Algebraic & Geometric Topology Articles (Project Euclid)
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The latest articles from Algebraic & Geometric Topology on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 19 Oct 2017 12:44 EDTThu, 19 Oct 2017 12:44 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Indecomposable nonorientable $\mathrm{PD}_3$–complexes
https://projecteuclid.org/euclid.agt/1508431438
<strong>Jonathan Hillman</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 17, Number 2, 645--656.</p><p><strong>Abstract:</strong><br/>
We show that the orientable double covering space of an indecomposable, nonorientable [math] –complex has torsion-free fundamental group.
</p>projecteuclid.org/euclid.agt/1508431438_20171019124413Thu, 19 Oct 2017 12:44 EDTThe motivic Mahowald invarianthttps://projecteuclid.org/euclid.agt/1572055261<strong>J D Quigley</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 5, 2485--2534.</p><p><strong>Abstract:</strong><br/>
The classical Mahowald invariant is a method for producing nonzero classes in the stable homotopy groups of spheres from classes in lower stems. We study the Mahowald invariant in the setting of motivic stable homotopy theory over [math] . We compute a motivic version of the [math] –Tate construction for various motivic spectra, and show that this construction produces “blueshift” in these cases. We use these computations to show that for [math] , the Mahowald invariant of [math] is the first element in Adams filtration [math] of the [math] –periodic families constructed by Andrews (2018). This provides an exotic periodic analog of the computation of Mahowald and Ravenel (1993) that for [math] , the classical Mahowald invariant of [math] , is the first element in Adams filtration [math] of the [math] –periodic families constructed by Adams (1966).
</p>projecteuclid.org/euclid.agt/1572055261_20191025220123Fri, 25 Oct 2019 22:01 EDTConnected Heegaard Floer homology of sums of Seifert fibrationshttps://projecteuclid.org/euclid.agt/1572055267<strong>Irving Dai</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 5, 2535--2574.</p><p><strong>Abstract:</strong><br/>
We compute the connected Heegaard Floer homology (defined by Hendricks, Hom, and Lidman) for a large class of [math] –manifolds, including all linear combinations of Seifert fibered homology spheres. We show that for such manifolds, the connected Floer homology completely determines the local equivalence class of the associated [math] –complex. Some identities relating the rank of the connected Floer homology to the Rokhlin invariant and the Neumann–Siebenmann invariant are also derived. Our computations are based on combinatorial models inspired by the work of Némethi on lattice homology.
</p>projecteuclid.org/euclid.agt/1572055267_20191025220123Fri, 25 Oct 2019 22:01 EDTGeneralized Kuperberg invariants of $3$–manifoldshttps://projecteuclid.org/euclid.agt/1572055268<strong>Rinat Kashaev</strong>, <strong>Alexis Virelizier</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 5, 2575--2624.</p><p><strong>Abstract:</strong><br/>
In the 1990s, based on presentations of [math] –manifolds by Heegaard diagrams, Kuperberg associated a scalar invariant of [math] –manifolds to each finite-dimensional involutory Hopf algebra over a field. We generalize this construction to the case of involutory Hopf algebras in arbitrary symmetric monoidal categories admitting certain pairs of morphisms called good pairs . We construct examples of such good pairs for involutory Hopf algebras whose distinguished grouplike elements are central. The generalized construction is illustrated by an example of an involutory super-Hopf algebra.
</p>projecteuclid.org/euclid.agt/1572055268_20191025220123Fri, 25 Oct 2019 22:01 EDTTreewidth, crushing and hyperbolic volumehttps://projecteuclid.org/euclid.agt/1572055269<strong>Clément Maria</strong>, <strong>Jessica S Purcell</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 5, 2625--2652.</p><p><strong>Abstract:</strong><br/>
The treewidth of a [math] –manifold triangulation plays an important role in algorithmic [math] –manifold theory, and so it is useful to find bounds on the treewidth in terms of other properties of the manifold. We prove that there exists a universal constant [math] such that any closed hyperbolic [math] –manifold admits a triangulation of treewidth at most the product of [math] and the volume. The converse is not true: we show there exists a sequence of hyperbolic [math] –manifolds of bounded treewidth but volume approaching infinity. Along the way, we prove that crushing a normal surface in a triangulation does not increase the carving-width, and hence crushing any number of normal surfaces in a triangulation affects treewidth by at most a constant multiple.
</p>projecteuclid.org/euclid.agt/1572055269_20191025220123Fri, 25 Oct 2019 22:01 EDTNew hyperbolic $4$–manifolds of low volumehttps://projecteuclid.org/euclid.agt/1572055270<strong>Stefano Riolo</strong>, <strong>Leone Slavich</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 5, 2653--2676.</p><p><strong>Abstract:</strong><br/>
We prove that there are at least two commensurability classes of (cusped, arithmetic) minimal-volume hyperbolic [math] –manifolds. Moreover, by applying a well-known technique due to Gromov and Piatetski-Shapiro, we build the smallest known nonarithmetic hyperbolic [math] –manifold.
</p>projecteuclid.org/euclid.agt/1572055270_20191025220123Fri, 25 Oct 2019 22:01 EDTA combinatorial model for the known Bousfield classeshttps://projecteuclid.org/euclid.agt/1572314538<strong>Neil Patrick Strickland</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 6, 2677--2713.</p><p><strong>Abstract:</strong><br/>
We give a combinatorial construction of an ordered semiring [math] , and show that it can be identified with a certain subquotient of the semiring of [math] –local Bousfield classes, containing almost all of the classes that have previously been named and studied. This is a convenient way to encapsulate most of the known results about Bousfield classes.
</p>projecteuclid.org/euclid.agt/1572314538_20191028220235Mon, 28 Oct 2019 22:02 EDTOn Ruan's cohomological crepant resolution conjecture for the complexified Bianchi orbifoldshttps://projecteuclid.org/euclid.agt/1572314539<strong>Fabio Perroni</strong>, <strong>Alexander D Rahm</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 6, 2715--2762.</p><p><strong>Abstract:</strong><br/>
We give formulae for the Chen–Ruan orbifold cohomology for the orbifolds given by a Bianchi group acting on complex hyperbolic [math] –space.
The Bianchi groups are the arithmetic groups [math] , where [math] is the ring of integers in an imaginary quadratic number field. The underlying real orbifolds which help us in our study, given by the action of a Bianchi group on real hyperbolic [math] –space (which is a model for its classifying space for proper actions), have applications in physics.
We then prove that, for any such orbifold, its Chen–Ruan orbifold cohomology ring is isomorphic to the usual cohomology ring of any crepant resolution of its coarse moduli space. By vanishing of the quantum corrections, we show that this result fits in with Ruan’s cohomological crepant resolution conjecture.
</p>projecteuclid.org/euclid.agt/1572314539_20191028220235Mon, 28 Oct 2019 22:02 EDT$C^{1,0}$ foliation theoryhttps://projecteuclid.org/euclid.agt/1572314540<strong>William H Kazez</strong>, <strong>Rachel Roberts</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 6, 2763--2794.</p><p><strong>Abstract:</strong><br/>
Transverse [math] –dimensional foliations play an important role in the study of codimension-one foliations. In Geom. Topol. Monogr. 19 (2015) 21–72, the authors introduced the notion of flow box decomposition of a [math] –manifold [math] . This is a combinatorial decomposition of [math] that reflects both the structure of a given codimension-one foliation and that of a given transverse dimension-one foliation, and that is amenable to inductive strategies.
In this paper, flow box decompositions are used to extend some classical foliation results to foliations that are not [math] . Enhancements of well-known results of Calegari on smoothing leaves, Dippolito on Denjoy blowup of leaves, and Tischler on approximations by fibrations are obtained. The methods developed are not intrinsically [math] –dimensional techniques, and should generalize to prove corresponding results for codimension-one foliations in [math] –dimensional manifolds.
</p>projecteuclid.org/euclid.agt/1572314540_20191028220235Mon, 28 Oct 2019 22:02 EDTLocal cut points and splittings of relatively hyperbolic groupshttps://projecteuclid.org/euclid.agt/1572314542<strong>Matthew Haulmark</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 6, 2795--2836.</p><p><strong>Abstract:</strong><br/>
We show that the existence of a nonparabolic local cut point in the Bowditch boundary [math] of a relatively hyperbolic group [math] implies that [math] splits over a [math] –ended subgroup. This theorem generalizes a theorem of Bowditch from the setting of hyperbolic groups to relatively hyperbolic groups. As a consequence we are able to generalize a theorem of Kapovich and Kleiner by classifying the homeomorphism type of [math] –dimensional Bowditch boundaries of relatively hyperbolic groups which satisfy certain properties, such as no splittings over [math] –ended subgroups and no peripheral splittings.
In order to prove the boundary classification result we require a notion of ends of a group which is more general than the standard notion. We show that if a finitely generated discrete group acts properly and cocompactly on two generalized Peano continua [math] and [math] , then [math] is homeomorphic to [math] . Thus we propose an alternative definition of [math] which increases the class of spaces on which [math] can act.
</p>projecteuclid.org/euclid.agt/1572314542_20191028220235Mon, 28 Oct 2019 22:02 EDTNonorientable Lagrangian surfaces in rational $4$–manifoldshttps://projecteuclid.org/euclid.agt/1572314543<strong>Bo Dai</strong>, <strong>Chung-I Ho</strong>, <strong>Tian-Jun Li</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 6, 2837--2854.</p><p><strong>Abstract:</strong><br/>
We show that for any nonzero class [math] in [math] in a rational [math] manifold [math] , [math] is represented by a nonorientable embedded Lagrangian surface [math] (for some symplectic structure) if and only if [math] , where [math] denotes the mod [math] valued Pontryagin square of [math] .
</p>projecteuclid.org/euclid.agt/1572314543_20191028220235Mon, 28 Oct 2019 22:02 EDTAlgebraic filling inequalities and cohomological widthhttps://projecteuclid.org/euclid.agt/1572314544<strong>Meru Alagalingam</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 6, 2855--2898.</p><p><strong>Abstract:</strong><br/>
In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real-valued map on the [math] –torus admits a fibre whose homological size is bounded below by some universal constant depending on [math] . He obtained similar estimates for maps with values in finite-dimensional complexes, by a Lusternik–Schnirelmann-type argument.
We describe a new homological filling technique which enables us to derive sharp lower bounds in these theorems in certain situations. This partly realises a programme envisaged by Gromov.
In contrast to previous approaches, our methods imply similar lower bounds for maps defined on products of higher-dimensional spheres.
</p>projecteuclid.org/euclid.agt/1572314544_20191028220235Mon, 28 Oct 2019 22:02 EDTTwisted differential generalized cohomology theories and their Atiyah–Hirzebruch spectral sequencehttps://projecteuclid.org/euclid.agt/1572314545<strong>Daniel Grady</strong>, <strong>Hisham Sati</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 6, 2899--2960.</p><p><strong>Abstract:</strong><br/>
We construct the Atiyah–Hirzebruch spectral sequence (AHSS) for twisted differential generalized cohomology theories. This generalizes to the twisted setting the authors’ corresponding earlier construction for differential cohomology theories, as well as to the differential setting the AHSS for twisted generalized cohomology theories, including that of twisted [math] –theory by Rosenberg and by Atiyah and Segal. In describing twisted differential spectra we build on the work of Bunke and Nikolaus, but we find it useful for our purposes to take an approach that highlights direct analogies with classical bundles and that is at the same time amenable for calculations. We will, in particular, establish that twisted differential spectra are bundles of spectra equipped with a flat connection. Our prominent case will be twisted differential [math] –theory, for which we work out the differentials in detail. This involves differential refinements of primary and secondary cohomology operations the authors developed in earlier papers. We illustrate our constructions and computational tools with examples.
</p>projecteuclid.org/euclid.agt/1572314545_20191028220235Mon, 28 Oct 2019 22:02 EDTCoproducts in brane topologyhttps://projecteuclid.org/euclid.agt/1572314546<strong>Shun Wakatsuki</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 6, 2961--2988.</p><p><strong>Abstract:</strong><br/>
We extend the loop product and the loop coproduct to the mapping space from the [math] –dimensional sphere, or more generally from any [math] –manifold, to a [math] –connected space with finite-dimensional rational homotopy group for [math] . The key to extending the loop coproduct is the fact that the embedding [math] is of “finite codimension” in the sense of Gorenstein spaces. Moreover, we prove the associativity, commutativity and Frobenius compatibility of them.
</p>projecteuclid.org/euclid.agt/1572314546_20191028220235Mon, 28 Oct 2019 22:02 EDTQuasi-right-veering braids and nonloose linkshttps://projecteuclid.org/euclid.agt/1572314548<strong>Tetsuya Ito</strong>, <strong>Keiko Kawamuro</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 6, 2989--3032.</p><p><strong>Abstract:</strong><br/>
We introduce a notion of quasi-right-veering for closed braids, which plays an analogous role to right-veering for open books. We show that a transverse link [math] in a contact [math] –manifold [math] is nonloose if and only if every braid representative of [math] with respect to every open book decomposition that supports [math] is quasi-right-veering. We also show that several definitions of right-veering closed braids are equivalent.
</p>projecteuclid.org/euclid.agt/1572314548_20191028220235Mon, 28 Oct 2019 22:02 EDTCoarse homology theories and finite decomposition complexityhttps://projecteuclid.org/euclid.agt/1572314549<strong>Ulrich Bunke</strong>, <strong>Alexander Engel</strong>, <strong>Daniel Kasprowski</strong>, <strong>Christoph Winges</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 6, 3033--3074.</p><p><strong>Abstract:</strong><br/>
Using the language of coarse homology theories, we provide an axiomatic account of vanishing results for the fibres of forget-control maps associated to spaces with equivariant finite decomposition complexity.
</p>projecteuclid.org/euclid.agt/1572314549_20191028220235Mon, 28 Oct 2019 22:02 EDTOn the coarse geometry of certain right-angled Coxeter groupshttps://projecteuclid.org/euclid.agt/1572314550<strong>Hoang Thanh Nguyen</strong>, <strong>Hung Cong Tran</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 6, 3075--3118.</p><p><strong>Abstract:</strong><br/>
Let [math] be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph [math] is [math] , we prove that the right-angled Coxeter group [math] is virtually a Seifert manifold group or virtually a graph manifold group and we give a complete quasi-isometry classification of these groups. Furthermore, we prove that [math] is hyperbolic relative to a collection of [math] right-angled Coxeter subgroups of [math] . Consequently, the divergence of [math] is linear, quadratic or exponential. We also generalize right-angled Coxeter groups which are virtually graph manifold groups to certain high-dimensional right-angled Coxeter groups (our families exist in every dimension) and study the coarse geometry of this collection. We prove that strongly quasiconvex, torsion-free, infinite-index subgroups in certain graph of groups are free and we apply this result to our right-angled Coxeter groups.
</p>projecteuclid.org/euclid.agt/1572314550_20191028220235Mon, 28 Oct 2019 22:02 EDTThe $\infty$–categorical Eckmann–Hilton argumenthttps://projecteuclid.org/euclid.agt/1572314551<strong>Tomer M Schlank</strong>, <strong>Lior Yanovski</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 6, 3119--3170.</p><p><strong>Abstract:</strong><br/>
We define a reduced [math] –operad [math] to be [math] –connected if the spaces [math] of [math] –ary operations are [math] –connected for all [math] . Let [math] and [math] be two reduced [math] –operads. We prove that if [math] is [math] –connected and [math] is [math] –connected, then their Boardman–Vogt tensor product [math] is [math] –connected. We consider this to be a natural [math] –categorical generalization of the classical Eckmann–Hilton argument.
</p>projecteuclid.org/euclid.agt/1572314551_20191028220235Mon, 28 Oct 2019 22:02 EDTCategories and orbispaceshttps://projecteuclid.org/euclid.agt/1572314552<strong>Stefan Schwede</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 6, 3171--3215.</p><p><strong>Abstract:</strong><br/>
Constructing and manipulating homotopy types from categorical input data has been an important theme in algebraic topology for decades. Every category gives rise to a “classifying space”, the geometric realization of the nerve. Up to weak homotopy equivalence, every space is the classifying space of a small category. More is true: the entire homotopy theory of topological spaces and continuous maps can be modeled by categories and functors. We establish a vast generalization of the equivalence of the homotopy theories of categories and spaces: small categories represent refined homotopy types of orbispaces whose underlying coarse moduli space is the traditional homotopy type hitherto considered.
A global equivalence is a functor [math] between small categories with the following property: for every finite group [math] , the functor [math] induced on categories of [math] –objects is a weak equivalence. We show that the global equivalences are part of a model structure on the category of small categories, which is moreover Quillen equivalent to the homotopy theory of orbispaces in the sense of Gepner and Henriques. Every cofibrant category in this global model structure is opposite to a complex of groups in the sense of Haefliger.
</p>projecteuclid.org/euclid.agt/1572314552_20191028220235Mon, 28 Oct 2019 22:02 EDTThe universality of the Rezk nervehttps://projecteuclid.org/euclid.agt/1578020518<strong>Aaron Mazel-Gee</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3217--3260.</p><p><strong>Abstract:</strong><br/>
We functorially associate to each relative [math] –category [math] a simplicial space [math] , called its Rezk nerve (a straightforward generalization of Rezk’s “classification diagram” construction for relative categories). We prove the following local and global universal properties of this construction: (i) that the complete Segal space generated by the Rezk nerve [math] is precisely the one corresponding to the localization [math] ; and (ii) that the Rezk nerve functor defines an equivalence [math] from a localization of the [math] –category of relative [math] –categories to the [math] –category of [math] –categories.
</p>projecteuclid.org/euclid.agt/1578020518_20200102220213Thu, 02 Jan 2020 22:02 ESTSymplectic structure perturbations and continuity of symplectic invariantshttps://projecteuclid.org/euclid.agt/1578020519<strong>Jun Zhang</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3261--3314.</p><p><strong>Abstract:</strong><br/>
This paper studies how symplectic invariants created from Hamiltonian Floer theory change under the perturbations of symplectic structures, not necessarily in the same cohomology class . These symplectic invariants include spectral invariants, boundary depth, and (partial) symplectic quasistates. This paper can split into two parts. In the first part, we prove some energy estimations which control the shifts of symplectic action functionals. These directly imply positive conclusions on the continuity of spectral invariants and boundary depth in some important cases, including any symplectic surface [math] and any closed symplectic manifold [math] with [math] . This follows by applications on some rigidity of the subsets of a symplectic manifold in terms of heaviness and superheaviness, as well as on the continuity property of some symplectic capacities. In the second part, we generalize the construction in the first part to any closed symplectic manifold. In particular, to deal with the change of Novikov rings from symplectic structure perturbations, we construct a family of variant Floer chain complexes over a common Novikov-type ring. In this setup, we define a new family of spectral invariants called [math] –spectral invariants, and prove that they are upper semicontinuous under the symplectic structure perturbations. This implies a quasi-isometric embedding from [math] to [math] under some dynamical assumption, imitating the main result of Usher (Ann. Sci. Éc. Norm. Supér. 46 (2013) 57–128).
</p>projecteuclid.org/euclid.agt/1578020519_20200102220213Thu, 02 Jan 2020 22:02 ESTUpsilon-type concordance invariantshttps://projecteuclid.org/euclid.agt/1578020520<strong>Antonio Alfieri</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3315--3334.</p><p><strong>Abstract:</strong><br/>
To a region [math] of the plane satisfying a suitable convexity condition we associate a knot concordance invariant [math] . For appropriate choices of the domain this construction gives back some known knot Floer concordance invariants like Rasmussen’s [math] invariants, and the Ozsváth–Stipsicz–Szabó upsilon invariant. Furthermore, to three such regions [math] , [math] and [math] we associate invariants [math] generalizing the Kim–Livingston secondary invariant. We show how to compute these invariants for some interesting classes of knots (including alternating and torus knots), and we use them to obstruct concordances to Floer thin knots and algebraic knots.
</p>projecteuclid.org/euclid.agt/1578020520_20200102220213Thu, 02 Jan 2020 22:02 ESTRopelength, crossing number and finite-type invariants of linkshttps://projecteuclid.org/euclid.agt/1578020521<strong>Rafal Komendarczyk</strong>, <strong>Andreas Michaelides</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3335--3357.</p><p><strong>Abstract:</strong><br/>
Ropelength and embedding thickness are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of [math] –component links in terms of the Milnor linking numbers. The main goal of the current paper is to provide such estimates, and thus generalize the known linking number bound. In the process, we collect several facts about finite-type invariants and ropelength/crossing number of knots. We give examples of families of knots where such estimates behave better than the well-known knot–genus estimate.
</p>projecteuclid.org/euclid.agt/1578020521_20200102220213Thu, 02 Jan 2020 22:02 ESTOn rational homological stability for block automorphisms of connected sums of products of sphereshttps://projecteuclid.org/euclid.agt/1578020522<strong>Matthias Grey</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3359--3407.</p><p><strong>Abstract:</strong><br/>
We show rational homological stability for the classifying spaces of the monoid of homotopy self-equivalences and the block diffeomorphism group of iterated connected sums of products of spheres. The spheres can have different dimensions, but need to satisfy a certain connectivity assumption. The main theorems of this paper extend homological stability results for automorphism spaces of connected sums of products of spheres of the same dimension by Berglund and Madsen.
</p>projecteuclid.org/euclid.agt/1578020522_20200102220213Thu, 02 Jan 2020 22:02 ESTOn properties of Bourgeois contact structureshttps://projecteuclid.org/euclid.agt/1578020523<strong>Samuel Lisi</strong>, <strong>Aleksandra Marinković</strong>, <strong>Klaus Niederkrüger</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3409--3451.</p><p><strong>Abstract:</strong><br/>
The Bourgeois construction associates to every contact open book on a manifold [math] a contact structure on [math] . We study some of the properties of [math] that are inherited by [math] and some that are not.
Giroux has provided recently a better framework to work with contact open books. In the appendix, we quickly review this formalism, and we work out a few classical examples of contact open books to illustrate how to use this new language.
</p>projecteuclid.org/euclid.agt/1578020523_20200102220213Thu, 02 Jan 2020 22:02 ESTOn Kauffman bracket skein modules of marked $3$–manifolds and the Chebyshev–Frobenius homomorphismhttps://projecteuclid.org/euclid.agt/1578020524<strong>Thang T Q Lê</strong>, <strong>Jonathan Paprocki</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3453--3509.</p><p><strong>Abstract:</strong><br/>
We study the skein algebras of marked surfaces and the skein modules of marked [math] –manifolds. Muller showed that skein algebras of totally marked surfaces may be embedded in easy-to-study algebras known as quantum tori. We first extend Muller’s result to permit marked surfaces with unmarked boundary components. The addition of unmarked components allows us to develop a surgery theory which enables us to extend the Chebyshev homomorphism of Bonahon and Wong between skein algebras of unmarked surfaces to a “Chebyshev–Frobenius homomorphism” between skein modules of marked [math] –manifolds. We show that the image of the Chebyshev–Frobenius homomorphism is either transparent or skew-transparent. In addition, we make use of the Muller algebra method to calculate the center of the skein algebra of a marked surface when the quantum parameter is not a root of unity.
</p>projecteuclid.org/euclid.agt/1578020524_20200102220213Thu, 02 Jan 2020 22:02 ESTFour-genera of links and Heegaard Floer homologyhttps://projecteuclid.org/euclid.agt/1578020525<strong>Beibei Liu</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3511--3540.</p><p><strong>Abstract:</strong><br/>
For links with vanishing pairwise linking numbers, the link components bound pairwise disjoint oriented surfaces in [math] . We use the [math] –function which is a link invariant from the Heegaard Floer homology to give lower bounds for the [math] –genus of the link. For [math] –space links, the [math] –function is explicitly determined by the Alexander polynomials of the link and its sublinks. We show some [math] –space links where the lower bounds are sharp, and also describe all possible genera of disjoint oriented surfaces bounded by such links.
</p>projecteuclid.org/euclid.agt/1578020525_20200102220213Thu, 02 Jan 2020 22:02 ESTAn algebraic model for rational toral $G$–spectrahttps://projecteuclid.org/euclid.agt/1578020526<strong>David Barnes</strong>, <strong>John Greenlees</strong>, <strong>Magdalena Kędziorek</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3541--3599.</p><p><strong>Abstract:</strong><br/>
For [math] a compact Lie group, toral [math] –spectra are those rational [math] –spectra whose geometric isotropy consists of subgroups of a maximal torus of [math] . The homotopy category of rational toral [math] –spectra is a retract of the category of all rational [math] –spectra.
We show that the abelian category of Greenlees (Algebr. Geom. Topol. 16 (2016) 1953–2019) gives an algebraic model for the toral part of rational [math] –spectra. This is a major step in establishing an algebraic model for all rational [math] –spectra for any compact Lie group [math] .
</p>projecteuclid.org/euclid.agt/1578020526_20200102220213Thu, 02 Jan 2020 22:02 ESTAn upper bound on the LS category in presence of the fundamental grouphttps://projecteuclid.org/euclid.agt/1578020527<strong>Alexander Dranishnikov</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3601--3614.</p><p><strong>Abstract:</strong><br/>
We prove that
cat
LS
X
≤
1
2
(
cd
(
π
1
(
X
)
)
+
dim
X
)
for every CW complex [math] , where [math] denotes the cohomological dimension of the fundamental group of [math] . We obtain this as a corollary of the inequality
cat
LS
X
≤
1
2
(
cat
LS
(
u
X
)
+
dim
X
)
,
where [math] is a classifying map for the universal covering of [math] .
</p>projecteuclid.org/euclid.agt/1578020527_20200102220213Thu, 02 Jan 2020 22:02 ESTOn the local homology of Artin groups of finite and affine typehttps://projecteuclid.org/euclid.agt/1578020528<strong>Giovanni Paolini</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3615--3639.</p><p><strong>Abstract:</strong><br/>
We study the local homology of Artin groups using weighted discrete Morse theory. In all finite and affine cases, we are able to construct Morse matchings of a special type (we call them “precise matchings”). The existence of precise matchings implies that the homology has a squarefree torsion. This property was known for Artin groups of finite type, but not in general for Artin groups of affine type. We also use the constructed matchings to compute the local homology in all exceptional cases, correcting some results in the literature.
</p>projecteuclid.org/euclid.agt/1578020528_20200102220213Thu, 02 Jan 2020 22:02 ESTOn equivariant and motivic sliceshttps://projecteuclid.org/euclid.agt/1578020529<strong>Drew Heard</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3641--3681.</p><p><strong>Abstract:</strong><br/>
Let [math] be a field with a real embedding. We compare the motivic slice filtration of a motivic spectrum over [math] with the [math] –equivariant slice filtration of its equivariant Betti realization, giving conditions under which realization induces an equivalence between the associated slice towers. In particular, we show that, up to reindexing, the towers agree for all spectra obtained from localized quotients of [math] and [math] , and for motivic Landweber exact spectra and their realizations. As a consequence, we deduce that equivariant spectra obtained from localized quotients of [math] are even in the sense of Hill and Meier, and give a computation of the slice spectral sequence converging to [math] for [math] .
</p>projecteuclid.org/euclid.agt/1578020529_20200102220213Thu, 02 Jan 2020 22:02 ESTFunctoriality of the $\mathrm{EH}$ class and the LOSS invariant under Lagrangian concordanceshttps://projecteuclid.org/euclid.agt/1578020530<strong>Marco Golla</strong>, <strong>András Juhász</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3683--3699.</p><p><strong>Abstract:</strong><br/>
We show that the [math] class and the LOSS invariant of Legendrian knots in contact [math] –manifolds are functorial under regular Lagrangian concordances in Weinstein cobordisms. This gives computable obstructions to the existence of regular Lagrangian concordances.
</p>projecteuclid.org/euclid.agt/1578020530_20200102220213Thu, 02 Jan 2020 22:02 ESTOn local tameness of certain graphs of groupshttps://projecteuclid.org/euclid.agt/1578020531<strong>Rita Gitik</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3701--3710.</p><p><strong>Abstract:</strong><br/>
Let [math] be the fundamental group of a finite graph of groups with Noetherian edge groups and locally tame vertex groups. We prove that [math] is locally tame. It follows that if a finitely presented group [math] has a nontrivial [math] –decomposition over the class of its [math] subgroups for [math] or [math] , and all the vertex groups in the decomposition are flexible, then [math] is locally tame.
</p>projecteuclid.org/euclid.agt/1578020531_20200102220213Thu, 02 Jan 2020 22:02 ESTSplittings and calculational techniques for higher $\mathsf{THH}$https://projecteuclid.org/euclid.agt/1578020532<strong>Irina Bobkova</strong>, <strong>Eva Höning</strong>, <strong>Ayelet Lindenstrauss</strong>, <strong>Kate Poirier</strong>, <strong>Birgit Richter</strong>, <strong>Inna Zakharevich</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 7, 3711--3753.</p><p><strong>Abstract:</strong><br/>
Tensoring finite pointed simplicial sets [math] with commutative ring spectra [math] yields important homology theories such as (higher) topological Hochschild homology and torus homology. We prove several structural properties of these constructions relating [math] to [math] and we establish splitting results. This allows us, among other important examples, to determine [math] for all [math] and for all [math] .
</p>projecteuclid.org/euclid.agt/1578020532_20200102220213Thu, 02 Jan 2020 22:02 ESTOn spectral sequences from Khovanov homologyhttps://projecteuclid.org/euclid.agt/1588212069<strong>Andrew Lobb</strong>, <strong>Raphael Zentner</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 531--564.</p><p><strong>Abstract:</strong><br/>
There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realised as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary [math] –handle movie moves induce a morphism of spectral sequences. These morphisms remain unexploited in the literature, perhaps because there is still an open question concerning the naturality of maps induced by general movies.
Here we focus on the spectral sequence due to Kronheimer and Mrowka from Khovanov homology to instanton knot Floer homology, and on that due to Ozsváth and Szabó to the Heegaard Floer homology of the branched double cover. For example, we use the [math] –handle morphisms to give new information about the filtrations on the instanton knot Floer homology of the [math] –torus knot, determining these up to an ambiguity in a pair of degrees; to determine the Ozsváth–Szabó spectral sequence for an infinite class of prime knots; and to show that higher differentials of both the Kronheimer–Mrowka and the Ozsváth–Szabó spectral sequences necessarily lower the delta grading for all pretzel knots.
</p>projecteuclid.org/euclid.agt/1588212069_20200429220122Wed, 29 Apr 2020 22:01 EDTHofer–Zehnder capacity and Bruhat graphhttps://projecteuclid.org/euclid.agt/1588212070<strong>Alexander Caviedes Castro</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 565--600.</p><p><strong>Abstract:</strong><br/>
We find bounds for the Hofer–Zehnder capacity of spherically monotone coadjoint orbits of compact Lie groups with respect to the Kostant–Kirillov–Souriau symplectic form in terms of the combinatorics of their Bruhat graphs. We show that our bounds are sharp for coadjoint orbits of the unitary group and equal in that case to the diameter of a weighted Cayley graph.
</p>projecteuclid.org/euclid.agt/1588212070_20200429220122Wed, 29 Apr 2020 22:01 EDTTopological properties of spaces admitting a coaxial homeomorphismhttps://projecteuclid.org/euclid.agt/1588212071<strong>Ross Geoghegan</strong>, <strong>Craig Guilbault</strong>, <strong>Michael Mihalik</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 601--642.</p><p><strong>Abstract:</strong><br/>
Wright (1992) showed that, if a [math] –ended, simply connected, locally compact ANR [math] with pro-monomorphic fundamental group at infinity (ie representable by an inverse sequence of monomorphisms) admits a [math] –action by covering transformations, then that fundamental group at infinity can be represented by an inverse sequence of finitely generated free groups. Geoghegan and Guilbault (2012) strengthened that result, proving that [math] also satisfies the crucial semistability condition (ie representable by an inverse sequence of epimorphisms).
Here we get a stronger theorem with weaker hypotheses. We drop the “pro-monomorphic hypothesis” and simply assume that the [math] –action is generated by what we call a “coaxial” homeomorphism. In the pro-monomorphic case every [math] –action by covering transformations is generated by a coaxial homeomorphism, but coaxials occur in far greater generality (often embedded in a cocompact action). When the generator is coaxial, we obtain the sharp conclusion: [math] is proper [math] –equivalent to the product of a locally finite tree with [math] . Even in the pro-monomorphic case this is new: it says that, from the viewpoint of the fundamental group at infinity, the “end” of [math] looks like the suspension of a totally disconnected compact set.
</p>projecteuclid.org/euclid.agt/1588212071_20200429220122Wed, 29 Apr 2020 22:01 EDTThe Reidemeister graph is a complete knot invarianthttps://projecteuclid.org/euclid.agt/1588212072<strong>Agnese Barbensi</strong>, <strong>Daniele Celoria</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 643--698.</p><p><strong>Abstract:</strong><br/>
We describe two locally finite graphs naturally associated to each knot type [math] , called Reidemeister graphs . We determine several local and global properties of these graphs and prove that in one case the graph-isomorphism type is a complete knot invariant up to mirroring. Lastly, we introduce another object, relating the Reidemeister and Gordian graphs, and determine some of its properties.
</p>projecteuclid.org/euclid.agt/1588212072_20200429220122Wed, 29 Apr 2020 22:01 EDTNonabelian reciprocity laws and higher Brauer–Manin obstructionshttps://projecteuclid.org/euclid.agt/1588212073<strong>Jonathan P Pridham</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 699--756.</p><p><strong>Abstract:</strong><br/>
We reinterpret Kim’s nonabelian reciprocity maps for algebraic varieties as obstruction towers of mapping spaces of étale homotopy types, removing technical hypotheses such as global basepoints and cohomological constraints. We then extend the theory by considering alternative natural series of extensions, one of which gives an obstruction tower whose first stage is the Brauer–Manin obstruction, allowing us to determine when Kim’s maps recover the Brauer–Manin locus. A tower based on relative completions yields nontrivial reciprocity maps even for Shimura varieties; for the stacky modular curve, these take values in Galois cohomology of modular forms, and give obstructions to an adèlic elliptic curve with global Tate module underlying a global elliptic curve.
</p>projecteuclid.org/euclid.agt/1588212073_20200429220122Wed, 29 Apr 2020 22:01 EDTThe prism manifold realization problemhttps://projecteuclid.org/euclid.agt/1588212074<strong>William Ballinger</strong>, <strong>Chloe Ching-Yun Hsu</strong>, <strong>Wyatt Mackey</strong>, <strong>Yi Ni</strong>, <strong>Tynan Ochse</strong>, <strong>Faramarz Vafaee</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 757--816.</p><p><strong>Abstract:</strong><br/>
The spherical manifold realization problem asks which spherical three-manifolds arise from surgeries on knots in [math] . In recent years, the realization problem for C–, T–, O– and I–type spherical manifolds has been solved, leaving the D–type manifolds (also known as the prism manifolds) as the only remaining case. Every prism manifold can be parametrized as [math] for a pair of relatively prime integers [math] and [math] . We determine a list of prism manifolds [math] that can possibly be realized by positive integral surgeries on knots in [math] when [math] . Based on the forthcoming work of Berge and Kang, we are confident that this list is complete. The methodology undertaken to obtain the classification is similar to that of Greene for lens spaces.
</p>projecteuclid.org/euclid.agt/1588212074_20200429220122Wed, 29 Apr 2020 22:01 EDTOn the Brun spectral sequence for topological Hochschild homologyhttps://projecteuclid.org/euclid.agt/1588212075<strong>Eva Höning</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 817--863.</p><p><strong>Abstract:</strong><br/>
We generalize a spectral sequence of Brun for the computation of topological Hochschild homology. The generalized version computes the [math] –homology of [math] , where [math] is a ring spectrum, [math] is a commutative [math] –algebra and [math] is a connective commutative [math] –algebra. The input of the spectral sequence are the topological Hochschild homology groups of [math] with coefficients in the [math] –homology groups of [math] . The mod [math] and [math] topological Hochschild homology of connective complex [math] –theory has been computed by Ausoni and later again by Rognes, Sagave and Schlichtkrull. We present an alternative, short computation using the generalized Brun spectral sequence.
</p>projecteuclid.org/euclid.agt/1588212075_20200429220122Wed, 29 Apr 2020 22:01 EDTRational homology $3$–spheres and simply connected definite boundinghttps://projecteuclid.org/euclid.agt/1588212076<strong>Kouki Sato</strong>, <strong>Masaki Taniguchi</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 865--882.</p><p><strong>Abstract:</strong><br/>
For each rational homology [math] –sphere [math] which bounds simply connected definite [math] –manifolds of both signs, we construct an infinite family of irreducible rational homology [math] –spheres which are homology cobordant to [math] but cannot bound any simply connected definite [math] –manifold. As a corollary, for any coprime integers [math] and [math] , we obtain an infinite family of irreducible rational homology [math] –spheres which are homology cobordant to the lens space [math] but cannot be obtained by a knot surgery.
</p>projecteuclid.org/euclid.agt/1588212076_20200429220122Wed, 29 Apr 2020 22:01 EDTOn the mod-$\ell$ homology of the classifying space for commutativityhttps://projecteuclid.org/euclid.agt/1588212077<strong>Cihan Okay</strong>, <strong>Ben Williams</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 883--923.</p><p><strong>Abstract:</strong><br/>
We study the mod- [math] homotopy type of classifying spaces for commutativity, [math] , at a prime [math] . We show that the mod- [math] homology of [math] depends on the mod- [math] homotopy type of [math] when [math] is a compact connected Lie group, in the sense that a mod- [math] homology isomorphism [math] for such groups induces a mod- [math] homology isomorphism [math] . In order to prove this result, we study a presentation of [math] as a homotopy colimit over a topological poset of closed abelian subgroups, expanding on an idea of Adem and Gómez. We also study the relationship between the mod- [math] type of a Lie group [math] and the locally finite group [math] , where [math] is a Chevalley group. We see that the naïve analogue for [math] of the celebrated Friedlander–Mislin result cannot hold, but we show that it does hold after taking the homotopy quotient of a [math] action on [math] .
</p>projecteuclid.org/euclid.agt/1588212077_20200429220122Wed, 29 Apr 2020 22:01 EDTA new nonarithmetic lattice in ${\rm PU}(3,1)$https://projecteuclid.org/euclid.agt/1588212078<strong>Martin Deraux</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 925--963.</p><p><strong>Abstract:</strong><br/>
We study the arithmeticity of the Couwenberg–Heckman–Looijenga lattices in [math] , and show that they contain a nonarithmetic lattice in [math] which is not commensurable to the nonarithmetic Deligne–Mostow lattice in [math] .
</p>projecteuclid.org/euclid.agt/1588212078_20200429220122Wed, 29 Apr 2020 22:01 EDTThe Segal conjecture for infinite discrete groupshttps://projecteuclid.org/euclid.agt/1588212079<strong>Wolfgang Lück</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 965--986.</p><p><strong>Abstract:</strong><br/>
We formulate and prove a version of the Segal conjecture for infinite groups. For finite groups it reduces to the original version. The condition that [math] is finite is replaced in our setting by the assumption that there exists a finite model for the classifying space [math] for proper actions. This assumption is satisfied for instance for word hyperbolic groups or cocompact discrete subgroups of Lie groups with finitely many path components. As a consequence we get for such groups [math] that the zeroth stable cohomotopy of the classifying space [math] is isomorphic to the [math] –adic completion of the ring given by the zeroth equivariant stable cohomotopy of [math] for [math] the augmentation ideal.
</p>projecteuclid.org/euclid.agt/1588212079_20200429220122Wed, 29 Apr 2020 22:01 EDTState graphs and fibered state surfaceshttps://projecteuclid.org/euclid.agt/1588212080<strong>Darlan Girão</strong>, <strong>Jessica S Purcell</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 987--1014.</p><p><strong>Abstract:</strong><br/>
Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph is a checkerboard surface, and each state surface is a fiber if and only if all of its associated checkerboard surfaces are fibers. We give an algebraic condition that characterizes which checkerboard surfaces are fibers directly from their state graphs. We use this to classify fibering of checkerboard surfaces for several families of planar graphs, including those associated with [math] –bridge links. This characterizes fibering for many families of state surfaces.
</p>projecteuclid.org/euclid.agt/1588212080_20200429220122Wed, 29 Apr 2020 22:01 EDTTrisections, intersection forms and the Torelli grouphttps://projecteuclid.org/euclid.agt/1588212081<strong>Peter Lambert-Cole</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 1015--1040.</p><p><strong>Abstract:</strong><br/>
We apply mapping class group techniques and trisections to study intersection forms of smooth [math] –manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology [math] –sphere can be obtained from the standard Heegaard decomposition of [math] by regluing according to a map in the kernel of this homomorphism. We prove an analogous result for trisections of [math] –manifolds. Specifically, if [math] and [math] admit handle decompositions without [math] – or [math] –handles and have isomorphic intersection forms, then a trisection of [math] can be obtained from a trisection of [math] by cutting and regluing by an element of the Johnson kernel. We also describe how invariants of homology [math] –spheres can be applied, via this result, to obstruct intersection forms of smooth [math] –manifolds. As an application, we use the Casson invariant to recover Rohlin’s theorem on the signature of spin [math] –manifolds.
</p>projecteuclid.org/euclid.agt/1588212081_20200429220122Wed, 29 Apr 2020 22:01 EDTRibbon distance and Khovanov homologyhttps://projecteuclid.org/euclid.agt/1588212082<strong>Sucharit Sarkar</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 1041--1058.</p><p><strong>Abstract:</strong><br/>
We study a notion of distance between knots, defined in terms of the number of saddles in ribbon concordances connecting the knots. We construct a lower bound on this distance using the [math] –action on Lee’s perturbation of Khovanov homology.
</p>projecteuclid.org/euclid.agt/1588212082_20200429220122Wed, 29 Apr 2020 22:01 EDTImmersed Möbius bands in knot complementshttps://projecteuclid.org/euclid.agt/1588212083<strong>Mark C Hughes</strong>, <strong>Seungwon Kim</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 2, 1059--1072.</p><p><strong>Abstract:</strong><br/>
We study the [math] –dimensional immersed crosscap number of a knot, which is a nonorientable analogue of the immersed Seifert genus. We study knots with immersed crosscap number [math] , and show that a knot has immersed crosscap number [math] if and only if it is a nontrivial [math] –torus or [math] –cable knot. We show that unlike in the orientable case the immersed crosscap number can differ from the embedded crosscap number by arbitrarily large amounts, and that it is neither bounded below nor above by the [math] –dimensional crosscap number.
</p>projecteuclid.org/euclid.agt/1588212083_20200429220122Wed, 29 Apr 2020 22:01 EDTRational homology cobordisms of plumbed manifoldshttps://projecteuclid.org/euclid.agt/1591374776<strong>Paolo Aceto</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 3, 1073--1126.</p><p><strong>Abstract:</strong><br/>
We investigate rational homology cobordisms of [math] –manifolds with nonzero first Betti number. This is motivated by the natural generalization of the slice-ribbon conjecture to multicomponent links. In particular we consider the problem of which rational homology [math] ’s bound rational homology [math] ’s. We give a simple procedure to construct rational homology cobordisms between plumbed [math] –manifolds. We introduce a family of plumbed [math] –manifolds with [math] . By adapting an obstruction based on Donaldson’s diagonalization theorem we characterize all manifolds in our family that bound rational homology [math] ’s. For all these manifolds a rational homology cobordism to [math] can be constructed via our procedure. Our family is large enough to include all Seifert fibered spaces over the [math] –sphere with vanishing Euler invariant. In a subsequent paper we describe applications to arborescent link concordance.
</p>projecteuclid.org/euclid.agt/1591374776_20200605123309Fri, 05 Jun 2020 12:33 EDTOn the homotopy theory for Lie $\infty$–groupoids, with an application to integrating $L_\infty$–algebrashttps://projecteuclid.org/euclid.agt/1591374777<strong>Christopher L Rogers</strong>, <strong>Chenchang Zhu</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 3, 1127--1219.</p><p><strong>Abstract:</strong><br/>
Lie [math] –groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie [math] –groupoids called “Lie [math] –groups” by integrating finite type Lie [math] –algebras. In order to study the compatibility between this integration procedure and the homotopy theory of Lie [math] –algebras introduced in the companion paper (1371–1429), we present a homotopy theory for Lie [math] –groupoids. Unlike Kan simplicial sets and the higher geometric groupoids of Behrend and Getzler, Lie [math] –groupoids do not form a category of fibrant objects (CFO), since the category of manifolds lacks pullbacks. Instead, we show that Lie [math] –groupoids form an “incomplete category of fibrant objects” in which the weak equivalences correspond to “stalkwise” weak equivalences of simplicial sheaves. This homotopical structure enjoys many of the same properties as a CFO, such as having, in the presence of functorial path objects, a convenient realization of its simplicial localization. We further prove that the acyclic fibrations are precisely the hypercovers, which implies that many of Behrend and Getzler’s results also hold in this more general context. As an application, we show that Henriques’ integration functor is an exact functor with respect to a class of distinguished fibrations, which we call “quasisplit fibrations”. Such fibrations include acyclic fibrations as well as fibrations that arise in string-like extensions. In particular, integration sends [math] –quasi-isomorphisms to weak equivalences and quasisplit fibrations to Kan fibrations, and preserves acyclic fibrations, as well as pullbacks of acyclic/quasisplit fibrations.
</p>projecteuclid.org/euclid.agt/1591374777_20200605123309Fri, 05 Jun 2020 12:33 EDTUnboundedness of some higher Euler classeshttps://projecteuclid.org/euclid.agt/1591374778<strong>Kathryn Mann</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 3, 1221--1234.</p><p><strong>Abstract:</strong><br/>
We study Euler classes in groups of homeomorphisms of Seifert-fibered [math] –manifolds. In contrast to the familiar Euler class for [math] as a discrete group, we show that these Euler classes for [math] as a discrete group are unbounded classes. In fact, we give examples of flat topological [math] –bundles over a genus [math] surface whose Euler class takes arbitrary values.
</p>projecteuclid.org/euclid.agt/1591374778_20200605123309Fri, 05 Jun 2020 12:33 EDTTowards the $K(2)$–local homotopy groups of $Z$https://projecteuclid.org/euclid.agt/1591374779<strong>Prasit Bhattacharya</strong>, <strong>Philip Egger</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 3, 1235--1277.</p><p><strong>Abstract:</strong><br/>
Previously (Adv. Math. 360 (2020) art. id. 106895), we introduced a class [math] of [math] –local finite spectra and showed that all spectra [math] admit a [math] –self-map of periodicity [math] . The aim here is to compute the [math] –local homotopy groups [math] of all spectra [math] using a homotopy fixed point spectral sequence, and we give an almost complete answer. The incompleteness lies in the fact that we are unable to eliminate one family of [math] –differentials and a few potential hidden [math] –extensions, though we conjecture that all these differentials and hidden extensions are trivial.
</p>projecteuclid.org/euclid.agt/1591374779_20200605123309Fri, 05 Jun 2020 12:33 EDTExponential functors, $R$–matrices and twistshttps://projecteuclid.org/euclid.agt/1591374780<strong>Ulrich Pennig</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 3, 1279--1324.</p><p><strong>Abstract:</strong><br/>
We show that each polynomial exponential functor on complex finite-dimensional inner product spaces is defined up to equivalence of monoidal functors by an involutive solution to the Yang–Baxter equation (an involutive [math] –matrix), which determines an extremal character on [math] . These characters are classified by Thoma parameters, and Thoma parameters resulting from polynomial exponential functors are of a special kind. Moreover, we show that each [math] –matrix with Thoma parameters of this kind yield a corresponding polynomial exponential functor.
In the second part of the paper we use these functors to construct a higher twist over [math] for a localisation of [math] –theory that generalises the one classified by the basic gerbe. We compute the indecomposable part of the rational characteristic classes of these twists in terms of the Thoma parameters of their [math] –matrices.
</p>projecteuclid.org/euclid.agt/1591374780_20200605123309Fri, 05 Jun 2020 12:33 EDTRoller boundaries for median spaces and algebrashttps://projecteuclid.org/euclid.agt/1591374781<strong>Elia Fioravanti</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 3, 1325--1370.</p><p><strong>Abstract:</strong><br/>
We construct compactifications for median spaces with compact intervals, generalising Roller boundaries of [math] cube complexes. Examples of median spaces with compact intervals include all finite-rank median spaces and all proper median spaces of infinite rank. Our methods also apply to general median algebras, where we recover the zero-completions of Bandelt and Meletiou (1993). Along the way, we prove various properties of halfspaces in finite-rank median spaces and a duality result for locally convex median spaces.
</p>projecteuclid.org/euclid.agt/1591374781_20200605123309Fri, 05 Jun 2020 12:33 EDTAn explicit model for the homotopy theory of finite-type Lie $n$–algebrashttps://projecteuclid.org/euclid.agt/1591374782<strong>Christopher L Rogers</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 3, 1371--1429.</p><p><strong>Abstract:</strong><br/>
Lie [math] –algebras are the [math] analogs of chain Lie algebras from rational homotopy theory. Henriques showed that finite-type Lie [math] –algebras can be integrated to produce certain simplicial Banach manifolds, known as Lie [math] –groups, via a smooth analog of Sullivan’s realization functor. We provide an explicit proof that the category of finite-type Lie [math] –algebras and (weak) [math] –morphisms admits the structure of a category of fibrant objects (CFO) for a homotopy theory. Roughly speaking, this CFO structure can be thought of as the transfer of the classical projective CFO structure on nonnegatively graded chain complexes via the tangent functor. In particular, the weak equivalences are precisely the [math] –quasi-isomorphisms. Along the way, we give explicit constructions for pullbacks and factorizations of [math] –morphisms between finite-type Lie [math] –algebras. We also analyze Postnikov towers and Maurer–Cartan/deformation functors associated to such Lie [math] –algebras. The main application of this work is our joint paper with C Zhu (1127–1219), which characterizes the compatibility of Henriques’ integration functor with the homotopy theory of Lie [math] –algebras and that of Lie [math] –groups.
</p>projecteuclid.org/euclid.agt/1591374782_20200605123309Fri, 05 Jun 2020 12:33 EDTRelative recognition principlehttps://projecteuclid.org/euclid.agt/1591374783<strong>Renato Vasconcellos Vieira</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 3, 1431--1486.</p><p><strong>Abstract:</strong><br/>
We prove the recognition principle for relative [math] –loop pairs of spaces for [math] . If [math] , this states that a pair of spaces homotopy equivalent to CW–complexes [math] is homotopy equivalent to [math] for a functorially determined relative space [math] if and only if [math] is a grouplike [math] –space, where [math] is any cofibrant resolution of the Swiss-cheese relative operad [math] . The relative recognition principle for relative [math] –loop pairs of spaces states that a pair of spaces [math] homotopy equivalent to CW–complexes is homotopy equivalent to [math] for a functorially determined relative spectrum [math] if and only if [math] is a grouplike [math] –algebra, where [math] is a contractible cofibrant relative operad or equivalently a cofibrant resolution of the terminal relative operad [math] of continuous homomorphisms of commutative monoids. These principles are proved as equivalences of homotopy categories.
</p>projecteuclid.org/euclid.agt/1591374783_20200605123309Fri, 05 Jun 2020 12:33 EDTCohomological correspondence categorieshttps://projecteuclid.org/euclid.agt/1591374784<strong>Andrei Druzhinin</strong>, <strong>Håkon Kolderup</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 3, 1487--1541.</p><p><strong>Abstract:</strong><br/>
We prove that homotopy invariance and cancellation properties are satisfied by any category of correspondences that is defined, via Calmès and Fasel’s construction, by an underlying cohomology theory. In particular, this includes any category of correspondences arising from the cohomology theory defined by an [math] –algebra.
</p>projecteuclid.org/euclid.agt/1591374784_20200605123309Fri, 05 Jun 2020 12:33 EDTModel structures for $(\infty,n)$–categories on (pre)stratified simplicial sets and prestratified simplicial spaceshttps://projecteuclid.org/euclid.agt/1591374785<strong>Viktoriya Ozornova</strong>, <strong>Martina Rovelli</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 20, Number 3, 1543--1600.</p><p><strong>Abstract:</strong><br/>
We prove the existence of a model structure on the category of stratified simplicial sets whose fibrant objects are precisely [math] –complicial sets, which are a proposed model for [math] –categories, based on previous work of Verity and Riehl. We then construct a Quillen equivalent model based on simplicial presheaves over a category that can facilitate the comparison with other established models.
</p>projecteuclid.org/euclid.agt/1591374785_20200605123309Fri, 05 Jun 2020 12:33 EDT