Topic overview
Topic snapshot
Next steps
The graph preview below keeps the nearby papers, people and communities visible in the same reading flow.
Topic graph
Inspect nearby papers, researchers, institutions and communities without opening a separate graph page.
BZPEER is loading the nearby papers, people, topics and institutions for this page.
Papers in this area
We give an elementary proof of a theorem that characterizes quasisymmetric maps of the unit circle in terms of shear coordinates on the Farey tesselation. The proof only uses the normal family argument for quasisymmetric maps and some elementary hyperbolic geometry.
This article in memory of Patrick Dehornoy (1952-2019) is an invitation to Garside theory for mainstream geometric group theorists interested in mapping class groups, curve complexes, and the geometry of Artin-Tits groups.
In this note, we provide a generalization for the definition of a trisection of a 4-manifold with boundary. We demonstrate the utility of this more general definition by finding a trisection diagram for the Cacime Surface, and also by finding a trisection-theoretic way to perform logarithmic surgery. In addition, we describe how to perform 1-surgery on closed trisections. The insight gained from this description leads us to the classification of an infinite family of genus three trisections. We include an appendix where we extend two classic results for relative trisections for the case when the trisection surface is closed.
The virtual braid groups are generalizations of the classical braid groups. This paper gives an elementary proof that the classical braid group injects into the virtual braid group over the same number of strands.
The present paper show that Conway-Coxeter friezes of zigzag type are characterized by (unoriented) rational links. As an application of this characterization Jones polynomial can be defined for Conway-Coxeter friezes of zigzag type. This gives a new method for computing the Jones polynomial for oriented rational links.
In this paper, a generalized cusp is a properly convex manifold with strictly convex boundary that is diffeomorphic to $M \times [0, \infty)$ where $M$ is a closed Euclidean manifold. These are classified in [2]. The marked moduli space is homeomorphic to a subspace of the space of conjugacy classes of representations of $π_1(M)$. It has one description as a generalization of a trace-variety, and another description involving weight data that is similar to that used to describe semi-simple Lie groups. It is also a bundle over the space of Euclidean similarity (conformally flat) structures on $M$, and the fiber is a closed cone in the space of cubic differentials. For 3-dimensional orientable generalized cusps, the fiber is homeomorphic to a cone on a solid torus.
We give a brief survey of some facts about homotopy $4$-spheres \cite{a1}, then give a proof that the curious homotopy sphere constructed in \cite{a2} is in fact diffeomorphic to the standard $S^4$, and discuss its relation to infinite order loose corks and anti-corks.
For a classical link, Milnor defined a family of isotopy invariants, called Milnor $\overlineμ$-invariants. Recently, Chrisman extended Milnor $\overlineμ$-invariants to welded links by a topological approach. The aim of this paper is to show that Milnor $\overlineμ$-invariants can be extended to welded links by a combinatorial approach. The proof contains an alternative proof for the invariance of the original $\overlineμ$-invariants of classical links.
This paper is a continuation of our previous work with Margalit where we studied group actions on projection complexes. In that paper, we demonstrated sufficient conditions so that the normal closure of a family of subgroups of vertex stabilizers is a free product of certain conjugates of these subgroups. In this paper, we study both the quotient of the projection complex by this normal subgroup and the action of the quotient group on the quotient of the projection complex. We show that under certain conditions that the quotient complex is $δ$-hyperbolic. Additionally, under certain circumstances, we show that if the original action on the projection complex was a non-elementary WPD action, then so is the action of the quotient group on the quotient of the projection complex. This implies that the quotient group is acylindrically hyperbolic.
The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Examples for which the rank coincides with familiar quantities include: the dimension of maximal Dehn twist flats for mapping class groups, the maximal rank of a free abelian subgroup for right-angled Coxeter and Artin groups, and, for the Weil--Petersson metric, the rank is the integer part of half the complex dimension of Teichmüller space. We prove that any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants (under a mild condition satisfied by all natural examples). This resolves outstanding conjectures when applied to various examples. For mapping class group, we verify a conjecture of Farb; for Teichmüller space we answer a question of Brock; for CAT(0) cubical groups, we handle special cases including right-angled Coxeter groups. An important ingredient in the proof is that the hull of any finite set in an HHS is quasi-isometric to a CAT(0) cube complex of dimension bounded by the rank. We deduce a number of applica
We introduce an unknotting-type number of knot projections that gives an upper bound of the crosscap number of knots. We determine the set of knot projections with the unknotting-type number at most two, and this result implies classical and new results that determine the set of alternating knots with the crosscap number at most two.
We describe each multiple curve on the orientable surface of genus-$g$ with $n$ punctures and one boundary component by using this multiple curve's geometric intersection number with the embedded curves in this surface.
Let $M$ be a closed smooth Riemannian manifold $M$, and let $f:M\to M$ be a diffeomorphism. Herein, we demonstrate that (i) if $f$ has the $C^1$ robustly inverse shadowing property on the chain recurrent set $\mathcal{CR}(f)$, then $\mathcal{CR}(f)$ is hyperbolic and (ii) if $f$ has the $C^1$ robustly inverse shadowing property on a nontrivial transitive set $Λ\subset M$, then $Λ$ is hyperbolic for $f$. Especially, the item (ii) is a proof of the conjecture of Lee and Lee \cite{LL}.
In arXiv:1905.07734 we presented a construction that is an analogue of Pontryagin's for proper maps in stable dimensions. This gives a bijection between the cobordism set of framed embedded compact submanifolds in $W\times\mathbb{R}^n$ for a given manifold $W$ and a large enough number $n$, and the homotopy classes of proper maps from $W\times\mathbb{R}^n$ to $\mathbb{R}^{k+n}$. In the present paper we generalise this result in a similar way as Thom's construction generalises Pontryagin's. In other words, we present a bijection between the cobordism set of submanifolds embedded in $W\times\mathbb{R}^n$ with normal bundles induced from a given bundle $ξ\oplus\varepsilon^n$, and the homotopy classes of proper maps from $W\times\mathbb{R}^n$ to a space $U(ξ\oplus\varepsilon^n)$ that depends on the given bundle. An important difference between Thom's construction and ours is that we also consider cobordisms of non-compact manifolds after indroducing a suitable notion of cobordism relation for these.
We provide a geometric interpretation of the formulas for Steenrod's $\cup_i$ products, giving an explicit construction for a conjecture of Thorngren. We construct from a simplex and a branching structure a special frame of vector fields inside each simplex that allow us to interpret cochain-level formulas for the $\cup_i$ as a generalized intersection product on the dual cellular decomposition. It can be thought of as measuring the intersection between a collection of dual cells and thickened, shifted version of another collection, where the vector field frame determines the thickening and shifting. Defining this vector field frame in a neighborhood of the dual 1-skeleton of a simplicial complex allows us to combinatorially define $Spin$ and $Pin^\pm$ structures on triangulated manifolds. We use them to geometrically interpret the `Grassmann Integral' of Gu-Wen/Gaiotto-Kapustin, without using Grassmann variables. In particular, we find that the `quadratic refinement' property of Gaiotto-Kapustin can be derived geometrically using our vector fields and interpretation of $\cup_i$, together with a certain trivalent resolution of the dual 1-skeleton. This lets us extend th
Let $K\subset S^3$ be a Fox $p$-colored knot and assume $K$ bounds a locally flat surface $S\subset B^4$ over which the given $p$-coloring extends. This coloring of $S$ induces a dihedral branched cover $X\to S^4$. Its branching set is a closed surface embedded in $S^4$ locally flatly away from one singularity whose link is $K$. When $S$ is homotopy ribbon and $X$ a definite four-manifold, a condition relating the signature of $X$ and the Murasugi signature of $K$ guarantees that $S$ in fact realizes the four-genus of $K$. We exhibit an infinite family of knots $K_m$ with this property, each with a {Fox 3-}colored surface of minimal genus $m$. As a consequence, we classify the signatures of manifolds $X$ which arise as dihedral covers of $S^4$ in the above sense.
We present the complete classification of the subgroup of the classical knot concordance group generated by knots with eight or fewer crossings. Proofs are presented in summary. We also describe extensions of this work to the case of nine crossing knots.
This article presents a whirlwind tour of some results surrounding the Koebe-Andre'ev-Thurston Theorem, Bill Thurston's seminal circle packing theorem that appears in Chapter 13 of The Geometry and Topology of Three-Manifolds. It will appear as a chapter in the volume: In the tradition of Thurston: geometry and topology (ed. K. Ohshika and A. Papadopoulos), Springer, 2020.
We describe the second integral cohomology group of a surface bundle as the group of Chern classes of fiberwise holomorphic complex line bundles and use this to obtain information on this group.
We explicitly show that the Poisson bracket on the set of shear coordinates introduced by V.V. Fock in 1997 induces the Fenchel--Nielsen bracket on the set of gluing parameters (length and twist parameters) for pairs of pants decomposition for Riemann surfaces with holes $Σ_{g,s}$. We generalize these structures to the case of Riemann surfaces $Σ_{g,s,n}$ with holes and bordered cusps.
If a knot $K$ in $S^3$ admits a pair of truly cosmetic surgeries, we show that the surgery slopes are either $\pm 2$ or $\pm 1/q$ for some value of $q$ that is explicitly determined by the knot Floer homology of $K$. Moreover, in the former case the genus of $K$ must be two, and in the latter case there is bound relating $q$ to the genus and the Heegaard Floer thickness of $K$. As a consequence, we show that the cosmetic crossing conjecture holds for alternating knots (or more generally, Heegaard Floer thin knots) with genus not equal to two. We also show that the conjecture holds for any knot $K$ for which each prime summand of $K$ has at most 16 crossings; our techniques rule out cosmetic surgeries in this setting except for slopes $\pm 1$ and $\pm 2$ on a small number of knots, and these remaining examples can be checked by comparing hyperbolic invariants. These results make use of the surgery formula for Heegaard Floer homology, which has already proved to be a powerful tool for obstructing cosmetic surgeries; we get stronger obstructions than previously known by considering the full graded theory. We make use of a new graphical interpretation of knot Floer homology and the sur
A cork is a smooth, contractible, oriented, compact 4-manifold $W$ together with a self-diffeomorphism $f$ of the boundary 3-manifold that cannot extend to a self-diffeomorphism of $W$; the cork is said to be strong if $f$ cannot extend to a self-diffeomorphism of any smooth integer homology ball bounded by $\partial W$. Surprising recent work of Dai, Hedden, and Mallick showed that most of the well-known corks in the literature are strong. We construct the first non-strong corks, which also give rise to new examples of absolutely exotic Mazur manifolds. Additionally we give the first examples of corks where the diffeomorphism of $\partial W$ can be taken to be orientation-reversing.
Let $X^4$ and $Y^4$ be smooth manifolds and $f: X\to Y$ a branched cover with branching set $B$. Classically, if $B$ is smoothly embedded in $Y$, the signature $σ(X)$ can be computed from data about $Y$, $B$ and the local degrees of $f$. When $f$ is an irregular dihedral cover and $B\subset Y$ smoothly embedded away from a cone singularity whose link is $K$, the second author gave a formula for the contribution $Ξ(K)$ to $σ(X)$ resulting from the non-smooth point. We extend the above results to the case where $Y$ is a {\it topological} four-manifold and $B$ is locally flat, away from the possible singularity. Owing to the presence of non-locally-flat points on $B$, $X$ in this setting is a stratified pseudomanifold, and we use the Intersection Homology signature of $X$, $σ_{IH}(X)$. For any knot $K$ whose determinant is not $\pm 1$, a homotopy ribbon obstruction is derived from $Ξ(K)$, providing a new technique to potentially detect slice knots that are not ribbon.
In this paper we study colored triangulations of compact PL 4-manifolds with empty or connected boundary which induce handle decompositions lacking in 1-handles or in 1- and 3-handles, thus facing also the problem, posed by Kirby, of the existence of {\it special handlebody decompositions} for any simply-connected closed PL 4-manifold. In particular, we detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary, which admit such decompositions and, therefore, can be represented by (undotted) framed links. Moreover, this class includes any compact simply-connected PL 4-manifold with empty or connected boundary having colored triangulations that minimize the combinatorially defined PL invariant {\em regular genus, gem-complexity} or {\em G-degree} among all such manifolds with the same second Betti number.
People in this topic
Researcher
Taras Banakh contributes to research discovery and scholarly infrastructure.
Open to collaborateResearcher
Indranil Biswas contributes to research discovery and scholarly infrastructure.
Open to collaborateResearcher
Robert Osburn contributes to research discovery and scholarly infrastructure.
Open to collaborateResearcher
A. Quintero contributes to research discovery and scholarly infrastructure.
Open to collaborateResearcher
Shing-Tung Yau contributes to research discovery and scholarly infrastructure.
Open to collaborateResearcher
Juven Wang contributes to research discovery and scholarly infrastructure.
Open to collaborateResearcher
Tao Yu contributes to research discovery and scholarly infrastructure.
Open to collaborateResearcher
Wei Lin contributes to research discovery and scholarly infrastructure.
Open to collaborateResearcher
Dušan D. Repovš contributes to research discovery and scholarly infrastructure.
Open to collaborateResearcher
Feng Zhu contributes to research discovery and scholarly infrastructure.
Open to collaborateResearcher
Yu Pan contributes to research discovery and scholarly infrastructure.
Open to collaborateResearcher
Yi Shi contributes to research discovery and scholarly infrastructure.
Open to collaborate