1525312835<strong>Javier Aramayona</strong>, <strong>Daniel Juan-Pineda</strong>, <strong>Alejandra Trujillo-Negrete</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 2471--2495.</p><p><strong>Abstract:</strong><br/>
We calculate the virtually cyclic dimension of the mapping class group of a sphere with at most six punctures. As an immediate consequence, we obtain the virtually cyclic dimension of the mapping class group of the twice-holed torus and of the closed genus-two surface.
For spheres with an arbitrary number of punctures, we give a new upper bound for the virtually cyclic dimension of their mapping class group, improving the recent bound of Degrijse and Petrosyan (2015).
</p>projecteuclid.org/euclid.agt/1525312835_20180502220029Wed, 02 May 2018 22:00 EDTLink invariants derived from multiplexing of crossingshttps://projecteuclid.org/euclid.agt/1525312836<strong>Haruko Aida Miyazawa</strong>, <strong>Kodai Wada</strong>, <strong>Akira Yasuhara</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 4, 2497--2507.</p><p><strong>Abstract:</strong><br/>
We introduce the multiplexing of a crossing, replacing a classical crossing of a virtual link diagram with a mixture of classical and virtual crossings.
For integers [math] [math] and an ordered [math] –component virtual link diagram [math] , a new virtual link diagram [math] is obtained from [math] by the multiplexing of all crossings. For welded isotopic virtual link diagrams [math] and [math] , the virtual link diagrams [math] and [math] are welded isotopic. From the point of view of classical link theory, it seems very interesting that new classical link invariants are obtained from welded link invariants via the multiplexing of crossings.
</p>projecteuclid.org/euclid.agt/1525312836_20180502220029Wed, 02 May 2018 22:00 EDTThe geometry of the knot concordance spacehttps://projecteuclid.org/euclid.agt/1535594416<strong>Tim D Cochran</strong>, <strong>Shelly Harvey</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 2509--2540.</p><p><strong>Abstract:</strong><br/>
Most of the [math] years of study of the set of knot concordance classes, [math] , has focused on its structure as an abelian group. Here we take a different approach, namely we study [math] as a metric space admitting many natural geometric operators. We focus especially on the coarse geometry of satellite operators. We consider several knot concordance spaces, corresponding to different categories of concordance, and two different metrics. We establish the existence of quasi- [math] –flats for every [math] , implying that [math] admits no quasi-isometric embedding into a finite product of (Gromov) hyperbolic spaces. We show that every satellite operator is a quasihomomorphism [math] . We show that winding number one satellite operators induce quasi-isometries with respect to the metric induced by slice genus. We prove that strong winding number one patterns induce isometric embeddings for certain metrics. By contrast, winding number zero satellite operators are bounded functions and hence quasicontractions. These results contribute to the suggestion that [math] is a fractal space. We establish various other results about the large-scale geometry of arbitrary satellite operators.
</p>projecteuclid.org/euclid.agt/1535594416_20180829220047Wed, 29 Aug 2018 22:00 EDTThe factorization theory of Thom spectra and twisted nonabelian Poincaré dualityhttps://projecteuclid.org/euclid.agt/1535594417<strong>Inbar Klang</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 2541--2592.</p><p><strong>Abstract:</strong><br/>
We give a description of the factorization homology and [math] topological Hochschild cohomology of Thom spectra arising from [math] –fold loop maps [math] , where [math] is an [math] –fold loop space. We describe the factorization homology [math] as the Thom spectrum associated to a certain map [math] , where [math] is the factorization homology of [math] with coefficients in [math] . When [math] is framed and [math] is [math] –connected, this spectrum is equivalent to a Thom spectrum of a virtual bundle over the mapping space [math] ; in general, this is a Thom spectrum of a virtual bundle over a certain section space. This can be viewed as a twisted form of the nonabelian Poincaré duality theorem of Segal, Salvatore and Lurie, which occurs when [math] is nullhomotopic. This result also generalizes the results of Blumberg, Cohen and Schlichtkrull on the topological Hochschild homology of Thom spectra, and of Schlichtkrull on higher topological Hochschild homology of Thom spectra. We use this description of the factorization homology of Thom spectra to calculate the factorization homology of the classical cobordism spectra, spectra arising from systems of groups and the Eilenberg–Mac Lane spectra [math] , [math] and [math] . We build upon the description of the factorization homology of Thom spectra to study the ( [math] and higher) topological Hochschild cohomology of Thom spectra, which enables calculations and a description in terms of sections of a parametrized spectrum. If [math] is a closed manifold, Atiyah duality for parametrized spectra allows us to deduce a duality between [math] topological Hochschild homology and [math] topological Hochschild cohomology, recovering string topology operations when [math] is nullhomotopic. In conjunction with the higher Deligne conjecture, this gives [math] –structures on a certain family of Thom spectra, which were not previously known to be ring spectra.
</p>projecteuclid.org/euclid.agt/1535594417_20180829220047Wed, 29 Aug 2018 22:00 EDTA May-type spectral sequence for higher topological Hochschild homologyhttps://projecteuclid.org/euclid.agt/1535594418<strong>Gabe Angelini-Knoll</strong>, <strong>Andrew Salch</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 2593--2660.</p><p><strong>Abstract:</strong><br/>
Given a filtration of a commutative monoid [math] in a symmetric monoidal stable model category [math] , we construct a spectral sequence analogous to the May spectral sequence whose input is the higher order topological Hochschild homology of the associated graded commutative monoid of [math] , and whose output is the higher order topological Hochschild homology of [math] . We then construct examples of such filtrations and derive some consequences: for example, given a connective commutative graded ring [math] , we get an upper bound on the size of the [math] –groups of [math] –ring spectra [math] such that [math] .
</p>projecteuclid.org/euclid.agt/1535594418_20180829220047Wed, 29 Aug 2018 22:00 EDTWild translation surfaces and infinite genushttps://projecteuclid.org/euclid.agt/1535594419<strong>Anja Randecker</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 2661--2699.</p><p><strong>Abstract:</strong><br/>
The Gauss–Bonnet formula for classical translation surfaces relates the cone angle of the singularities (geometry) to the genus of the surface (topology). When considering more general translation surfaces, we observe so-called wild singularities for which the notion of cone angle is not applicable any more.
We study whether there still exist relations between the geometry and the topology for translation surfaces with wild singularities. By considering short saddle connections, we determine under which conditions the existence of a wild singularity implies infinite genus. We apply this to show that parabolic or essentially finite translation surfaces with wild singularities have infinite genus.
</p>projecteuclid.org/euclid.agt/1535594419_20180829220047Wed, 29 Aug 2018 22:00 EDTModulo $2$ counting of Klein-bottle leaves in smooth taut foliationshttps://projecteuclid.org/euclid.agt/1535594420<strong>Boyu Zhang</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 2701--2727.</p><p><strong>Abstract:</strong><br/>
We prove a modulo [math] invariance for the number of Klein-bottle leaves in taut foliations. Given two smooth cooriented taut foliations, assume that every Klein-bottle leaf has nontrivial linear holonomy, and assume that the two foliations can be smoothly deformed to each other through taut foliations. We prove that the numbers of Klein-bottle leaves in these two foliations must have the same parity.
</p>projecteuclid.org/euclid.agt/1535594420_20180829220047Wed, 29 Aug 2018 22:00 EDTSelf-dual binary codes from small covers and simple polytopeshttps://projecteuclid.org/euclid.agt/1535594421<strong>Bo Chen</strong>, <strong>Zhi Lü</strong>, <strong>Li Yu</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 2729--2767.</p><p><strong>Abstract:</strong><br/>
The work of Volker Puppe and Matthias Kreck exhibited some intriguing connections between the algebraic topology of involutions on closed manifolds and the combinatorics of self-dual binary codes. On the other hand, the work of Michael Davis and Tadeusz Januszkiewicz brought forth a topological analogue of smooth, real toric varieties, known as “small covers”, which are closed smooth manifolds equipped with some actions of elementary abelian [math] –groups whose orbit spaces are simple convex polytopes. Building on these works, we find various new connections between all these topological and combinatorial objects and obtain some new applications to the study of self-dual binary codes, as well as colorability of polytopes. We first show that a small cover [math] over a simple [math] –polytope [math] produces a self-dual code in the sense of Kreck and Puppe if and only if [math] is [math] –colorable and [math] is odd. Then we show how to describe such a self-dual binary code in terms of the combinatorics of [math] . Moreover, we can construct a family of binary codes [math] , for [math] , from an arbitrary simple [math] –polytope [math] . Then we give some necessary and sufficient conditions for [math] to be self-dual. A spinoff of our study of such binary codes gives some new ways to judge whether a simple [math] –polytope [math] is [math] –colorable in terms of the associated binary codes [math] . In addition, we prove that the minimum distance of the self-dual binary code obtained from a [math] –colorable simple [math] –polytope is always [math] .
</p>projecteuclid.org/euclid.agt/1535594421_20180829220047Wed, 29 Aug 2018 22:00 EDTHomological stability for diffeomorphism groups of high-dimensional handlebodieshttps://projecteuclid.org/euclid.agt/1535594422<strong>Nathan Perlmutter</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 2769--2820.</p><p><strong>Abstract:</strong><br/>
We prove a homological stability theorem for the diffeomorphism groups of high-dimensional manifolds with boundary, with respect to forming the boundary connected sum with the product [math] for [math] . In a recent joint paper with B Botvinnik, we prove that there is an isomorphism
colim
g
→
∞
H
∗
(
BDiff
(
(
D
n
+
1
×
S
n
)
♮
g
,
D
2
n
)
;
ℤ
)
≅
H
∗
(
Q
0
B
O
(
2
n
+
1
)
〈
n
〉
+
;
ℤ
)
in the case that [math] . By combining this “stable homology” calculation with the homological stability theorem of this paper, we obtain the isomorphism
H
k
(
BDiff
(
(
D
n
+
1
×
S
n
)
♮
g
,
D
2
n
)
;
ℤ
)
≅
H
k
(
Q
0
B
O
(
2
n
+
1
)
〈
n
〉
+
;
ℤ
)
in the case that [math] .
</p>projecteuclid.org/euclid.agt/1535594422_20180829220047Wed, 29 Aug 2018 22:00 EDTFramed cobordism and flow category moveshttps://projecteuclid.org/euclid.agt/1535594423<strong>Andrew Lobb</strong>, <strong>Patrick Orson</strong>, <strong>Dirk Schütz</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 2821--2858.</p><p><strong>Abstract:</strong><br/>
Framed flow categories were introduced by Cohen, Jones and Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the resulting complex, recovering the Floer cohomology as its singular cohomology. Such a framed flow category was produced, for example, by Lipshitz and Sarkar from the input of a knot diagram, resulting in a stable homotopy type generalising Khovanov cohomology.
We give moves that change a framed flow category without changing the associated stable homotopy type. These are inspired by moves that can be performed in the Morse–Smale case without altering the underlying smooth manifold. We posit that if two framed flow categories represent the same stable homotopy type then a finite sequence of these moves is sufficient to connect the two categories. This is directed towards the goal of reducing the study of framed flow categories to a combinatorial calculus.
We provide examples of calculations performed with these moves (related to the Khovanov framed flow category), and prove some general results about the simplification of framed flow categories via these moves.
</p>projecteuclid.org/euclid.agt/1535594423_20180829220047Wed, 29 Aug 2018 22:00 EDTThe space of short ropes and the classifying space of the space of long knotshttps://projecteuclid.org/euclid.agt/1535594424<strong>Syunji Moriya</strong>, <strong>Keiichi Sakai</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 2859--2873.</p><p><strong>Abstract:</strong><br/>
We prove affirmatively the conjecture raised by J Mostovoy (Topology 41 (2002) 435–450); the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in [math] . We make use of techniques developed by S Galatius and O Randal-Williams (Geom. Topol. 14 (2010) 1243–1302) to construct a manifold space model of the classifying space of the space of long knots, and we give an explicit map from the space of short ropes to the model in a geometric way.
</p>projecteuclid.org/euclid.agt/1535594424_20180829220047Wed, 29 Aug 2018 22:00 EDTThe action of matrix groups on aspherical manifoldshttps://projecteuclid.org/euclid.agt/1535594425<strong>Shengkui Ye</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 2875--2895.</p><p><strong>Abstract:</strong><br/>
Let [math] for [math] be the special linear group and [math] be a closed aspherical manifold. It is proved that when [math] , a group action of [math] on [math] by homeomorphisms is trivial if and only if the induced group homomorphism [math] is trivial. For (almost) flat manifolds, we prove a similar result in terms of holonomy groups. In particular, when [math] is nilpotent, the group [math] cannot act nontrivially on [math] when [math] . This confirms a conjecture related to Zimmer’s program for these manifolds.
</p>projecteuclid.org/euclid.agt/1535594425_20180829220047Wed, 29 Aug 2018 22:00 EDTKakimizu complexes of Seifert fibered spaceshttps://projecteuclid.org/euclid.agt/1535594426<strong>Jennifer Schultens</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 2897--2918.</p><p><strong>Abstract:</strong><br/>
Kakimizu complexes of Seifert fibered spaces can be described as either horizontal or vertical, depending on what type of surfaces represent their vertices. Horizontal Kakimizu complexes are shown to be trivial. Each vertical Kakimizu complex is shown to be isomorphic to a Kakimizu complex of the base orbifold minus its singular points.
</p>projecteuclid.org/euclid.agt/1535594426_20180829220047Wed, 29 Aug 2018 22:00 EDTEncoding equivariant commutativity via operadshttps://projecteuclid.org/euclid.agt/1535594427<strong>Javier J Gutiérrez</strong>, <strong>David White</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 2919--2962.</p><p><strong>Abstract:</strong><br/>
We prove a conjecture of Blumberg and Hill regarding the existence of [math] –operads associated to given sequences [math] of families of subgroups of [math] . For every such sequence, we construct a model structure on the category of [math] –operads, and we use these model structures to define [math] –operads, generalizing the notion of an [math] –operad, and to prove the Blumberg–Hill conjecture. We then explore questions of admissibility, rectification, and preservation under left Bousfield localization for these [math] –operads, obtaining some new results as well for [math] –operads.
</p>projecteuclid.org/euclid.agt/1535594427_20180829220047Wed, 29 Aug 2018 22:00 EDTOn the commutative algebra of categorieshttps://projecteuclid.org/euclid.agt/1535594428<strong>John D Berman</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 2963--3012.</p><p><strong>Abstract:</strong><br/>
We discuss what it means for a symmetric monoidal category to be a module over a commutative semiring category. Each of the categories of (1) cartesian monoidal categories, (2) semiadditive categories, and (3) connective spectra can be recovered in this way as categories of modules over a commutative semiring category (or [math] –category in the last case). This language provides a simultaneous generalization of the formalism of algebraic theories (operads, PROPs, Lawvere theories) and stable homotopy theory, with essentially a variant of algebraic K–theory bridging between the two.
</p>projecteuclid.org/euclid.agt/1535594428_20180829220047Wed, 29 Aug 2018 22:00 EDTThe profinite completions of knot groups determine the Alexander polynomialshttps://projecteuclid.org/euclid.agt/1535594429<strong>Jun Ueki</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 3013--3030.</p><p><strong>Abstract:</strong><br/>
We study several properties of the completed group ring [math] and the completed Alexander modules of knots. Then we prove that if the profinite completions of the groups of two knots [math] and [math] are isomorphic, then their Alexander polynomials [math] and [math] coincide.
</p>projecteuclid.org/euclid.agt/1535594429_20180829220047Wed, 29 Aug 2018 22:00 EDTThe fundamental group of locally standard $T$–manifoldshttps://projecteuclid.org/euclid.agt/1535594430<strong>Haozhi Zeng</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 3031--3035.</p><p><strong>Abstract:</strong><br/>
We calculate the fundamental group of locally standard [math] –manifolds under the assumption that the principal [math] –bundle obtained from the free [math] –orbits is trivial. This family of manifolds contains nonsingular toric varieties which may be noncompact, quasitoric manifolds and toric origami manifolds with coörientable folding hypersurface. Although the fundamental groups of the above three kinds of manifolds are well-studied, we give a uniform and simple method to generalize the formulas of their fundamental groups.
</p>projecteuclid.org/euclid.agt/1535594430_20180829220047Wed, 29 Aug 2018 22:00 EDTA refinement of Betti numbers and homology in the presence of a continuous function, II: The case of an angle-valued maphttps://projecteuclid.org/euclid.agt/1535594431<strong>Dan Burghelea</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 3037--3087.</p><p><strong>Abstract:</strong><br/>
For [math] a continuous angle-valued map defined on a compact ANR [math] , [math] a field and any integer [math] , one proposes a refinement [math] of the Novikov–Betti numbers of the pair [math] and a refinement [math] of the Novikov homology of [math] , where [math] denotes the integral degree one cohomology class represented by [math] . The refinement [math] is a configuration of points, with multiplicity located in [math] identified to [math] , whose total cardinality is the [math] Novikov–Betti number of the pair. The refinement [math] is a configuration of submodules of the [math] Novikov homology whose direct sum is isomorphic to the Novikov homology and with the same support as of [math] . When [math] , the configuration [math] is convertible into a configuration of mutually orthogonal closed Hilbert submodules of the [math] –homology of the infinite cyclic cover of [math] defined by [math] , which is an [math] –Hilbert module. One discusses the properties of these configurations, namely robustness with respect to continuous perturbation of the angle-values map and the Poincaré duality and one derives some computational applications in topology. The main results parallel the results for the case of real-valued map but with Novikov homology and Novikov–Betti numbers replacing standard homology and standard Betti numbers.
</p>projecteuclid.org/euclid.agt/1535594431_20180829220047Wed, 29 Aug 2018 22:00 EDTEnds of Schreier graphs of hyperbolic groupshttps://projecteuclid.org/euclid.agt/1535594432<strong>Audrey Vonseel</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 3089--3118.</p><p><strong>Abstract:</strong><br/>
We study the number of ends of a Schreier graph of a hyperbolic group. Let [math] be a hyperbolic group and let [math] be a subgroup of [math] . In general, there is no algorithm to compute the number of ends of a Schreier graph of the pair [math] . However, assuming that [math] is a quasiconvex subgroup of [math] , we construct an algorithm.
</p>projecteuclid.org/euclid.agt/1535594432_20180829220047Wed, 29 Aug 2018 22:00 EDTA note on knot concordancehttps://projecteuclid.org/euclid.agt/1535594433<strong>Eylem Zeliha Yildiz</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 3119--3128.</p><p><strong>Abstract:</strong><br/>
We use classical techniques to answer some questions raised by Daniele Celoria about almost-concordance of knots in arbitrary closed [math] –manifolds. We first prove that, given [math] , for any nontrivial element [math] there are infinitely many distinct smooth almost-concordance classes in the free homotopy class of the unknot. In particular we consider these distinct smooth almost-concordance classes on the boundary of a Mazur manifold and we show none of these distinct classes bounds a PL–disk in the Mazur manifold, but all the representatives we construct are topologically slice. We also prove that all knots in the free homotopy class of [math] in [math] are smoothly concordant.
</p>projecteuclid.org/euclid.agt/1535594433_20180829220047Wed, 29 Aug 2018 22:00 EDTCorrection to the article Gorenstein duality for real spectrahttps://projecteuclid.org/euclid.agt/1535594434<strong>John P C Greenlees</strong>, <strong>Lennart Meier</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 5, 3129--3131.</p><p><strong>Abstract:</strong><br/>
This paper reports and corrects a mistake in the authors’ paper (Algebr. Geom. Topol. 17 (2017) 3547–3619).
</p>projecteuclid.org/euclid.agt/1535594434_20180829220047Wed, 29 Aug 2018 22:00 EDT