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 EDTOn periodic groups of homeomorphisms of the $2$–dimensional spherehttps://projecteuclid.org/euclid.agt/1545102065<strong>Jonathan Conejeros</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4093--4107.</p><p><strong>Abstract:</strong><br/>
We prove that every finitely generated group of homeomorphisms of the [math] –dimensional sphere all of whose elements have a finite order which is a power of [math] and is such that there exists a uniform bound for the orders of the group elements is finite. We prove a similar result for groups of area-preserving homeomorphisms without the hypothesis that the orders of group elements are powers of [math] provided there is an element of even order.
</p>projecteuclid.org/euclid.agt/1545102065_20181217220124Mon, 17 Dec 2018 22:01 ESTAlgebraic and topological properties of big mapping class groupshttps://projecteuclid.org/euclid.agt/1545102066<strong>Priyam Patel</strong>, <strong>Nicholas G Vlamis</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4109--4142.</p><p><strong>Abstract:</strong><br/>
Let [math] be an orientable, connected topological surface of infinite type (that is, with infinitely generated fundamental group). The main theorem states that if the genus of [math] is finite and at least [math] , then the isomorphism type of the pure mapping class group associated to [math] , denoted by [math] , detects the homeomorphism type of [math] . As a corollary, every automorphism of [math] is induced by a homeomorphism, which extends a theorem of Ivanov from the finite-type setting. In the process of proving these results, we show that [math] is residually finite if and only if [math] has finite genus, demonstrating that the algebraic structure of [math] can distinguish finite- and infinite-genus surfaces. As an independent result, we also show that [math] fails to be residually finite for any infinite-type surface [math] . In addition, we give a topological generating set for [math] equipped with the compact-open topology. In particular, if [math] has at most one end accumulated by genus, then [math] is topologically generated by Dehn twists, otherwise it is topologically generated by Dehn twists along with handle shifts.
</p>projecteuclid.org/euclid.agt/1545102066_20181217220124Mon, 17 Dec 2018 22:01 ESTEquivariant cohomology Chern numbers determine equivariant unitary bordism for torus groupshttps://projecteuclid.org/euclid.agt/1545102068<strong>Zhi Lü</strong>, <strong>Wei Wang</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4143--4160.</p><p><strong>Abstract:</strong><br/>
We show that the integral equivariant cohomology Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary [math] –manifolds, which gives an affirmative answer to a conjecture posed by Guillemin, Ginzburg and Karshon ( Moment maps, cobordisms, and Hamiltonian group actions , Remark H.5 in Appendix H.3), where [math] is a torus. As a further application, we also obtain a satisfactory solution of their Question (A) (Appendix H.1.1) on unitary Hamiltonian [math] –manifolds. Our key ingredients in the proof are the universal toric genus defined by Buchstaber, Panov and Ray and the Kronecker pairing of bordism and cobordism. Our approach heavily exploits Quillen’s geometric interpretation of homotopic unitary cobordism theory. Moreover, this method can also be applied to the study of [math] –equivariant unoriented bordism and can still derive the classical result of tom Dieck.
</p>projecteuclid.org/euclid.agt/1545102068_20181217220124Mon, 17 Dec 2018 22:01 ESTSpaces of orders of some one-relator groupshttps://projecteuclid.org/euclid.agt/1545102069<strong>Juan Alonso</strong>, <strong>Joaquín Brum</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4161--4185.</p><p><strong>Abstract:</strong><br/>
We show that certain left-orderable groups admit no isolated left orders. The groups we consider are cyclic amalgamations of a free group with a general left-orderable group, the HNN extensions of free groups over cyclic subgroups, and a particular class of one-relator groups. In order to prove the results about orders, we develop perturbation techniques for actions of these groups on the line.
</p>projecteuclid.org/euclid.agt/1545102069_20181217220124Mon, 17 Dec 2018 22:01 ESTOn the asymptotic expansion of the quantum $\mathrm{SU}(2)$ invariant at $q = \exp(4\pi\sqrt{-1}/N)$ for closed hyperbolic $3$–manifolds obtained by integral surgery along the figure-eight knothttps://projecteuclid.org/euclid.agt/1545102070<strong>Tomotada Ohtsuki</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4187--4274.</p><p><strong>Abstract:</strong><br/>
It is known that the quantum [math] invariant of a closed [math] –manifold at [math] is of polynomial order as [math] . Recently, Chen and Yang conjectured that the quantum [math] invariant of a closed hyperbolic [math] –manifold at [math] is of order [math] , where [math] is a normalized complex volume of [math] . We can regard this conjecture as a kind of “volume conjecture”, which is an important topic from the viewpoint that it relates quantum topology and hyperbolic geometry.
In this paper, we give a concrete presentation of the asymptotic expansion of the quantum [math] invariant at [math] for closed hyperbolic [math] –manifolds obtained from the [math] –sphere by integral surgery along the figure-eight knot. In particular, the leading term of the expansion is [math] , which gives a proof of the Chen–Yang conjecture for such [math] –manifolds. Further, the semiclassical part of the expansion is a constant multiple of the square root of the Reidemeister torsion for such [math] –manifolds. We expect that the higher-order coefficients of the expansion would be “new” invariants, which are related to “quantization” of the hyperbolic structure of a closed hyperbolic [math] –manifold.
</p>projecteuclid.org/euclid.agt/1545102070_20181217220124Mon, 17 Dec 2018 22:01 ESTNotes on open book decompositions for Engel structureshttps://projecteuclid.org/euclid.agt/1545102071<strong>Vincent Colin</strong>, <strong>Francisco Presas</strong>, <strong>Thomas Vogel</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4275--4303.</p><p><strong>Abstract:</strong><br/>
We relate open book decompositions of a [math] –manifold [math] with its Engel structures. Our main result is, given an open book decomposition of [math] whose binding is a collection of [math] –tori and whose monodromy preserves a framing of a page, the construction of an Engel structure whose isotropic foliation is transverse to the interior of the pages and tangent to the binding.
In particular, the pages are contact manifolds and the monodromy is a compactly supported contactomorphism. As a consequence, on a parallelizable closed [math] –manifold, every open book with toric binding carries in the previous sense an Engel structure. Moreover, we show that among the supported Engel structures we construct, there are loose Engel structures.
</p>projecteuclid.org/euclid.agt/1545102071_20181217220124Mon, 17 Dec 2018 22:01 ESTAnick spaces and Kac–Moody groupshttps://projecteuclid.org/euclid.agt/1545102072<strong>Stephen Theriault</strong>, <strong>Jie Wu</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4305--4328.</p><p><strong>Abstract:</strong><br/>
For primes [math] we prove an approximation to Cohen, Moore and Neisendorfer’s conjecture that the loops on an Anick space retracts off the double loops on a mod- [math] Moore space. The approximation is then used to answer a question posed by Kitchloo regarding the topology of Kac–Moody groups. We show that, for certain rank- [math] Kac–Moody groups [math] , the based loops on [math] is [math] –locally homotopy equivalent to the product of the loops on a [math] –sphere and the loops on an Anick space.
</p>projecteuclid.org/euclid.agt/1545102072_20181217220124Mon, 17 Dec 2018 22:01 ESTLogarithmic Hennings invariants for restricted quantum ${\mathfrak{sl}}(2)$https://projecteuclid.org/euclid.agt/1545102073<strong>Anna Beliakova</strong>, <strong>Christian Blanchet</strong>, <strong>Nathan Geer</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4329--4358.</p><p><strong>Abstract:</strong><br/>
We construct a Hennings-type logarithmic invariant for restricted quantum [math] at a [math] root of unity. This quantum group [math] is not quasitriangular and hence not ribbon, but factorizable. The invariant is defined for a pair: a [math] –manifold [math] and a colored link [math] inside [math] . The link [math] is split into two parts colored by central elements and by trace classes, or elements in the [math] Hochschild homology of [math] , respectively. The two main ingredients of our construction are the universal invariant of a string link with values in tensor powers of [math] , and the modified trace introduced by the third author with his collaborators and computed on tensor powers of the regular representation. Our invariant is a colored extension of the logarithmic invariant constructed by Jun Murakami.
</p>projecteuclid.org/euclid.agt/1545102073_20181217220124Mon, 17 Dec 2018 22:01 ESTNonarithmetic hyperbolic manifolds and trace ringshttps://projecteuclid.org/euclid.agt/1545102074<strong>Olivier Mila</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 18, Number 7, 4359--4373.</p><p><strong>Abstract:</strong><br/>
We give a sufficient condition on the hyperplanes used in the Belolipetsky–Thomson inbreeding construction to obtain nonarithmetic manifolds. We explicitly construct infinitely many examples of such manifolds that are pairwise noncommensurable and estimate their volume.
</p>projecteuclid.org/euclid.agt/1545102074_20181217220124Mon, 17 Dec 2018 22:01 ESTPretty rational models for Poincaré duality pairshttps://projecteuclid.org/euclid.agt/1549940428<strong>Hector Cordova Bulens</strong>, <strong>Pascal Lambrechts</strong>, <strong>Don Stanley</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 1--30.</p><p><strong>Abstract:</strong><br/>
We prove that a large class of Poincaré duality pairs of spaces admit rational models (in the sense of Sullivan) of a convenient form associated to some Poincaré duality CDGA.
</p>projecteuclid.org/euclid.agt/1549940428_20190211220040Mon, 11 Feb 2019 22:00 ESTTopological Hochschild homology of maximal orders in simple $\mathbb{Q}$–algebrashttps://projecteuclid.org/euclid.agt/1549940429<strong>Henry Yi-Wei Chan</strong>, <strong>Ayelet Lindenstrauss</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 31--75.</p><p><strong>Abstract:</strong><br/>
We calculate the topological Hochschild homology groups of a maximal order in a simple algebra over the rationals. Since the positive-dimensional [math] groups consist only of torsion, we do this one prime ideal at a time for all the nonzero prime ideals in the center of the maximal order. This allows us to reduce the problem to studying the topological Hochschild homology groups of maximal orders [math] in simple [math] –algebras. We show that the topological Hochschild homology of [math] splits as the tensor product of its Hochschild homology with [math] . We use this result in Brun’s spectral sequence to calculate [math] , and then we analyze the torsion to get [math] .
</p>projecteuclid.org/euclid.agt/1549940429_20190211220040Mon, 11 Feb 2019 22:00 ESTA simplicial James–Hopf map and decompositions of the unstable Adams spectral sequence for suspensionshttps://projecteuclid.org/euclid.agt/1549940430<strong>Fedor Pavutnitskiy</strong>, <strong>Jie Wu</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 77--108.</p><p><strong>Abstract:</strong><br/>
We use combinatorial group theory methods to extend the definition of the classical James–Hopf invariant to a simplicial group setting. This allows us to realize certain coalgebra idempotents at an [math] level and obtain a functorial decomposition of the spectral sequence, associated with the lower [math] –central series filtration on a free simplicial group.
</p>projecteuclid.org/euclid.agt/1549940430_20190211220040Mon, 11 Feb 2019 22:00 ESTDimensional reduction and the equivariant Chern characterhttps://projecteuclid.org/euclid.agt/1549940431<strong>Augusto Stoffel</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 109--150.</p><p><strong>Abstract:</strong><br/>
We propose a dimensional reduction procedure for [math] –dimensional supersymmetric euclidean field theories (EFTs) in the sense of Stolz and Teichner. Our construction is well suited in the presence of a finite gauge group or, more generally, for field theories over an orbifold. As an illustration, we give a geometric interpretation of the Chern character for manifolds with an action by a finite group.
</p>projecteuclid.org/euclid.agt/1549940431_20190211220040Mon, 11 Feb 2019 22:00 ESTConstructing the virtual fundamental class of a Kuranishi atlashttps://projecteuclid.org/euclid.agt/1549940432<strong>Dusa McDuff</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 151--238.</p><p><strong>Abstract:</strong><br/>
Consider a space [math] , such as a compact space of [math] –holomorphic stable maps, that is the zero set of a Kuranishi atlas. This note explains how to define the virtual fundamental class of [math] by representing [math] via the zero set of a map [math] , where [math] is a finite-dimensional vector space and the domain [math] is an oriented, weighted branched topological manifold. Moreover, [math] is equivariant under the action of the global isotropy group [math] on [math] and [math] . This tuple [math] together with a homeomorphism from [math] to [math] forms a single finite-dimensional model (or chart) for [math] . The construction assumes only that the atlas satisfies a topological version of the index condition that can be obtained from a standard, rather than a smooth, gluing theorem. However, if [math] is presented as the zero set of an [math] –Fredholm operator on a strong polyfold bundle, we outline a much more direct construction of the branched manifold [math] that uses an [math] –smooth partition of unity.
</p>projecteuclid.org/euclid.agt/1549940432_20190211220040Mon, 11 Feb 2019 22:00 EST$E_2$ structures and derived Koszul duality in string topologyhttps://projecteuclid.org/euclid.agt/1549940433<strong>Andrew J Blumberg</strong>, <strong>Michael A Mandell</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 239--279.</p><p><strong>Abstract:</strong><br/>
We construct an equivalence of [math] algebras between two models for the Thom spectrum of the free loop space that are related by derived Koszul duality. To do this, we describe the functoriality and invariance properties of topological Hochschild cohomology.
</p>projecteuclid.org/euclid.agt/1549940433_20190211220040Mon, 11 Feb 2019 22:00 ESTVanishing theorems for representation homology and the derived cotangent complexhttps://projecteuclid.org/euclid.agt/1549940434<strong>Yuri Berest</strong>, <strong>Ajay C Ramadoss</strong>, <strong>Wai-kit Yeung</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 281--339.</p><p><strong>Abstract:</strong><br/>
Let [math] be a reductive affine algebraic group defined over a field [math] of characteristic zero. We study the cotangent complex of the derived [math] –representation scheme [math] of a pointed connected topological space [math] . We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of [math] to the representation homology [math] to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in [math] and generalized lens spaces. In particular, for any finitely generated virtually free group [math] , we show that [math] for all [math] . For a closed Riemann surface [math] of genus [math] , we have [math] for all [math] . The sharp vanishing bounds for [math] actually depend on the genus: we conjecture that if [math] , then [math] for [math] , and if [math] , then [math] for [math] , where [math] is the center of [math] . We prove these bounds locally on the smooth locus of the representation scheme [math] in the case of complex connected reductive groups. One important consequence of our results is the existence of a well-defined [math] –theoretic virtual fundamental class for [math] in the sense of Ciocan-Fontanine and Kapranov (Geom. Topol. 13 (2009) 1779–1804). We give a new “Tor formula” for this class in terms of functor homology.
</p>projecteuclid.org/euclid.agt/1549940434_20190211220040Mon, 11 Feb 2019 22:00 ESTRelative phantom mapshttps://projecteuclid.org/euclid.agt/1549940435<strong>Kouyemon Iriye</strong>, <strong>Daisuke Kishimoto</strong>, <strong>Takahiro Matsushita</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 341--362.</p><p><strong>Abstract:</strong><br/>
The de Bruijn–Erdős theorem states that the chromatic number of an infinite graph equals the maximum of the chromatic numbers of its finite subgraphs. Such determination by finite subobjects appears in the definition of a phantom map, which is classical in algebraic topology. The topological method in combinatorics connects these two, which leads us to define the relative version of a phantom map: a map [math] is called a relative phantom map to a map [math] if the restriction of [math] to any finite subcomplex of [math] lifts to [math] through [math] , up to homotopy. There are two kinds of maps which are obviously relative phantom maps: (1) the composite of a map [math] with [math] ; (2) a usual phantom map [math] . A relative phantom map of type (1) is called trivial, and a relative phantom map out of a suspension which is a sum of (1) and (2) is called relatively trivial. We study the (relative) triviality of relative phantom maps and, in particular, we give rational homology conditions for the (relative) triviality.
</p>projecteuclid.org/euclid.agt/1549940435_20190211220040Mon, 11 Feb 2019 22:00 ESTDistortion of surfaces in graph manifoldshttps://projecteuclid.org/euclid.agt/1549940436<strong>G Christopher Hruska</strong>, <strong>Hoang Thanh Nguyen</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 363--395.</p><p><strong>Abstract:</strong><br/>
Let [math] be an immersed horizontal surface in a [math] –dimensional graph manifold. We show that the fundamental group of the surface [math] is quadratically distorted whenever the surface is virtually embedded (ie separable) and is exponentially distorted when the surface is not virtually embedded.
</p>projecteuclid.org/euclid.agt/1549940436_20190211220040Mon, 11 Feb 2019 22:00 ESTArrow calculus for welded and classical linkshttps://projecteuclid.org/euclid.agt/1549940437<strong>Jean-Baptiste Meilhan</strong>, <strong>Akira Yasuhara</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 397--456.</p><p><strong>Abstract:</strong><br/>
We develop a calculus for diagrams of knotted objects. We define arrow presentations, which encode the crossing information of a diagram into arrows in a way somewhat similar to Gauss diagrams, and more generally [math] –tree presentations, which can be seen as “higher-order Gauss diagrams”. This arrow calculus is used to develop an analogue of Habiro’s clasper theory for welded knotted objects, which contain classical link diagrams as a subset. This provides a “realization” of Polyak’s algebra of arrow diagrams at the welded level, and leads to a characterization of finite-type invariants of welded knots and long knots. As a corollary, we recover several topological results due to Habiro and Shima and to Watanabe on knotted surfaces in [math] –space. We also classify welded string links up to homotopy, thus recovering a result of the first author with Audoux, Bellingeri and Wagner.
</p>projecteuclid.org/euclid.agt/1549940437_20190211220040Mon, 11 Feb 2019 22:00 ESTTorsion homology and cellular approximationhttps://projecteuclid.org/euclid.agt/1549940438<strong>Ramón Flores</strong>, <strong>Fernando Muro</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 457--476.</p><p><strong>Abstract:</strong><br/>
We describe the role of the Schur multiplier in the structure of the [math] –torsion of discrete groups. More concretely, we show how the knowledge of [math] allows us to approximate many groups by colimits of copies of [math] –groups. Our examples include interesting families of noncommutative infinite groups, including Burnside groups, certain solvable groups and branch groups. We also provide a counterexample for a conjecture of Emmanuel Farjoun.
</p>projecteuclid.org/euclid.agt/1549940438_20190211220040Mon, 11 Feb 2019 22:00 ESTThe verbal width of acylindrically hyperbolic groupshttps://projecteuclid.org/euclid.agt/1549940439<strong>Mladen Bestvina</strong>, <strong>Kenneth Bromberg</strong>, <strong>Koji Fujiwara</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 477--489.</p><p><strong>Abstract:</strong><br/>
We show that the verbal width is infinite for acylindrically hyperbolic groups, which include hyperbolic groups, mapping class groups and [math] .
</p>projecteuclid.org/euclid.agt/1549940439_20190211220040Mon, 11 Feb 2019 22:00 ESTOn the homotopy types of $\mathrm{Sp}(n)$ gauge groupshttps://projecteuclid.org/euclid.agt/1549940440<strong>Daisuke Kishimoto</strong>, <strong>Akira Kono</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 491--502.</p><p><strong>Abstract:</strong><br/>
Let [math] be the gauge group of the principal [math] –bundle over [math] corresponding to [math] . We refine the result of Sutherland on the homotopy types of [math] and relate it to the order of a certain Samelson product in [math] . Then we classify the [math] –local homotopy types of [math] for [math] .
</p>projecteuclid.org/euclid.agt/1549940440_20190211220040Mon, 11 Feb 2019 22:00 ESTCohomology rings of compactifications of toric arrangementshttps://projecteuclid.org/euclid.agt/1549940441<strong>Corrado De Concini</strong>, <strong>Giovanni Gaiffi</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 1, 503--532.</p><p><strong>Abstract:</strong><br/>
We previously (Adv. Math. 327 (2018) 390–409) constructed some projective wonderful models for the complement of a toric arrangement in an [math] –dimensional algebraic torus [math] . In this paper we describe their integer cohomology rings by generators and relations.
</p>projecteuclid.org/euclid.agt/1549940441_20190211220040Mon, 11 Feb 2019 22:00 ESTA note on the $(\infty,n)$–category of cobordismshttps://projecteuclid.org/euclid.agt/1552960819<strong>Damien Calaque</strong>, <strong>Claudia Scheimbauer</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 533--655.</p><p><strong>Abstract:</strong><br/>
In this extended note we give a precise definition of fully extended topological field theories à la Lurie. Using complete [math] –fold Segal spaces as a model, we construct an [math] –category of [math] –dimensional bordisms, possibly with tangential structure. We endow it with a symmetric monoidal structure and show that we can recover the usual category of bordisms.
</p>projecteuclid.org/euclid.agt/1552960819_20190318220045Mon, 18 Mar 2019 22:00 EDTParametrized homology via zigzag persistencehttps://projecteuclid.org/euclid.agt/1552960823<strong>Gunnar Carlsson</strong>, <strong>Vin de Silva</strong>, <strong>Sara Kališnik</strong>, <strong>Dmitriy Morozov</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 657--700.</p><p><strong>Abstract:</strong><br/>
This paper introduces parametrized homology, a continuous-parameter generalization of levelset zigzag persistent homology that captures the behavior of the homology of the fibers of a real-valued function on a topological space. This information is encoded as a “barcode” of real intervals, each corresponding to a homological feature supported over that interval; or, equivalently, as a persistence diagram. Points in the persistence diagram are classified algebraically into four classes; geometrically, the classes identify the distinct ways in which homological features perish at the boundaries of their interval of persistence. We study the conditions under which spaces fibered over the real line have a well-defined parametrized homology; we establish the stability of these invariants and we show how the four classes of persistence diagram correspond to the four diagrams that appear in the theory of extended persistence.
</p>projecteuclid.org/euclid.agt/1552960823_20190318220045Mon, 18 Mar 2019 22:00 EDTTopology of (small) Lagrangian cobordismshttps://projecteuclid.org/euclid.agt/1552960827<strong>Mads R Bisgaard</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 701--742.</p><p><strong>Abstract:</strong><br/>
We study the following quantitative phenomenon in symplectic topology: in many situations, if a Lagrangian cobordism is sufficiently small (in a sense we specify) then its topology is to a large extend determined by its boundary. This principle allows us to derive several homological uniqueness results for small Lagrangian cobordisms. In particular, under the smallness assumption, we prove homological uniqueness of the class of Lagrangian cobordisms, which, by Biran and Cornea’s Lagrangian cobordism theory, induces operations on a version of the derived Fukaya category. We also establish a link between our results and Vassilyev’s theory of Lagrange characteristic classes. Most currently known constructions of Lagrangian cobordisms yield small Lagrangian cobordisms in many examples.
</p>projecteuclid.org/euclid.agt/1552960827_20190318220045Mon, 18 Mar 2019 22:00 EDT$2$–associahedrahttps://projecteuclid.org/euclid.agt/1552960828<strong>Nathaniel Bottman</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 743--806.</p><p><strong>Abstract:</strong><br/>
For any [math] and [math] we construct a poset [math] called a [math] –associahedron . The [math] –associahedra arose in symplectic geometry, where they are expected to control maps between Fukaya categories of different symplectic manifolds. We prove that the completion [math] is an abstract polytope of dimension [math] . There are forgetful maps [math] , where [math] is the [math] –dimensional associahedron, and the [math] –associahedra specialize to the associahedra (in two ways) and to the multiplihedra. In an appendix, we work out the [math] – and [math] –dimensional [math] –associahedra in detail.
</p>projecteuclid.org/euclid.agt/1552960828_20190318220045Mon, 18 Mar 2019 22:00 EDTOn $\mathrm{BP}\langle 2\rangle$–cooperationshttps://projecteuclid.org/euclid.agt/1552960829<strong>Dominic Leon Culver</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 807--862.</p><p><strong>Abstract:</strong><br/>
We develop techniques to compute the cooperations algebra for the second truncated Brown–Peterson spectrum [math] . We prove that the cooperations algebra [math] decomposes as a direct sum of an [math] –vector space concentrated in Adams filtration [math] and an [math] –module which is concentrated in even degrees and is [math] –torsion-free. We also develop a recursive method which produces a basis for the [math] –torsion-free component.
</p>projecteuclid.org/euclid.agt/1552960829_20190318220045Mon, 18 Mar 2019 22:00 EDTHigher cyclic operadshttps://projecteuclid.org/euclid.agt/1552960830<strong>Philip Hackney</strong>, <strong>Marcy Robertson</strong>, <strong>Donald Yau</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 863--940.</p><p><strong>Abstract:</strong><br/>
We introduce a convenient definition for weak cyclic operads, which is based on unrooted trees and Segal conditions. More specifically, we introduce a category [math] of trees, which carries a tight relationship to the Moerdijk–Weiss category of rooted trees [math] . We prove a nerve theorem exhibiting colored cyclic operads as presheaves on [math] which satisfy a Segal condition. Finally, we produce a Quillen model category whose fibrant objects satisfy a weak Segal condition, and we consider these objects as an up-to-homotopy generalization of the concept of cyclic operad.
</p>projecteuclid.org/euclid.agt/1552960830_20190318220045Mon, 18 Mar 2019 22:00 EDTLeast dilatation of pure surface braidshttps://projecteuclid.org/euclid.agt/1552960831<strong>Marissa Loving</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 941--964.</p><p><strong>Abstract:</strong><br/>
We study the minimal dilatation of pseudo-Anosov pure surface braids and provide upper and lower bounds as a function of genus and the number of punctures. For a fixed number of punctures, these bounds tend to infinity as the genus does. We also bound the dilatation of pseudo-Anosov pure surface braids away from zero and give a constant upper bound in the case of a sufficient number of punctures.
</p>projecteuclid.org/euclid.agt/1552960831_20190318220045Mon, 18 Mar 2019 22:00 EDTTopological Hochschild homology and higher characteristicshttps://projecteuclid.org/euclid.agt/1552960832<strong>Jonathan A Campbell</strong>, <strong>Kate Ponto</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 965--1017.</p><p><strong>Abstract:</strong><br/>
We show that an important classical fixed-point invariant, the Reidemeister trace, arises as a topological Hochschild homology transfer. This generalizes a corresponding classical result for the Euler characteristic and is a first step in showing the Reidemeister trace is in the image of the cyclotomic trace. The main result follows from developing the relationship between shadows (see Astérisque 333, Soc. Math. France, Paris (2010)), topological Hochschild homology and Morita-invariance in bicategorical generality.
</p>projecteuclid.org/euclid.agt/1552960832_20190318220045Mon, 18 Mar 2019 22:00 EDTSpecies substitution, graph suspension, and graded Hopf algebras of painted tree polytopeshttps://projecteuclid.org/euclid.agt/1552960833<strong>Lisa Berry</strong>, <strong>Stefan Forcey</strong>, <strong>Maria Ronco</strong>, <strong>Patrick Showers</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 2, 1019--1078.</p><p><strong>Abstract:</strong><br/>
Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here include ordered and unordered binary trees and ordered lists (labeled corollas). Some of the polytopes that constitute our main results are well known in other contexts. First we see the classical permutohedra, and then certain generalized permutohedra : specifically the graph associahedra of suspensions of certain simple graphs. As an aside we show that the stellohedra also appear as liftings of generalized permutohedra: graph composihedra for complete graphs. Thus our results give examples of Hopf algebras of tubings and marked tubings of graphs. We also show an alternative associative algebra structure on the graph tubings of star graphs.
</p>projecteuclid.org/euclid.agt/1552960833_20190318220045Mon, 18 Mar 2019 22:00 EDTHomotopical intersection theory, III: Multirelative intersection problemshttps://projecteuclid.org/euclid.agt/1559095419<strong>John R Klein</strong>, <strong>Bruce Williams</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1079--1134.</p><p><strong>Abstract:</strong><br/>
We extend some results of Hatcher and Quinn (1974) beyond the metastable range. We give a bordism-theoretic obstruction [math] to deforming a map [math] between manifolds simultaneously off of a collection of pairwise disjoint submanifolds [math] under the assumption that [math] can be deformed off of any proper subcollection in a homotopy coherent way. In a certain range of dimensions, [math] is a complete obstruction to finding the desired deformation. We apply this machinery to embedding problems and to the study of linking phenomena.
</p>projecteuclid.org/euclid.agt/1559095419_20190528220354Tue, 28 May 2019 22:03 EDTRips filtrations for quasimetric spaces and asymmetric functions with stability resultshttps://projecteuclid.org/euclid.agt/1559095421<strong>Katharine Turner</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1135--1170.</p><p><strong>Abstract:</strong><br/>
The Rips filtration over a finite metric space and its corresponding persistent homology are prominent methods in topological data analysis to summarise the “shape” of data. Crucial to their use is the stability result that says if [math] and [math] are finite metric spaces then the (bottleneck) distance between the persistence diagrams constructed via the Rips filtration is bounded by [math] (where [math] is the Gromov–Hausdorff distance). A generalisation of the Rips filtration to any symmetric function [math] was defined by Chazal, de Silva and Oudot (Geom. Dedicata 173 (2014) 193–214), where they showed it was stable with respect to the correspondence distortion distance. Allowing asymmetry, we consider four different persistence modules, definable for pairs [math] where [math] is any real valued function. These generalise the persistent homology of the symmetric Rips filtration in different ways. The first method is through symmetrisation. For each [math] we can construct a symmetric function [math] . We can then apply the standard theory for symmetric functions and get stability as a corollary. The second method is to construct a filtration [math] of ordered tuple complexes where [math] if [math] for all [math] . Both our first two methods have the same persistent homology as the standard Rips filtration when applied to a metric space, or more generally to a symmetric function. We then consider two constructions using an associated filtration of directed graphs or preorders. For each [math] we can define a directed graph [math] where directed edges [math] are included in [math] whenever [math] (note this is when [math] for [math] a quasimetric). From this we construct a preorder where [math] if there is a path from [math] to [math] in [math] . We build persistence modules using the strongly connected components of the graphs [math] , which are also the equivalence classes of the associated preorders. We also consider persistence modules using a generalisation of poset topology to preorders.
The Gromov–Hausdorff distance, when expressed via correspondence distortions, can be naturally extended as a correspondence distortion distance to set–function pairs [math] . We prove that all these new constructions enjoy the same stability as persistence modules built via the original persistent homology for symmetric functions.
</p>projecteuclid.org/euclid.agt/1559095421_20190528220354Tue, 28 May 2019 22:03 EDT$C^*$–algebraic drawings of dendroidal setshttps://projecteuclid.org/euclid.agt/1559095422<strong>Snigdhayan Mahanta</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1171--1206.</p><p><strong>Abstract:</strong><br/>
In recent years the theory of dendroidal sets has emerged as an important framework for higher algebra. We introduce the concept of a [math] –algebraic drawing of a dendroidal set. It depicts a dendroidal set as an object in the category of presheaves on [math] –algebras. We show that the construction is functorial and, in fact, it is the left adjoint of a Quillen adjunction between combinatorial model categories. We use this construction to produce a bridge between the two prominent paradigms of noncommutative geometry via adjunctions of presentable [math] –categories, which is the primary motivation behind this article. As a consequence we obtain a single mechanism to construct bivariant homology theories in both paradigms. We propose a (conjectural) roadmap to harmonize algebraic and analytic (or topological) bivariant [math] –theory. Finally, a method to analyze graph algebras in terms of trees is sketched.
</p>projecteuclid.org/euclid.agt/1559095422_20190528220354Tue, 28 May 2019 22:03 EDTExamples of nontrivial contact mapping classes for overtwisted contact manifolds in all dimensionshttps://projecteuclid.org/euclid.agt/1559095423<strong>Fabio Gironella</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1207--1227.</p><p><strong>Abstract:</strong><br/>
We construct (infinitely many) examples in all dimensions of contactomorphisms of closed overtwisted contact manifolds that are smoothly isotopic but not contact-isotopic to the identity.
</p>projecteuclid.org/euclid.agt/1559095423_20190528220354Tue, 28 May 2019 22:03 EDTUniform exponential growth for CAT(0) square complexeshttps://projecteuclid.org/euclid.agt/1559095424<strong>Aditi Kar</strong>, <strong>Michah Sageev</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1229--1245.</p><p><strong>Abstract:</strong><br/>
We start the inquiry into proving uniform exponential growth in the context of groups acting on CAT(0) cube complexes. We address free group actions on CAT(0) square complexes and prove a more general statement. This says that if [math] is a finite collection of hyperbolic automorphisms of a CAT(0) square complex [math] , then either there exists a pair of words of length at most [math] in [math] which freely generate a free semigroup, or all elements of [math] stabilize a flat (of dimension [math] or [math] in [math] ). As a corollary, we obtain a lower bound for the growth constant, [math] , which is uniform not just for a given group acting freely on a given CAT(0) cube complex, but for all groups which are not virtually abelian and have a free action on a CAT(0) square complex.
</p>projecteuclid.org/euclid.agt/1559095424_20190528220354Tue, 28 May 2019 22:03 EDTCommensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groupshttps://projecteuclid.org/euclid.agt/1559095425<strong>Matthew C B Zaremsky</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1247--1264.</p><p><strong>Abstract:</strong><br/>
We prove that if a right-angled Artin group [math] is abstractly commensurable to a group splitting nontrivially as an amalgam or HNN extension over [math] , then [math] must itself split nontrivially over [math] for some [math] . Consequently, if two right-angled Artin groups [math] and [math] are commensurable and [math] has no separating [math] –cliques for any [math] , then neither does [math] , so “smallest size of separating clique” is a commensurability invariant. We also discuss some implications for issues of quasi-isometry. Using similar methods we also prove that for [math] the braid group [math] is not abstractly commensurable to any group that splits nontrivially over a “free group–free” subgroup, and the same holds for [math] for the loop braid group [math] . Our approach makes heavy use of the Bieri–Neumann–Strebel invariant.
</p>projecteuclid.org/euclid.agt/1559095425_20190528220354Tue, 28 May 2019 22:03 EDTOccupants in simplicial complexeshttps://projecteuclid.org/euclid.agt/1559095426<strong>Steffen Tillmann</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1265--1298.</p><p><strong>Abstract:</strong><br/>
Let [math] be a smooth manifold and [math] be a simplicial complex of codimension at least [math] . Functor calculus methods lead to a homotopical formula of [math] in terms of spaces [math] where [math] is a finite subset of [math] . This is a generalization of the author’s previous work with Michael Weiss (Contemp. Math. 682, Amer. Math. Soc., Providence, RI (2017) 237–259), where the subset [math] is assumed to be a smooth submanifold of [math] and uses his generalization of manifold calculus adapted for simplicial complexes.
</p>projecteuclid.org/euclid.agt/1559095426_20190528220354Tue, 28 May 2019 22:03 EDTOn uniqueness of end sums and $1$–handles at infinityhttps://projecteuclid.org/euclid.agt/1559095427<strong>Jack S Calcut</strong>, <strong>Robert E Gompf</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1299--1339.</p><p><strong>Abstract:</strong><br/>
For oriented manifolds of dimension at least [math] that are simply connected at infinity, it is known that end summing is a uniquely defined operation. Calcut and Haggerty showed that more complicated fundamental group behavior at infinity can lead to nonuniqueness. We examine how and when uniqueness fails. Examples are given, in the categories top, pl and diff, of nonuniqueness that cannot be detected in a weaker category (including the homotopy category). In contrast, uniqueness is proved for Mittag-Leffler ends, and generalized to allow slides and cancellation of (possibly infinite) collections of [math] – and [math] –handles at infinity. Various applications are presented, including an analysis of how the monoid of smooth manifolds homeomorphic to [math] acts on the smoothings of any noncompact [math] –manifold.
</p>projecteuclid.org/euclid.agt/1559095427_20190528220354Tue, 28 May 2019 22:03 EDTThe topology of arrangements of ideal typehttps://projecteuclid.org/euclid.agt/1559095428<strong>Nils Amend</strong>, <strong>Gerhard Röhrle</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1341--1358.</p><p><strong>Abstract:</strong><br/>
In 1962, Fadell and Neuwirth showed that the configuration space of the braid arrangement is aspherical. Having generalized this to many real reflection groups, Brieskorn conjectured this for all finite Coxeter groups. This in turn follows from Deligne’s seminal work from 1972, where he showed that the complexification of every real simplicial arrangement is a [math] –arrangement.
We study the [math] –property for a certain class of subarrangements of Weyl arrangements, the so-called arrangements of ideal type [math] . These stem from ideals [math] in the set of positive roots of a reduced root system. We show that the [math] –property holds for all arrangements [math] if the underlying Weyl group is classical and that it extends to most of the [math] if the underlying Weyl group is of exceptional type. Conjecturally this holds for all [math] . In general, the [math] are neither simplicial nor is their complexification of fiber type.
</p>projecteuclid.org/euclid.agt/1559095428_20190528220354Tue, 28 May 2019 22:03 EDTTopological complexity of unordered configuration spaces of surfaceshttps://projecteuclid.org/euclid.agt/1559095429<strong>Andrea Bianchi</strong>, <strong>David Recio-Mitter</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1359--1384.</p><p><strong>Abstract:</strong><br/>
We determine the topological complexity of unordered configuration spaces on almost all punctured surfaces (both orientable and nonorientable). We also give improved bounds for the topological complexity of unordered configuration spaces on all aspherical closed surfaces, reducing it to three possible values. The main methods used in the proofs were developed in 2015 by Grant, Lupton and Oprea to give bounds for the topological complexity of aspherical spaces. As such this paper is also part of the current effort to study the topological complexity of aspherical spaces and it presents many further examples where these methods strongly improve upon the lower bounds given by zero-divisor cup-length.
</p>projecteuclid.org/euclid.agt/1559095429_20190528220354Tue, 28 May 2019 22:03 EDTHyperbolic extensions of free groups from atoroidal ping-ponghttps://projecteuclid.org/euclid.agt/1559095430<strong>Caglar Uyanik</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1385--1411.</p><p><strong>Abstract:</strong><br/>
We prove that all atoroidal automorphisms of [math] act on the space of projectivized geodesic currents with generalized north–south dynamics. As an application, we produce new examples of nonvirtually cyclic, free and purely atoroidal subgroups of [math] such that the corresponding free group extension is hyperbolic. Moreover, these subgroups are not necessarily convex cocompact.
</p>projecteuclid.org/euclid.agt/1559095430_20190528220354Tue, 28 May 2019 22:03 EDTSymmetric spectra model global homotopy theory of finite groupshttps://projecteuclid.org/euclid.agt/1559095431<strong>Markus Hausmann</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1413--1452.</p><p><strong>Abstract:</strong><br/>
We show that the category of symmetric spectra can be used to model global equivariant homotopy theory of finite groups.
</p>projecteuclid.org/euclid.agt/1559095431_20190528220354Tue, 28 May 2019 22:03 EDTRepresenting the deformation $\infty$–groupoidhttps://projecteuclid.org/euclid.agt/1559095432<strong>Daniel Robert-Nicoud</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1453--1476.</p><p><strong>Abstract:</strong><br/>
Our goal is to introduce a smaller, but equivalent version of the deformation [math] –groupoid associated to a homotopy Lie algebra. In the case of differential graded Lie algebras, we represent it by a universal cosimplicial object.
</p>projecteuclid.org/euclid.agt/1559095432_20190528220354Tue, 28 May 2019 22:03 EDTClassifying spaces from Ore categories with Garside familieshttps://projecteuclid.org/euclid.agt/1559095433<strong>Stefan Witzel</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1477--1524.</p><p><strong>Abstract:</strong><br/>
We describe how an Ore category with a Garside family can be used to construct a classifying space for its fundamental group(s). The construction simultaneously generalizes Brady’s classifying space for braid groups and the Stein–Farley complexes used for various relatives of Thompson’s groups. It recovers the fact that Garside groups have finite classifying spaces.
We describe the categories and Garside structures underlying certain Thompson groups. The indirect product of categories is introduced and used to construct new categories and groups from known ones. As an illustration of our methods we introduce the group braided [math] and show that it is of type [math] .
</p>projecteuclid.org/euclid.agt/1559095433_20190528220354Tue, 28 May 2019 22:03 EDTThe Lannes–Zarati homomorphism and decomposable elementshttps://projecteuclid.org/euclid.agt/1559095434<strong>Ngô A Tuấn</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1525--1539.</p><p><strong>Abstract:</strong><br/>
Let [math] be a pointed CW–complex. The generalized conjecture on spherical classes states that the Hurewicz homomorphism [math] vanishes on classes of [math] of Adams filtration greater than [math] . Let [math] denote the [math] Lannes–Zarati homomorphism for the unstable [math] –module [math] . When [math] , this homomorphism corresponds to an associated graded of the Hurewicz map. An algebraic version of the conjecture states that the [math] Lannes–Zarati homomorphism, [math] , vanishes in any positive stem for [math] and for any unstable [math] –module [math] .
We prove that, for [math] an unstable [math] –module of finite type, the [math] Lannes–Zarati homomorphism, [math] , vanishes on decomposable elements of the form [math] in positive stems, where [math] and [math] with either [math] , [math] and [math] , or [math] , [math] and [math] . Consequently, we obtain a theorem proved by Hung and Peterson in 1998. We also prove that the fifth Lannes–Zarati homomorphism for [math] vanishes on decomposable elements in positive stems.
</p>projecteuclid.org/euclid.agt/1559095434_20190528220354Tue, 28 May 2019 22:03 EDTHomotopy theory of unital algebrashttps://projecteuclid.org/euclid.agt/1559095435<strong>Brice Le Grignou</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 3, 1541--1618.</p><p><strong>Abstract:</strong><br/>
We provide an extensive study of the homotopy theory of types of algebras with units, for instance unital associative algebras or unital commutative algebras. To this purpose, we endow the Koszul dual category of curved coalgebras, where the notion of quasi-isomorphism barely makes sense, with a model category structure Quillen equivalent to that of unital algebras. To prove such a result, we use recent methods based on presentable categories. This allows us to describe the homotopy properties of unital algebras in a simpler and richer way. Moreover, we endow the various model categories with several enrichments which induce suitable models for the mapping spaces and describe the formal deformations of morphisms of algebras.
</p>projecteuclid.org/euclid.agt/1559095435_20190528220354Tue, 28 May 2019 22:03 EDTOn Lagrangian embeddings of closed nonorientable $3$–manifoldshttps://projecteuclid.org/euclid.agt/1566439271<strong>Toru Yoshiyasu</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 4, 1619--1630.</p><p><strong>Abstract:</strong><br/>
We prove that for any compact orientable connected [math] –manifold with torus boundary, a concatenation of it and the direct product of the circle and the Klein bottle with an open [math] –disk removed admits a Lagrangian embedding into the standard symplectic [math] –space. Moreover, the minimal Maslov number of the Lagrangian embedding is equal to [math] .
</p>projecteuclid.org/euclid.agt/1566439271_20190821220137Wed, 21 Aug 2019 22:01 EDTTwisted differential cohomologyhttps://projecteuclid.org/euclid.agt/1566439272<strong>Ulrich Bunke</strong>, <strong>Thomas Nikolaus</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 4, 1631--1710.</p><p><strong>Abstract:</strong><br/>
We construct multiplicative twisted versions of differential cohomology theories for all highly structured ring spectra and twists. We prove existence and give a full classification of differential refinements of twists under mild assumptions. Various concrete examples are discussed and related to earlier approaches.
</p>projecteuclid.org/euclid.agt/1566439272_20190821220137Wed, 21 Aug 2019 22:01 EDTA dynamical characterization of acylindrically hyperbolic groupshttps://projecteuclid.org/euclid.agt/1566439273<strong>Bin Sun</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 4, 1711--1745.</p><p><strong>Abstract:</strong><br/>
We give a dynamical characterization of acylindrically hyperbolic groups. As an application, we prove that nonelementary convergence groups are acylindrically hyperbolic.
</p>projecteuclid.org/euclid.agt/1566439273_20190821220137Wed, 21 Aug 2019 22:01 EDTHyperbolic structures on groupshttps://projecteuclid.org/euclid.agt/1566439274<strong>Carolyn Abbott</strong>, <strong>Sahana H Balasubramanya</strong>, <strong>Denis Osin</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 4, 1747--1835.</p><p><strong>Abstract:</strong><br/>
For every group [math] , we define the set of hyperbolic structures on [math] , denoted by [math] , which consists of equivalence classes of (possibly infinite) generating sets of [math] such that the corresponding Cayley graph is hyperbolic; two generating sets of [math] are equivalent if the corresponding word metrics on [math] are bi-Lipschitz equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded [math] –actions on hyperbolic spaces. We are especially interested in the subset [math] of acylindrically hyperbolic structures on [math] , ie hyperbolic structures corresponding to acylindrical actions. Elements of [math] can be ordered in a natural way according to the amount of information they provide about the group [math] . The main goal of this paper is to initiate the study of the posets [math] and [math] for various groups [math] . We discuss basic properties of these posets such as cardinality and existence of extremal elements, obtain several results about hyperbolic structures induced from hyperbolically embedded subgroups of [math] , and study to what extent a hyperbolic structure is determined by the set of loxodromic elements and their translation lengths.
</p>projecteuclid.org/euclid.agt/1566439274_20190821220137Wed, 21 Aug 2019 22:01 EDTOn negative-definite cobordisms among lens spaces of type $(m,1)$ and uniformization of four-orbifoldshttps://projecteuclid.org/euclid.agt/1566439276<strong>Yoshihiro Fukumoto</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 4, 1837--1880.</p><p><strong>Abstract:</strong><br/>
Connected sums of lens spaces which smoothly bound a rational homology ball are classified by P Lisca. In the classification, there is a phenomenon that a connected sum of a pair of lens spaces [math] appears in one of the typical cases of rational homology cobordisms. We consider smooth negative-definite cobordisms among several disjoint union of lens spaces and a rational homology [math] –sphere to give a topological condition for the cobordism to admit the above “pairing” phenomenon. By using Donaldson theory, we show that if [math] has a certain minimality condition concerning the Chern–Simons invariants of the boundary components, then any [math] must have a counterpart [math] in negative-definite cobordisms with a certain condition only on homology. In addition, we show an existence of a reducible flat connection through which the pair is related over the cobordism. As an application, we give a sufficient condition for a closed smooth negative-definite [math] –orbifold with two isolated singular points whose neighborhoods are homeomorphic to the cones over lens spaces [math] and [math] to admit a finite uniformization.
</p>projecteuclid.org/euclid.agt/1566439276_20190821220137Wed, 21 Aug 2019 22:01 EDTTruncated Heegaard Floer homology and knot concordance invariantshttps://projecteuclid.org/euclid.agt/1566439277<strong>Linh Truong</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 4, 1881--1901.</p><p><strong>Abstract:</strong><br/>
We construct a sequence of smooth concordance invariants [math] defined using truncated Heegaard Floer homology. The invariants generalize the concordance invariants [math] of Ozsváth and Szabó and [math] of Hom and Wu. We exhibit an example in which the gap between two consecutive elements in the sequence [math] can be arbitrarily large. We also prove that the sequence [math] contains more concordance information than [math] , [math] , [math] , [math] and [math] .
</p>projecteuclid.org/euclid.agt/1566439277_20190821220137Wed, 21 Aug 2019 22:01 EDTIntertwining for semidirect product operadshttps://projecteuclid.org/euclid.agt/1566439278<strong>Benjamin C Ward</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 4, 1903--1934.</p><p><strong>Abstract:</strong><br/>
We show that the semidirect product construction for [math] –operads and the levelwise Borel construction for [math] –cooperads are intertwined by the topological operadic bar construction. En route we give a generalization of the bar construction of M Ching from reduced to certain nonreduced topological operads.
</p>projecteuclid.org/euclid.agt/1566439278_20190821220137Wed, 21 Aug 2019 22:01 EDTInfinite staircases in the symplectic embedding problem for four-dimensional ellipsoids into polydiskshttps://projecteuclid.org/euclid.agt/1566439279<strong>Michael Usher</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 4, 1935--2022.</p><p><strong>Abstract:</strong><br/>
We study the symplectic embedding capacity function [math] for ellipsoids [math] into dilates of polydisks [math] as both [math] and [math] vary through [math] . For [math] , Frenkel and Müller showed that [math] has an infinite staircase accumulating at [math] , while for integer [math] , Cristofaro-Gardiner, Frenkel and Schlenk found that no infinite staircase arises. We show that for arbitrary [math] , the restriction of [math] to [math] is determined entirely by the obstructions from Frenkel and Müller’s work, leading [math] on this interval to have a finite staircase with the number of steps tending to [math] as [math] . On the other hand, in contrast to the results of Cristofaro-Gardiner, Frenkel and Schlenk, for a certain doubly indexed sequence of irrational numbers [math] we find that [math] has an infinite staircase; these [math] include both numbers that are arbitrarily large and numbers that are arbitrarily close to [math] , with the corresponding accumulation points respectively arbitrarily large and arbitrarily close to [math] .
</p>projecteuclid.org/euclid.agt/1566439279_20190821220137Wed, 21 Aug 2019 22:01 EDTBredon cohomology and robot motion planninghttps://projecteuclid.org/euclid.agt/1566439280<strong>Michael Farber</strong>, <strong>Mark Grant</strong>, <strong>Gregory Lupton</strong>, <strong>John Oprea</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 4, 2023--2059.</p><p><strong>Abstract:</strong><br/>
We study the topological invariant [math] reflecting the complexity of algorithms for autonomous robot motion. Here, [math] stands for the configuration space of a system and [math] is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in [math] . We focus on the case when the space [math] is aspherical; then the number [math] depends only on the fundamental group [math] and we denote it by [math] . We prove that [math] can be characterised as the smallest integer [math] such that the canonical [math] –equivariant map of classifying spaces
E
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π
)
can be equivariantly deformed into the [math] –dimensional skeleton of [math] . The symbol [math] denotes the classifying space for free actions and [math] denotes the classifying space for actions with isotropy in the family [math] of subgroups of [math] which are conjugate to the diagonal subgroup. Using this result we show how one can estimate [math] in terms of the equivariant Bredon cohomology theory. We prove that [math] , where [math] denotes the cohomological dimension of [math] with respect to the family of subgroups [math] . We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion-free hyperbolic groups as well as all torsion-free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher (2017) are exactly the classes having Bredon cohomology extensions with respect to the family [math] .
</p>projecteuclid.org/euclid.agt/1566439280_20190821220137Wed, 21 Aug 2019 22:01 EDTRepresenting a point and the diagonal as zero loci in flag manifoldshttps://projecteuclid.org/euclid.agt/1566439281<strong>Shizuo Kaji</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 4, 2061--2075.</p><p><strong>Abstract:</strong><br/>
The zero locus of a generic section of a vector bundle over a manifold defines a submanifold. A classical problem in geometry asks to realise a specified submanifold in this way. We study two cases: a point in a generalised flag manifold and the diagonal in the direct product of two copies of a generalised flag manifold. These cases are particularly interesting since they are related to ordinary and equivariant Schubert polynomials, respectively.
</p>projecteuclid.org/euclid.agt/1566439281_20190821220137Wed, 21 Aug 2019 22:01 EDTBoundaries of Baumslag–Solitar groupshttps://projecteuclid.org/euclid.agt/1566439282<strong>Craig R Guilbault</strong>, <strong>Molly A Moran</strong>, <strong>Carrie J Tirel</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 4, 2077--2097.</p><p><strong>Abstract:</strong><br/>
A [math] –structure on a group [math] was introduced by Bestvina in order to extend the notion of a group boundary beyond the realm of CAT(0) and hyperbolic groups. A refinement of this notion, introduced by Farrell and Lafont, includes a [math] –equivariance requirement, and is known as an [math] –structure. The general questions of which groups admit [math] – or [math] –structures remain open. Here we show that all Baumslag–Solitar groups admit [math] –structures and all generalized Baumslag–Solitar groups admit [math] –structures.
</p>projecteuclid.org/euclid.agt/1566439282_20190821220137Wed, 21 Aug 2019 22:01 EDTSemisimplicial spaceshttps://projecteuclid.org/euclid.agt/1566439283<strong>Johannes Ebert</strong>, <strong>Oscar Randal-Williams</strong>. <p><strong>Source: </strong>Algebraic & Geometric Topology, Volume 19, Number 4, 2099--2150.</p><p><strong>Abstract:</strong><br/>
This is an exposition of homotopical results on the geometric realisation of semisimplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The topics considered include: fibrancy conditions on topological categories; the effect on classifying spaces of freely adjoining units; approximate notions of units; Quillen’s Theorems A and B for nonunital topological categories; the effect on classifying spaces of changing the topology on the space of objects; the group-completion theorem.
</p>projecteuclid.org/euclid.agt/1566439283_20190821220137Wed, 21 Aug 2019 22:01 EDT