Showing papers in "Algebraic & Geometric Topology in 2016"

DAHA and iterated torus knots

Abstract: The theory of DAHA-Jones polynomials is extended from torus knots to their arbitrary iterations (for any reduced root systems and weights), which includes the polynomiality, duality and other properties of the DAHA superpolynomials. Presumably they coincide with the reduced stable Khovanov–Rozansky polynomials in the case of nonnegative coefficients. The new theory matches well the classical theory of algebraic knots and (unibranch) plane curve singularities; the Puiseux expansion naturally emerges. The corresponding DAHA superpolynomials are expected to coincide with the reduced ones in the Oblomkov–Shende–Rasmussen conjecture upon its generalization to arbitrary dominant weights. For instance, the DAHA uncolored superpolynomials at a = 0, q = 1 are conjectured to provide the Betti numbers of the Jacobian factors (compactified Jacobians) of the corresponding singularities.

Universality of multiplicative infinite loop space machines

Abstract: We establish a canonical and unique tensor product for commutative monoids and groups in an1‐category C which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that En ‐(semi)ring objects in C give rise to En ‐ring spectrum objects in C . In the case that C is the1‐category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic K‐theory of rings and ring spectra. The main tool we use to establish these results is the theory of smashing localizations of presentable 1‐categories. In particular, we identify preadditive and additive 1‐categories as the local objects for certain smashing localizations. A central theme is the stability of algebraic structures under basechange; for example, we show Ring.D C/’ Ring.D/ C . Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in1‐categories. 55P48; 55P43, 19D23

Obstructions to Lagrangian concordance

Abstract: We investigate the question of the existence of a Lagrangian concordance between two Legendrian knots in ℝ3. In particular, we give obstructions to a concordance from an arbitrary knot to the standard Legendrian unknot, in terms of normal rulings. We also place strong restrictions on knots that have concordances both to and from the unknot and construct an infinite family of knots with nonreversible concordances from the unknot. Finally, we use our obstructions to present a complete list of knots with up to 14 crossings that have Legendrian representatives that are Lagrangian slice.

Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial

Abstract: This is the first in a series of papers studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc). These are classes of knotted objects which are wider, but weaker than their “usual” counterparts. The group of w-braids was studied (under the name “welded braids”) by Fenn, Rimanyi and Rourke and was shown to be isomorphic to the McCool group of “basisconjugating” automorphisms of a free group Fn : the smallest subgroup of Aut.Fn/ that contains both braids and permutations. Brendle and Hatcher, in work that traces back to Goldsmith, have shown this group to be a group of movies of flying rings in R 3 . Satoh studied several classes of w-knotted objects (under the name “weaklyvirtual”) and has shown them to be closely related to certain classes of knotted surfaces in R 4 . So w-knotted objects are algebraically and topologically interesting. Here we study finite-type invariants of w-braids and w-knots. Following Berceanu and Papadima, we construct homomorphic universal finite-type invariants of w-braids. The universal finite-type invariant of w-knots is essentially the Alexander polynomial. Much as the spaces A of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, the spaces A w of “arrow diagrams” for w-knotted objects are related to not-necessarily-metrized Lie algebras. Many questions concerning w-knotted objects turn out to be equivalent to questions about Lie algebras. Later in this paper series we re-interpret the work of Alekseev and Torossian on Drinfel’d associators and the Kashiwara‐Vergne problem as a study of w-knotted trivalent graphs. 57M25, 57Q45

Morse theory for manifolds with boundary

Abstract: We develop Morse theory for manifolds with boundary. Besides standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that, under a topological assumption, a critical point in the interior of a Morse function can be moved to the boundary, where it splits into a pair of boundary critical points. As an application, we prove that every cobordism of manifolds with boundary splits as a union of left product cobordisms and right product cobordisms.

On the homotopy of Q(3) and Q(5) at the prime 2

Abstract: We study modular approximations Q(l), l = 3,5, of the K(2)-local sphere at the prime 2 that arise from l-power degree isogenies of elliptic curves. We develop Hopf algebroid level tools for working with Q(l) and record Hill, Hopkins, and Ravenel's computation of the homotopy groups of TMF_0(5). Using these tools and formulas of Mahowald and Rezk for Q(3) we determine the image of Shimomura's 2-primary divided beta-family in the Adams-Novikov spectral sequences for Q(3) and Q(5). Finally, we use low-dimensional computations of the homotopy of Q(3) and Q(5) to explore the role of these spectra as approximations to the K(2)-local sphere.

Volume bounds for weaving knots

Abstract: Weaving knots are alternating knots with the same projection as torus knots, and were conjectured by X.-S. Lin to be among the maximum volume knots for fixed crossing number. We provide the first asymptotically correct volume bounds for weaving knots, and we prove that the infinite weave is their geometric limit.

Bordered Heegaard Floer Homology and Graph Manifolds

Abstract: We perform two explicit computations of bordered Heegaard Floer invariants. The first is the type D trimodule associated to the trivial S^1 bundle over the pair of pants P. The second is a bimodule that is necessary for self-gluing, when two torus boundary components of a bordered manifold are glued to each other. Using the results of these two computations, we describe an algorithm for computing HF-hat of any graph manifold.

Satellite operators as group actions on knot concordance

Abstract: introduce a generalization of patterns that form a group (unlike traditional patterns), modulo a generalization of concordance. Generalized patterns induce functions, called generalized satellite operators, on concordance classes of knots in homology spheres; using this we recover the recent result of Cochran and the authors that patterns with strong winding number 1 induce injective satellite operators on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4‐dimensional Poincare conjecture. We also obtain a characterization of patterns inducing surjective satellite operators, as well as a sufficient condition for a generalized pattern to have an inverse. As a consequence, we are able to construct infinitely many nontrivial patterns P such that there is a pattern P for which P.P.K// is concordant to K (topologically as well as smoothly in a potentially exotic S 3 a0;1c) for all knots K ; we show that these patterns are distinct from all connected-sum patterns, even up to concordance, and that they induce bijective satellite operators on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4‐dimensional Poincare conjecture. 57M25

Geometry of the SL(3,ℂ)-character variety of torus knots.

Abstract: Let G be the fundamental group of the complement of the torus knot of type (m, n). It has a presentation G = . We find a geometric description of the character variety X(G) of characters of representations of G into SL(3,ℂ), GL(3,ℂ) and PGL(3,ℂ).

McCool groups of toral relatively hyperbolic groups

Abstract: The outer automorphism group Out(G) of a group G acts on the set of conjugacy classes of elements of G. McCool proved that the stabilizer $Mc(c_1,...,c_n)$ of a finite set of conjugacy classes is finitely presented when G is free. More generally, we consider the group $Mc(H_1,...,H_n)$ of outer automorphisms $\Phi$ of G acting trivially on a family of subgroups $H_i$, in the sense that $\Phi$ has representatives $\alpha_i$ with $\alpha_i$ equal to the identity on $H_i$. When G is a toral relatively hyperbolic group, we show that these two definitions lead to the same subgroups of Out(G), which we call "McCool groups" of G. We prove that such McCool groups are of type VF (some finite index subgroup has a finite classifying space). Being of type VF also holds for the group of automorphisms of G preserving a splitting of G over abelian groups. We show that McCool groups satisfy a uniform chain condition: there is a bound, depending only on G, for the length of a strictly decreasing sequence of McCool groups of G. Similarly, fixed subgroups of automorphisms of G satisfy a uniform chain condition.

The algebraic duality resolution at p = 2

Abstract: The goal of this paper is to develop some of the machinery necessary for doing $K(2)$-local computations in the stable homotopy category using duality resolutions at the prime $p=2$. The Morava stabilizer group $\mathbb{S}_2$ admits a norm whose kernel we denote by $\mathbb{S}_2^1$. The algebraic duality resolution is a finite resolution of the trivial $\mathbb{Z}_2[[\mathbb{S}_2^1]]$-module $\mathbb{Z}_2$ by modules induced from representations of finite subgroups of $\mathbb{S}_2^1$. Its construction is due to Goerss, Henn, Mahowald and Rezk. It is an analogue of their finite resolution of the trivial $\mathbb{Z}_3[[\mathbb{G}_2^1]]$-module $\mathbb{Z}_3$ at the prime $p=3$. The construction was never published and it is the main result in this paper. In the process, we give a detailed description of the structure of Morava stabilizer group $\mathbb{S}_2$ at the prime $2$. We also describe the maps in the algebraic duality resolution with the precision necessary for explicit computations.

Splitting line patterns in free groups

Abstract: We construct a boundary of a finite rank free group relative to a finite list of conjugacy classes of maximal cyclic subgroups. From the cut points and uncrossed cut pairs of this boundary we construct a simplicial tree on which the group acts cocompactly. We show that the quotient graph of groups is the JSJ decomposition of the group relative to the given collection of conjugacy classes. This provides a characterization of virtually geometric multiwords: they are the multiwords that are built from geometric pieces. In particular, a multiword is virtually geometric if and only if the relative boundary is planar.

The homotopy types of PU(3)– and PSp(2)–gauge groups

Abstract: Let G be a compact connected simple Lie group. Any principal G–bundle over S4 is classified by an integer k ∈ ℤ≅π3(G), and we denote the corresponding gauge group by Gk(G). We prove that Gk(PU(3)) ≃Gl(PU(3)) if and only if (24,k) = (24,l), and Gk(PSp(2)) ≃(p)Gl(PSp(2)) for any prime p if and only if (40,k) = (40,l), where (m,n) is the gcd of integers m,n.

Quasiflats in CAT.0/ 2-complexes

Abstract: We show that if X is a piecewise Euclidean 2‐complex with a cocompact isometry group, then every 2‐quasiflat in X is at finite Hausdorff distance from a subset Q which is locally flat outside a compact set, and asymptotically conical 20F65

Cup products, the Johnson homomorphism and surface bundles over surfaces with multiple fiberings

Abstract: Let Σg→E→Σh be a surface bundle over a surface with monodromy representation ρ:π1Σh→ Mod(Σg) contained in the Torelli group ℐg. We express the cup product structure in H∗(E,ℤ) in terms of the Johnson homomorphism τ:ℐg→∧3(H1(Σg,ℤ))∕H1(Σg,ℤ). This is applied to the question of obtaining an upper bound on the maximal n such that p1:E→Σh1,…,pn:E→Σhn are fibering maps realizing E as the total space of a surface bundle over a surface in n distinct ways. We prove that any nontrivial surface bundle over a surface with monodromy contained in the Johnson kernel Kg fibers in a unique way.

Homotopy theory of G-diagrams and equivariant excision

Abstract: Let $G$ be a finite group acting on a small category $I$. We study functors $X \colon I \to \mathscr{C}$ equipped with families of compatible natural transformations that give a kind of generalized $G$-action on $X$. Such objects are called $G$-diagrams. When $\mathscr{C}$ is a sufficiently nice model category we define a model structure on the category of $G$-diagrams in $\mathscr{C}$. There are natural $G$-actions on Bousfield-Kan style homotopy limits and colimits of $G$-diagrams. We prove that weak equivalences between point-wise (co)fibrant $G$-diagrams induce weak $G$-equivalences on homotopy (co)limits. A case of particular interest is when the indexing category is a cube. We use homotopy limits and colimits over such diagrams to produce loop and suspension spaces with respect to permutation representations of $G$. We go on to develop a theory of enriched equivariant homotopy functors and give an equivariant "linearity" condition in terms of cubical $G$-diagrams. In the case of $G$-topological spaces we prove that this condition is equivalent to Blumberg's notion of $G$-linearity. In particular we show that the Wirthm\"{u}ller isomorphism theorem is a direct consequence of the equivariant linearity of the identity functor on $G$-spectra.

The $L^2$–Alexander torsion is symmetric

Abstract: We show that the L2–Alexander torsion of a 3–manifold is a symmetric function. This can be viewed as a generalization of the symmetry of the Alexander polynomial of a knot.

Relative left properness of colored operads

Abstract: The categories of C-colored symmetric operads admits a cofibrantly generated model category structure. In this paper, we show that this model structure satisfies a weaker version of left properness. We also provide an example of Dwyer which shows that this category is not left proper.

Links of plane curve singularities are $L$–space links

Abstract: We prove that a sufficiently large surgery on any algebraic link is an L-space. For torus links we give a complete classification of integer surgery torus links we give a complete classification of integer surgery coefficients providing L-spaces. © 2016, Mathematical Sciences Publishers. All rights reserved.

Homotopy invariants of covers and KKM-type lemmas

Abstract: Given any (open or closed) cover of a space T , we associate certain homotopy classes of maps from T to n–spheres. These homotopy invariants can then be considered as obstructions for extending covers of a subspace A X to a cover of all of X . We use these obstructions to obtain generalizations of the classic KKM (Knaster– Kuratowski–Mazurkiewicz) and Sperner lemmas. In particular, we show that in the case when A is a k –sphere and X is a .kC 1/–disk there exist KKM-type lemmas for covers by nC 2 sets if and only if the homotopy group k.S/ is nontrivial.

On certain Lagrangian submanifolds of S2 ×S2 and ℂPn

Abstract: We consider various constructions of monotone Lagrangian submanifolds of ℂ Pn, S2 × S2, and quadric hypersurfaces of ℂ Pn. In S2 × S2 and ℂ P2 we show that several different known constructions of exotic monotone tori yield results that are Hamiltonian isotopic to each other, in particular answering a question of Wu by showing that the monotone fiber of a toric degeneration model of ℂ P2 is Hamiltonian isotopic to the Chekanov torus. Generalizing our constructions to higher dimensions leads us to consider monotone Lagrangian submanifolds (typically not tori) of quadrics and of ℂ Pn which can be understood either in terms of the geodesic flow on T∗Sn or in terms of the Biran circle bundle construction. Unlike previously known monotone Lagrangian submanifolds of closed simply connected symplectic manifolds, many of our higher-dimensional Lagrangian submanifolds are provably displaceable.

Homological stability for families of Coxeter groups

Abstract: We prove that certain families of Coxeter groups and inclusions $W_1\hookrightarrow W_2\hookrightarrow...$ satisfy homological stability, meaning that in each degree the homology $H_\ast(BW_n)$ is eventually independent of $n$. This gives a uniform treatment of homological stability for the families of Coxeter groups of type $A_n$, $B_n$ and $D_n$, recovering existing results in the first two cases, and giving a new result in the third. The key step in our proof is to show that a certain simplicial complex with $W_n$-action is highly connected. To do this we show that the barycentric subdivision is an instance of the 'basic construction', and then use Davis's description of the basic construction as an increasing union of chambers to deduce the required connectivity.

Bridge distance and plat projections

Abstract: We calculate the bridge distance for $m$-bridge knots/links in the $3$-sphere with sufficiently complicated $2m$-plat projections. In particular we show that if the underlying braid of the plat has $n - 1$ rows of twists and all its exponents have absolute value greater than or equal to three then the distance of the bridge sphere is exactly $\lceil n/(2(m - 2)) \rceil$, where $\lceil x \rceil$ is the smallest integer greater than or equal to $x$. As a corollary, we conclude that if such a diagram has more than $4m(m-2)$ rows then the bridge sphere defining the plat projection is the unique minimal bridge sphere for the knot.

An annular refinement of the transverse element in Khovanov homology

Abstract: We construct a braid conjugacy class invariant $\kappa$ by refining Plamenevskaya's transverse element $\psi$ in Khovanov homology via the annular grading. While $\kappa$ is not an invariant of transverse links, it distinguishes some braids whose closures share the same classical invariants but are not transversely isotopic. Using $\kappa$ we construct an obstruction to negative destabilization (stronger than $\psi$) and a solution to the word problem in braid groups. Also, $\kappa$ is a lower bound on the length of the spectral sequence from annular Khovanov homology to Khovanov homology, and we obtain concrete examples in which this spectral sequence does not collapse immediately. In addition, we study these constructions in reduced Khovanov homology and illustrate that the two reduced versions are fundamentally different with respect to the annular filtration.

A type A structure in Khovanov homology

Abstract: We describe the first part of a gluing theory for the bigraded Khovanov homology with integer coefficients. This part associates a type D structure to a tangle properly embedded in a half-space and proves that the homotopy class of the type D structure is an invariant of the isotopy class of the tangle. The construction is modeled off bordered Heegaard-Floer homology, but uses only combinatorial/diagrammatic methods

Exactly fourteen intrinsically knotted graphs have 21 edges

Abstract: Throughout the article we will take an embedded graph to mean a graph embedded in R . We call a graph G intrinsically knotted if every embedding of the graph contains a knotted cycle. Conway and Gordon [2] showed that K7 , the complete graph with seven vertices, is an intrinsically knotted graph. A graph H is minor of another graph G if it can be obtained from G by contracting or deleting some edges. An intrinsically knotted graph is minor minimal intrinsically knotted provided no proper minor is intrinsically knotted. Robertson and Seymour [9] proved that there are only finite minor minimal intrinsically knotted graphs, but finding the complete set of them is still an open problem. However, it is well known that K7 and the thirteen graphs obtained from this graph by rY moves are minor minimal intrinsically knotted; see Conway and Gordon [2], and Kohara and Suzuki [6].

Centralizers in good groups are good

Abstract: We modify the transchromatic character maps of [Sta13] to land in a faithfully flat extension of Morava E-theory. Our construction makes use of the interaction between topological and algebraic localization and completion. As an application we prove that centralizers of tuples of commuting prime-power order elements in good groups are good and we compute a new example.

Spin structures on loop spaces that characterize string manifolds

Abstract: Classically, a spin structure on the loop space of a manifold is a lift of the structure group of the looped frame bundle from the loop group to its universal central extension. Heuristically, the loop space of a manifold is spin if and only if the manifold itself is a string manifold, against which it is well-known that only the if-part is true in general. In this article we develop a new version of spin structures on loop spaces that exists if and only if the manifold is string, as desired. This new version consists of a classical spin structure plus a certain fusion product related to loops of frames in the manifold. We use the lifting gerbe theory of Carey-Murray, recent results of Stolz-Teichner on loop spaces, and some own results about string geometry and Brylinski-McLaughlin transgression.

Equivalence classes of augmentations and Morse complex sequences of Legendrian knots

Abstract: Let L be a Legendrian knot in R 3 with the standard contact structure. In earlier work of Henry, a map was constructed from equivalence classes of Morse complex sequences for L, which are combinatorial objects motivated by generating families, to homotopy classes of augmentations of the Legendrian contact homology algebra of L. Moreover, this map was shown to be a surjection. We show that this correspondence is, in fact, a bijection. As a corollary, homotopic augmentations determine the same graded normal ruling of L and have isomorphic linearized contact homology groups. A second corollary states that the count of equivalence classes of Morse complex sequences of a Legendrian knot is a Legendrian isotopy invariant. 57R17; 57M25, 53D40