Equivariant dendroidal Segal spaces and $G$–$\infty$–operads
Peter Bonventre, Luís A Pereira
KEYWORDS: operads, dendroidal sets, preoperads, equivariant homotopy theory, Reedy categories, 55U10, 55U35, 55U40, 18G30
We introduce the analogues of the notions of complete Segal space and of Segal category in the context of equivariant operads with norm maps, and build model categories with these as the fibrant objects. We then show that these model categories are Quillen equivalent to each other and to the model category for G–∞–operads built in a previous paper.
Moreover, we establish variants of these results for the Blumberg–Hill indexing systems.
In an appendix, we discuss Reedy categories in the equivariant context.
Horizontal vector fields and Seifert fiberings
Andy Hammerlindl
KEYWORDS: Seifert fiberings, vector fields, lens spaces, 55R05, 57R18, 57R25
We give a classification of the topology of vector fields which are nowhere tangent to the fibers of a Seifert fibering.
Reidemeister torsion form on character varieties
Léo Bénard
KEYWORDS: Reidemeister torsion, Character varieties, $3$–manifolds, Culler–Shalen theory, 57M25, 57M27
We define the adjoint Reidemeister torsion as a differential form on the character variety of a compact oriented 3–manifold with toral boundary, and prove it defines a rational volume form. Then we show that the torsion form has poles only at singular points of the character variety. In fact, if the singular point corresponds to a reducible character, we show that the torsion has no pole under a generic hypothesis on the Alexander polynomial; otherwise, we relate the order of the pole with the type of singularity. Finally we consider the ideal points added after compactification of the character variety. We bound the vanishing order of the torsion by the Euler characteristic of an essential surface associated to the ideal point by the Culler–Shalen theory. As a corollary we obtain an unexpected relation between the topology of those surfaces and the topology of the character variety.
Some results related to finiteness properties of groups for families of subgroups
Timm von Puttkamer, Xiaolei Wu
KEYWORDS: finiteness properties of groups for families of subgroups, Artin groups, conjugacy growth, CAT(0) cube group, virtually cyclic groups, poly-$\mathbb{Z}$–groups, 20B07, 20J05
Let E¯¯G be the classifying space of G for the family of virtually cyclic subgroups. We show that an Artin group admits a finite model for E¯¯G if and only if it is virtually cyclic. This solves a conjecture of Juan-Pineda and Leary and a question of Lück, Reich, Rognes and Varisco for Artin groups. We then study conjugacy growth of CAT(0) groups and show that if a CAT(0) group contains a free abelian group of rank two, its conjugacy growth is strictly faster than linear. This also yields an alternative proof for the fact that a CAT(0) cube group admits a finite model for E¯¯G if and only if it is virtually cyclic. Our last result deals with the homotopy type of the quotient space B¯¯G=E¯¯G∕G. We show, for a poly-ℤ–group G, that B¯¯G is homotopy equivalent to a finite CW–complex if and only if G is cyclic.
$\mathcal{E}_n$–Hopf invariants
Felix Wierstra
KEYWORDS: Hopf invariant, $E_n$–algebras, suspensions, 18D50, 55P40, 55P48, 55Q25
The classical Hopf invariant is an invariant of homotopy classes of maps from S4n−1 to S2n, and is an important invariant in homotopy theory. The goal of this paper is to use the Koszul duality theory for ℰn–operads to define a generalization of the classical Hopf invariant. One way of defining the classical Hopf invariant is by defining a pairing between the cohomology of the associative bar construction on the cochains of a space X and the homotopy groups of X. We will give a generalization of the classical Hopf invariant by defining a pairing between the cohomology of the ℰn–bar construction on the cochains of X and the homotopy groups of X. This pairing gives us a set of invariants of homotopy classes of maps from Sm to a simplicial set X; this pairing can detect more homotopy classes of maps than the classical Hopf invariant.
The second part of the paper is devoted to combining the ℰn–Hopf invariants with the Koszul duality theory for ℰn–operads to get a relation between the ℰn–Hopf invariants of a space X and the ℰn+1–Hopf invariants of the suspension of X. This is done by studying the suspension morphism for the ℰ∞–operad, which is a morphism from the ℰ∞–operad to the desuspension of the ℰ∞–operad. We show that it induces a functor from ℰ∞–algebras to ℰ∞–algebras, which has the property that it sends an ℰ∞–model for a simplicial set X to an ℰ∞–model for the suspension of X.
We use this result to give a relation between the ℰn–Hopf invariants of maps from Sm into X and the ℰn+1–Hopf invariants of maps from Sm+1 into the suspension of X. One of the main results we show here is that this relation can be used to define invariants of stable homotopy classes of maps.
Space of isospectral periodic tridiagonal matrices
Anton Ayzenberg
KEYWORDS: isospectral space, matrix spectrum, Toda flow, periodic tridiagonal matrix, discrete Schrödinger operator, permutohedral tiling, simplicial poset, face ring, equivariant cohomology, torus action, crystallization, 34L40, 52B70, 52C22, 55N91, 57R91, 05E45, 13F55, 14H70, 15A18‎, 37C80, 37K10, 51M20, 55R80, 55T10
A periodic tridiagonal matrix is a tridiagonal matrix with an additional two entries at the corners. We study the space Xn,λ of Hermitian periodic tridiagonal n×n matrices with a fixed simple spectrum λ. Using the discretized Schrödinger operator we describe all spectra λ for which Xn,λ is a topological manifold. The space Xn,λ carries a natural effective action of a compact (n−1)–torus. We describe the topology of its orbit space and, in particular, show that whenever the isospectral space is a manifold, its orbit space is homeomorphic to S4×Tn−3. There is a classical dynamical system: the flow of the periodic Toda lattice, acting on Xn,λ. Except for the degenerate locus Xn,λ0, the Toda lattice exhibits Liouville–Arnold behavior, so that the space Xn,λ∖Xn,λ0 is fibered into tori. The degenerate locus of the Toda system is described in terms of combinatorial geometry: its structure is encoded in the special cell subdivision of a torus, which is obtained from the regular tiling of the euclidean space by permutohedra. We apply methods of commutative algebra and toric topology to describe the cohomology and equivariant cohomology modules of Xn,λ.
The cohomology rings of the unordered configuration spaces of elliptic curves
Roberto Pagaria
KEYWORDS: configuration space, cohomology ring, Elliptic curve, Torus, representation theory, 55R80
We study the cohomology ring of the configuration space of unordered points in the two-dimensional torus. In particular, we compute the mixed Hodge structure on the cohomology, the action of the mapping class group, the structure of the cohomology ring and we prove the formality over the rationals.
Motivic random variables and representation stability, I: Configuration spaces
Sean Howe
KEYWORDS: representation stability, motivic stabilization, arithmetic statistics, configuration spaces, cohomological stability., 14G10, 18F30, 55R80
We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over ℂ attached to character polynomials. Our approach interprets the stabilization as a probabilistic phenomenon based on the asymptotic independence of certain motivic random variables, and gives explicit universal formulas for the limits in terms of the exponents of a motivic Euler product for the Kapranov zeta function. The result can be thought of as a weak but explicit version of representation stability for the cohomology of ordered configuration spaces. In the sequel, we find similar stability results in spaces of smooth hypersurface sections, providing new examples to be investigated through the lens of representation stability for symmetric, symplectic and orthogonal groups.
Section problems for configurations of points on the Riemann sphere
Lei Chen, Nick Salter
KEYWORDS: spherical braid group, configuration space, section, canonical reduction system, 20F36, 55S40
We prove a suite of results concerning the problem of adding m distinct new points to a configuration of n distinct points on the Riemann sphere, such that the new points depend continuously on the old. Altogether, these results provide a complete answer to the following question: given n≠5, for which m can one continuously add m points to a configuration of n points? For n≥6, we find that m must be divisible by n(n−1)(n−2), and we provide a construction based on the idea of cabling of braids. For n=3,4, we give some exceptional constructions based on the theory of elliptic curves.
Trees, dendrites and the Cannon–Thurston map
Elizabeth Field
KEYWORDS: Cannon–Thurston map, hyperbolic group, algebraic lamination, dendrite, Gromov boundary, 20F65, 20E07, 20F67, 57M07
When 1→H→G→Q→1 is a short exact sequence of three word-hyperbolic groups, Mahan Mj (formerly Mitra) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon–Thurston map. In this context, Mj associates to every point z in the Gromov boundary of Q an "ending lamination" on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich and Lustig and Dowdall, Kapovich and Taylor, who prove that in the case where H is a free group and Q is a convex cocompact purely atoroidal subgroup of Out(FN), one can identify the resultant quotient space with a certain ℝ–tree in the boundary of Culler and Vogtmann's Outer space.
Finite rigid sets in arc complexes
Emily Shinkle
KEYWORDS: arc complex, rigidity, finite rigid, arc, simplicial complex, geometric topology, 20F38, 20F65, 57M60, 57N05
For any compact, connected, orientable, finite-type surface with marked points other than the sphere with three marked points, we construct a finite rigid set of its arc complex: a finite simplicial subcomplex of its arc complex such that any locally injective map of this set into the arc complex of another surface with arc complex of the same or lower dimension is induced by a homeomorphism of the surfaces, unique up to isotopy in most cases. It follows that if the arc complexes of two surfaces are isomorphic, the surfaces are homeomorphic. We also give an exhaustion of the arc complex by finite rigid sets. This extends the results of Irmak and McCarthy (Turkish J. Math. 34 (2010) 339–354).
A relative $2$–nerve
Fernando Abellán García, Tobias Dyckerhoff, Walker H Stern
KEYWORDS: relative nerve, Grothendieck construction, bicategories, 18D30, 18E35, 18G30, 18G55
We introduce a 2–categorical variant of Lurie's relative nerve functor. We prove that it defines a right Quillen equivalence which, upon passage to ∞–categorical localizations, corresponds to Lurie's scaled unstraightening equivalence. In this ∞–bicategorical context, the relative 2–nerve provides a computationally tractable model for the Grothendieck construction which becomes equivalent, via an explicit comparison map, to Lurie's relative nerve when restricted to 1–categories.
Small $C^1$ actions of semidirect products on compact manifolds
Christian Bonatti, Sang-hyun Kim, Thomas Koberda, Michele Triestino
KEYWORDS: groups acting on manifolds, hyperbolic dynamics, fibered $3$–manifold, $C^1$–close to the identity, 37C85, 57M60, 20E22, 37D30, 57M50, 57R35
Let T be a compact fibered 3–manifold, presented as a mapping torus of a compact, orientable surface S with monodromy ψ, and let M be a compact Riemannian manifold. Our main result is that if the induced action ψ∗ on H1(S,ℝ) has no eigenvalues on the unit circle, then there exists a neighborhood 𝒰 of the trivial action in the space of C1 actions of π1(T) on M such that any action in 𝒰 is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group H provided that the conjugation action of the cyclic group on H1(H,ℝ)≠0 has no eigenvalues of modulus one. We thus generalize a result of A McCarthy, which addressed the case of abelian-by-cyclic groups acting on compact manifolds.
Correction to the article Floer homology and splicing knot complements
Eaman Eftekhary
KEYWORDS: Floer homology, Splicing, essential torus, 57M27, 57R58
This note corrects the mistakes in the splicing formulas of the paper "Floer homology and splicing knot complements" (Algebr. Geom. Topol. 15 (2015) 3155–3213). The mistakes are the result of the incorrect assumption that for a knot K inside a homology sphere Y, the involution on HFK̂(K) which corresponds to moving the basepoints by one full twist around K is trivial. The correction implicitly involves considering the contribution from this (possibly nontrivial) involution in a number of places.