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preprint2015arXiv

Contact open books with exotic pages

We consider a fixed contact 3-manifold that admits infinitely many compact Stein fillings which are all homeomorphic but pairwise non-diffeomorphic. Each of these fillings gives rise to a closed contact 5-manifold described as a contact open book whose page is the filling at hand and whose monodromy is the identity symplectomorphism. We show that the resulting infinitely many contact 5-manifolds are all diffeomorphic but pairwise non-contactomorphic. Moreover, we explicitly determine these contact 5-manifolds.

preprint2016arXiv

Concordance maps in knot Floer homology

We show that a decorated knot concordance $C$ from $K$ to $K'$ induces a homomorphism $F_C$ on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to $\widehat{HF}(S^3) \cong \mathbb{Z}_2$ that agrees with $F_C$ on the $E^1$ page and is the identity on the $E^\infty$ page. It follows that $F_C$ is non-vanishing on $\widehat{HFK}_0(K, τ(K))$. We also obtain an invariant of slice disks in homology 4-balls bounding $S^3$. If $C$ is invertible, then $F_C$ is injective, hence $\dim \widehat{HFK}_j(K,i) \le \dim \widehat{HFK}_j(K',i)$ for every $i$, $j \in \mathbb{Z}$. This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot $K$ to $K'$, then $g(K) \le g(K')$, where $g$ denotes the Seifert genus. Furthermore, if $g(K) = g(K')$ and $K'$ is fibred, then so is $K$.

preprint2015arXiv

Combinatorial tangle Floer homology

In this paper we extend the idea of bordered Floer homology to knots and links in $S^3$: Using a specific Heegaard diagram, we construct gluable combinatorial invariants of tangles in $S^3$, $D^3$ and $I\times S^2$. The special case of $S^3$ gives back a stabilized version of knot Floer homology.

preprint2013arXiv

Symplectic fillings of lens spaces as Lefschetz fibrations

We construct a positive allowable Lefschetz fibration over the disk on any minimal weak symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding complex two-dimensional cyclic quotient singularity.

preprint2010arXiv

Milnor open books of links of some rational surface singularities

We describe Milnor open books and Legendrian surgery diagrams for canonical contact structures of links of some rational surface singularities. We also describe an infinite family of Milnor fillable contact 3-manifolds so that the Milnor genus (resp. Milnor norm) is strictly greater than the support genus (resp. support norm) of the canonical contact structure, for each member of this family.

preprint2013arXiv

Trisecting 4-manifolds

We show that any smooth, closed, oriented, connected 4--manifold can be trisected into three copies of $\natural^k (S^1 \times B^3)$, intersecting pairwise in 3--dimensional handlebodies, with triple intersection a closed 2--dimensional surface. Such a trisection is unique up to a natural stabilization operation. This is analogous to the existence, and uniqueness up to stabilization, of Heegaard splittings of 3--manifolds. A trisection of a 4--manifold $X$ arises from a Morse 2--function $G:X \to B^2$ and the obvious trisection of $B^2$, in much the same way that a Heegaard splitting of a 3--manifold $Y$ arises from a Morse function $g : Y \to B^1$ and the obvious bisection of $B^1$.

preprint2012arXiv

Asymptotic formulas for curve operators in TQFT

Topological quantum field theories with gauge group $\textrm{SU}_2$ associate to each surface with marked points $Σ$ and each integer $r>0$ a vector space $V_r (Σ)$ and to each simple closed curve $γ$ in $Σ$ an Hermitian operator $T_r^γ$ acting on that space. We show that the matrix elements of the operators $T_r^γ$ have an asymptotic expansion in orders of $\frac{1}{r}$, and give a formula to compute the first two terms in terms of trace functions, generalizing results of Marché and Paul.

preprint2016arXiv

A 1-parameter family of spherical CR uniformizations of the figure eight knot complement

We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot complement. Finally, we construct an explicit 1-parameter family of deformations of the boundary unipotent holonomy group such that the boundary holonomy is twist-parabolic. For small values of the twist of these parabolic elements, this produces a 1-parameter family of pairwise non-conjugate spherical CR uniformizations of the figure eight knot complement.

preprint2017arXiv

Random subgroups of acylindrically hyperbolic groups and hyperbolic embeddings

Let G be an acylindrically hyperbolic group. We consider a random subgroup H in G, generated by a finite collection of independent random walks. We show that, with asymptotic probability one, such a random subgroup H of G is a free group, and the semidirect product of H acting on E(G) is hyperbolically embedded in G, where E(G) is the unique maximal finite normal subgroup of G.

preprint2016arXiv

Topological Rigidity for FJ by the Infinite Cyclic Group

We call a group FJ if it satisfies the $K$- and $L$-theoretic Farrell-Jones conjecture with coefficients in $\mathbb Z$. We show that if $G$ is FJ, then the simple Borel conjecture (in dimensions $\ge 5$) holds for every group of the form $G\rtimes\mathbb Z$. If in addition $Wh(G\times \mathbb Z)=0$, which is true for all known torsion free FJ groups, then the bordism Borel conjecture (in dimensions $n\ge 5$) holds for $G\rtimes\mathbb Z$. One of the key ingredients in proving these rigidity results is another main result, which says that if a torsion free group $G$ satisfies the $L$-theoretic Farrell-Jones conjecture with coefficients in $\mathbb Z$, then any semi-direct product $G\rtimes\mathbb Z$ also satisfies the $L$-theoretic Farrell-Jones conjecture with coefficients in $\mathbb Z$. Our result is indeed more general and implies the $L$-theoretic Farrell-Jones conjecture with coefficients in additive categories is closed under extensions of torsion free groups. This enables us to extend the class of groups which satisfy the Novikov conjecture.

preprint2016arXiv

Suspension theorems for links and link maps

We present a new short proof of the explicit formula for the group of links (and also link maps) in the 'quadruple point free' dimension. Denote by $L^m_{p,q}$ (respectively, $C^{m-p}_p$) the group of smooth embeddings $S^p\sqcup S^q\to S^m$ (respectively, $S^p\to S^m$) up to smooth isotopy. Denote by $LM^m_{p,q}$ the group of link maps $S^p\sqcup S^q\to S^m$ up to link homotopy. Theorem 1. If $p\le q\le m-3$ and $2p+2q\le 3m-6$ then \begin{equation*} L^m_{p,q}\cong π_p(S^{m-q-1})\oplusπ_{p+q+2-m}(SO/SO_{m-p-1})\oplus C^{m-p}_p\oplus C^{m-q}_q. \end{equation*} Theorem 2. If $p, q\le m-3$ and $2p+2q\le 3m-5$ then $LM^m_{p,q}\cong π^S_{p+q+1-m}$. Our approach is based on the use of the suspension operation for links and link maps, and suspension theorems for them.

preprint2016arXiv

On the growth of L2-invariants of locally symmetric spaces, II: exotic invariant random subgroups in rank one

In the first paper of this series (arxiv.org/abs/1210.2961) we studied the asymptotic behavior of Betti numbers, twisted torsion and other spectral invariants for sequences of lattices in Lie groups G. A key element of our work was the study of invariant random subgroups (IRSs) of G. Any sequence of lattices has a subsequence converging to an IRS, and when G has higher rank, the Nevo-Stuck-Zimmer theorem classifies all IRSs of G. Using the classification, one can deduce asymptotic statments about spectral invariants of lattices. When G has real rank one, the space of IRSs is more complicated. We construct here several uncountable families of IRSs in the groups SO(n,1). We give dimension-specific constructions when n=2,3, and also describe a general gluing construction that works for every n at least 2. Part of the latter construction is inspired by Gromov and Piatetski-Shapiro's construction of non-arithmetic lattices in SO(n,1).

preprint2017arXiv

Bounded cohomology and virtually free hyperbolically embedded subgroups

Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, ${\rm Out}(F_n)$, \dots) embeds via the natural restriction maps into the inverse limit of the second bounded cohomologies of its virtually free subgroups, and in fact even into the inverse limit of the second bounded cohomologies of its hyperbolically embedded virtually free subgroups. This result is new and non-trivial even in the case where $G$ is a (non-free) hyperbolic group. The corresponding statement fails in general for the third bounded cohomology, even for surface groups.

preprint2014arXiv

Reflection groups in non-negative curvature

We provide an equivariant description/classification of all complete (compact or not) non-negatively curved manifolds M together with a co-compact action by a reflection group W, and moreover, classify such W. In particular, we show that the building blocks consist of the classical constant curvature models and generalized open books with non negatively curved bundle pages, and derive a corresponding splitting theorem for the universal cover.

preprint2016arXiv

Automorphisms of the mapping class group of a nonorientable surface

Let $S$ be a nonorientable surface of genus $g\ge 5$ with $n\ge 0$ punctures, and $\Mcg(S)$ its mapping class group. We define the complexity of $S$ to be the maximum rank of a free abelian subgroup of $\Mcg(S)$. Suppose that $S_1$ and $S_2$ are two such surfaces of the same complexity. We prove that every isomorphism $\Mcg(S_1)\to\Mcg(S_2)$ is induced by a diffeomorphism $S_1\to S_2$. This is an analogue of Ivanov's theorem on automorphisms of the mapping class groups of an orientable surface, and also an extension and improvement of the first author's previous result.

preprint2017arXiv

Incompressible fillings of manifolds

We find boundaries of Borel-Serre compactifications of locally symmetric spaces, for which any filling is incompressible. We prove this result by showing that these boundaries have small singular models and using these models to obstruct compressions. We also show that small singular models of boundaries obstruct $S^1$-actions (and more generally homotopically trivial $\mathbb Z/p$-actions) on interiors of aspherical fillings. We use this to bound the symmetry of complete Riemannian metrics on such interiors in terms of the fundamental group. We also use small singular models to simplify the proofs of some already known theorems about moduli spaces (the minimal orbifold theorem and a topological analogue of Royden's theorem).

preprint2016arXiv

Number of Systoles of Once-Punctured Torus and Four-Punctured Sphere

We compute the number of systoles, the shortest simple closed geodesics and 2-systoles, the second shortest simple closed geodesics on hyperbolic surfaces homeomorphic to once-punctured torus and four-punctured sphere.

preprint2016arXiv

Involutions on surfaces

We use equivariant surgery to classify all involutions on closed surfaces, up to isomorphism. Work on this problem is classical, dating back to the nineteenth century, but some questions seem to have been left unanswered. We give a modern treatment that leads to a complete classification.

preprint2014arXiv

H_1-semistability for projective groups

We initiate the study of the asymptotic topology of groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers (these are called here as holomorphically convex groups). We prove the $H_1$-semistability conjecture of Geoghegan for holomorphically convex groups. In view of a theorem of Eyssidieux, Katzarkov, Pantev and Ramachandran \cite{ekpr}, this implies that linear projective groups satisfy the $H_1$-semistability conjecture.

preprint2016arXiv

Automorphisms and homology of non-positively curved cube complexes

We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the fundamental group of a special cube complex is either free abelian or surjects onto a non-cyclic free group. We also investigate automorphisms of special cube complexes, and give a new geometric proof that the Torelli subgroup for a right-angled Artin group is torsion-free.

preprint2014arXiv

Simple game induced manifolds

Starting by a simple game $Q $ as a combinatorial data, we build up a cell complex $M(Q)$, whose construction resembles combinatorics of the permutohedron. The cell complex proves to be a combinatorial manifold; we call it the \textit{ simple game induced manifold.} By some motivations coming from polygonal linkages, we think of $Q$ and of $M(Q)$ as of\textit{ a quasilinkage} and the \textit{moduli space of the quasilinkage} respectively. We present some examples of quasilinkages and show that the moduli space retains many properties of moduli space of polygonal linkages. In particular, we show that the moduli space $M(Q)$ is homeomorphic to the space of stable point configurations on $S^1$, for an associated with a quasilinkage notion of stability.

preprint2016arXiv

Conformally correct tilings

We discuss the art and science of producing conformally correct euclidean and hyperbolic tilings of compact surfaces. As an example, we present a tiling of the Chmutov surface by hyperbolic (2, 4, 6) triangles.

preprint2014arXiv

Nash Equilibria via Duality and Homological Selection

Given a multifunction from $X$ to the $k-$fold symmetric product $Sym_k(X)$, we use the Dold-Thom Theorem to establish a homological selection Theorem. This is used to establish existence of Nash equilibria. Cost functions in problems concerning the existence of Nash Equilibria are traditionally multilinear in the mixed strategies. The main aim of this paper is to relax the hypothesis of multilinearity. We use basic intersection theory, Poincaré Duality in addition to the Dold-Thom Theorem.

preprint2015arXiv

A splitting theorem for good complexifications

The purpose of this paper is to produce restrictions on fundamental groups of manifolds admitting good complexifications by proving the following Cheeger-Gromoll type splitting theorem: Any closed manifold $M$ admitting a good complexification has a finite-sheeted regular covering $M_1$ such that $M_1$ admits a fiber bundle structure with base $(S^1)^k$ and fiber $N$ that admits a good complexification and also has zero virtual first Betti number. We give several applications to manifolds of dimension at most 5.

473 works