Showing papers in "Geometry & Topology in 2003"
TL;DR: In this article, a series of compactness results for moduli spaces of holomorphic curves arising in Symplectic field theory is presented. But these results generalize Gromov's compactness theorem in (8) as well as compactness theorems in Floer homology theory, and in contact geometry, (9, 19).
Abstract: This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in (4). We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov's compactness theorem in (8) as well as compactness theorems in Floer homology theory, (6, 7), and in contact geometry, (9, 19).
575 citations
TL;DR: In this paper, the authors used the knot ltration on the Heegaard Floer complex d to derive an integer invariant (K) for knots, which gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus and hence also the unknotting number of a knot; but unlike the signature, gives sharp bounds on the four-ball genera of torus knots.
Abstract: We use the knot ltration on the Heegaard Floer complex d to dene an integer invariant (K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, gives sharp bounds on the four-ball genera of torus knots. As another illustration, we calculate the invariant for several ten-crossing knots.
444 citations
TL;DR: In this paper, a knot invariant for a null-homologous knot K in an oriented three-manifold Y, which is closely related to the Heegaard Floer homology of Y, was introduced.
Abstract: In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y , which is closely related to the Heegaard Floer homology of Y . In this paper we investigate some properties of these knot homology groups for knots in the three-sphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (generalizing a result of Rasmussen for two-bridge knots). Applications include new restrictions on the Alexander polynomial of alternating knots.
295 citations
TL;DR: In this article, a combination theorem for relatively hyperbolic groups seen as geometrically finite convergence groups is presented, and a proof of the Howson property for limit groups is given.
Abstract: We state and prove a combination theorem for relatively hyperbolic groups seen as geometrically finite convergence groups. For that, we explain how to contruct a boundary for a group that is an acylindrical amalgamation of relatively hyperbolic groups over a fully quasi-convex subgroup. We apply our result to Sela's theory on limit groups and prove their relative hyperbolicity. We also get a proof of the Howson property for limit groups.
264 citations
TL;DR: In this article, the authors calculate the Heegaard Floer homologies for three-manifolds obtained by plumbings of spheres specied by certain graphs, including the product of a circle with a genus two surface.
Abstract: We calculate the Heegaard Floer homologies for three-manifolds obtained by plumbings of spheres specied by certain graphs. Our class of graphs is sufciently large to describe, for example, all Seifert bered rational homology spheres. These calculations can be used to determine also these groups for other three-manifolds, including the product of a circle with a genus two surface.
241 citations
TL;DR: In this article, a universal n-excisive approximation of a functor from spaces to spaces or spectra that preserve weak homotopy equivalences is proposed. But it is not a universal functor.
Abstract: We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal n-excisive approximation, which may be thought of as its n-excisive part. Homogeneous functors, meaning n-excisive functors with trivial (n 1)-excisive part, can be classied: they correspond to symmetric functors of n variables that are reduced and 1-excisive in each variable. We discuss some important examples, including the identity functor and Waldhausen’s algebraic K -theory.
150 citations
TL;DR: The group Γ of automorphisms of the polynomial κ (x, y, z ) = x 2 + y 2 + z 2 − x y z − 2 is isomorphic to PGL ( 2, ℤ ) ⋉ ( Ω ∕ 2 ⊕ ℥ ∕ √ 2 ). For t ∈ ℝ, the Γ-action on κ−1(t)∩ℝ displays rich and varied dynamics.
Abstract: The group Γ of automorphisms of the polynomial κ ( x , y , z ) = x 2 + y 2 + z 2 − x y z − 2 is isomorphic to PGL ( 2 , ℤ ) ⋉ ( ℤ ∕ 2 ⊕ ℤ ∕ 2 ) . For t∈ℝ, the Γ–action on κ−1(t)∩ℝ displays rich and varied dynamics. The action of Γ preserves a Poisson structure defining a Γ–invariant area form on each κ−1(t)∩ℝ. For t 18, in which case the Γ–action on the complement (κ−1(t)∩ℝ)−Ω is ergodic.
138 citations
TL;DR: In this paper, the Wheels and Wheeling conjectures of [5, 9] and [10] were proved using elementary equalities between various cables of the unknot and the Hopf link.
Abstract: Using elementary equalities between various cables of the unknot and the Hopf link, we prove the Wheels and Wheeling conjectures of [5, 9], which give, respectively, the exact Kontsevich integral of the unknot and a map intertwining two natural products on a space of diagrams. It turns out that the Wheeling map is given by the Kontsevich integral of a cut Hopf link (a bead on a wire), and its intertwining property is analogous to the computation of 1 + 1 = 2 on an abacus. The Wheels conjecture is proved from the fact that the k-fold connected cover of the unknot is the unknot for all k. Along the way, we nd a formula for the invariant of the general (k;l) cable of a knot. Our results can also be interpreted as a new proof of the multiplicativity of the Duflo{Kirillov map S(g)! U (g) for metrized Lie (super-)algebras g. AMS Classication numbers Primary: 57M27 Secondary: 17B20, 17B37
105 citations
TL;DR: In this paper, a general theory of cosimplicial resolutions, homotopy spectral sequences, and completions for objects in model categories is developed, which is based on a generalized version of the Dwyer{Kan{Stover theory of resolution model categories.
Abstract: We develop a general theory of cosimplicial resolutions, homotopy spectral sequences, and completions for objects in model categories, extending work of Bouseld{Kan and Bendersky{Thompson for ordinary spaces. This is based on a generalized cosimplicial version of the Dwyer{Kan{Stover theory of resolution model categories, and we are able to construct our homotopy spectral sequences and completions using very flexible weak resolutions in the spirit of relative homological algebra. We deduce that our completion functors have triple structures and preserve certain ber squares up to homotopy. We also deduce that the Bendersky{Thompson completions over connective ring spectra are equivalent to Bouseld{Kan completions over solid rings. The present work allows us to show, in a subsequent paper, that the homotopy spectral sequences over arbitrary ring spectra have well-behaved composition pairings.
98 citations
TL;DR: In this paper, an invariant of homology 3{spheres which lives in the S 1 {equivariant graded suspension category was obtained using Furuta's idea of finite dimensional approximation in Seiberg{Witten theory.
Abstract: Using Furuta’s idea of nite dimensional approximation in Seiberg{Witten theory, we rene Seiberg{Witten Floer homology to obtain an invariant of homology 3{spheres which lives in the S 1 {equivariant graded suspension category. In particular, this gives a construction of Seiberg{Witten Floer homology that avoids the delicate transversality problems in the standard approach. We also dene a relative invariant of four-manifolds with boundary which generalizes the Bauer{Furuta stable homotopy invariant of closed four-manifolds.
93 citations
TL;DR: The Virtual Haken Conjecture as discussed by the authors states that a 3-manifold is Haken if it contains a topologically essential surface and every irreducible 3manifolds with a fundamental group has a finite cover which is virtually Haken.
Abstract: A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3-manifold with innite fundamental group has a nite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3-manifolds. We took the complete HodgsonWeeks census of 10,986 small-volume closed hyperbolic 3-manifolds, and for each of them found nite covers which are Haken. There are interesting and unexplained patterns in the data which may lead to a better understanding of this problem. Second, we discuss a method for transferring the virtual Haken property under Dehn lling. In particular, we show that if a 3-manifold with torus boundary has a Seifert bered Dehn lling with hyperbolic base orbifold, then most of the Dehn lled manifolds are virtually Haken. We use this to show that every non-trivial Dehn surgery on the gure-8 knot is virtually Haken. AMS Classication numbers Primary: 57M05, 57M10 Secondary: 57M27, 20E26, 20F05
TL;DR: In this paper, it was shown that every non-trivial Hamiltonian dieomor-phism of a closed oriented surface of genus at least one has periodic points of arbitrarily high period.
Abstract: The main result of this paper is that every non-trivial Hamiltonian dieomor- phism of a closed oriented surface of genus at least one has periodic points of arbitrarily high period. The same result is true for S 2 provided the dif- feomorphism has at least three xed points. In addition we show that up to isotopy relative to its xed point set, every orientation preserving dieomor- phism F : S! S of a closed orientable surface has a normal form. If the xed point set is nite this is just the Thurston normal form.
TL;DR: In this paper, it was shown that there are many manifolds that are homotopic equivalent to M but not homeomorphic to it, and that the {invariant is not a homotopy invariant for the manifolds in question.
Abstract: We prove that, if M is a compact oriented manifold of dimension 4k +3 , where k> 0, such that 1(M) is not torsion-free, then there are innitely many manifolds that are homotopic equivalent to M but not homeomorphic to it. To show the innite size of the structure set of M ,w e construct a secondary invariant (2) : S(M ) ! R that coincides with the {invariant of Cheeger{Gromov. In particular, our result shows that the {invariant is not a homotopy invariant for the manifolds in question. AMS Classication numbers Primary: 57R67 Secondary: 46L80, 58G10
TL;DR: In this paper, several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied, and the spaces underlying the basic operad are identied with open subsets of a combinatorial compactication due to Penner of a space closely related to Riemann's moduli space.
Abstract: Several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied. The spaces underlying the basic operad are identied with open subsets of a combinatorial compactication due to Penner of a space closely related to Riemann’s moduli space. Algebras over these operads are shown to be Batalin{Vilkovisky algebras, where the entire BV structure is realized simplicially. Furthermore, our basic operad contains the cacti operad up to homotopy. New operad structures on the circle are classied and combined with the basic operad to produce geometrically natural extensions of the algebraic structure of BV algebras, which are also computed.
TL;DR: In this paper, the p-primary homotopy of the smooth Whitehead spectrum Wh () is described, and a suspended copy of the cokernel-of-J spectrum splits o, and the torsion homo-graph of the remainder equals the Torso-Graph of the restricted S 1 -transfer map t : C P 1! S. The cohomology of Wh () can be expressed as an A-module in all degrees up to an extension.
Abstract: Let p be an odd regular prime, and assume that the Lichtenbaum{Quillen conjecture holds for K(Z[1=p]) at p. Then the p-primary homotopy type of the smooth Whitehead spectrum Wh () is described. A suspended copy of the cokernel-of-J spectrum splits o, and the torsion homotopy of the remainder equals the torsion homotopy of the ber of the restricted S 1 -transfer map t : C P 1 ! S. The homotopy groups of Wh () are determined in a range of degrees, and the cohomology of Wh () is expressed as an A-module in all degrees, up to an extension. These results have geometric topological interpretations, in terms of spaces of concordances or dieomorphisms of highly connected, high dimensional compact smooth manifolds.
TL;DR: In this paper, the Ricci flow is shown to converge in the Gromov-Hausdor sense to a space which is not a manifold but only a metric space.
Abstract: Consider a sequence of pointed n{dimensional complete Riemannian manifolds f(Mi;gi (t );Oi )g such that t2 [0;T] are solutions to the Ricci flow and gi(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an n{dimensional solution to the Ricci flow. We prove a generalization of this theorem where the initial metrics may collapse. Without injectivity radius bounds we must allow for convergence in the Gromov{Hausdor sense to a space which is not a manifold but only a metric space. We then look at the local geometry of the limit to understand how it relates to the Ricci flow.
TL;DR: For complete hyperbolic 3-manifolds N with finitely generated, freely indecomposable fundamental group and with bounded geometry, the authors gave a new construction of model geometries for the geometrically infinite ends of N, a key step in Minsky's proof of Thurston's ending lamination conjecture for such manifolds.
Abstract: We characterize which cobounded quasigeodesics in the Teichmuller space T of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path γ in T, we show that γ is a quasigeodesic with finite Hausdorff distance from some geodesic if and only if the canonical hyperbolic plane bundle over γ is a hyperbolic metric space. As an application, for complete hyperbolic 3–manifolds N with finitely generated, freely indecomposable fundamental group and with bounded geometry, we give a new construction of model geometries for the geometrically infinite ends of N, a key step in Minsky’s proof of Thurston’s ending lamination conjecture for such manifolds.
TL;DR: In this article, a canonical Thom isomorphism in Grojnowski's equivariant elliptic cohomology was constructed for virtual T-oriented T-equivariant spin bundles with vanishing Borel Equivariant second Chern class, which is natural under pull-back of vector bundles and exponential under Whitney sum.
Abstract: We construct a canonical Thom isomorphism in Grojnowski’s equivariant elliptic cohomology, for virtual T–oriented T–equivariant spin bundles with vanishing Borel-equivariant second Chern class, which is natural under pull-back of vector bundles and exponential under Whitney sum. It extends in the complex-analytic case the non-equivariant sigma orientation of Hopkins, Strickland, and the author. The construction relates the sigma orientation to the representation theory of loop groups and Looijenga’s weighted projective space, and sheds light even on the non-equivariant case. Rigidity theorems of Witten-Bott-Taubes including generalizations by Kefeng Liu follow.
TL;DR: The main result of as mentioned in this paper is that every action of G on a closed oriented surface by area preserving dieomorphisms factors through a finite group, and that any nite index subgroup of SL(3;Z) is such a group.
Abstract: Suppose G is an almost simple group containing a subgroup isomorphic to the three-dimensional integer Heisenberg group. For example any nite index subgroup of SL(3;Z) is such a group. The main result of this paper is that every action of G on a closed oriented surface by area preserving dieomorphisms factors through a nite group.
TL;DR: In this paper, it was shown that an innite family of virtually overtwisted tight contact structures discovered by Honda on certain circle bundles over surfaces admit no symplectic semi-llings.
Abstract: We prove that an innite family of virtually overtwisted tight contact structures discovered by Honda on certain circle bundles over surfaces admit no symplectic semi{llings. The argument uses results of Mrowka, Ozsv ath and Yu on the translation{invariant solutions to the Seiberg{Witten equations on cylinders and the non{triviality of the Kronheimer{Mrowka monopole invariants of symplectic llings.
TL;DR: In this paper, the universal equivariant Euler characteristic of a smooth proper cocompact G-manifold without boundary is shown to be a homomorphism from a group UG(M) to the equivariant KO-homology of M. The Euler operator defines via Kasparov theory an element, called the equivariance Euler class, which is defined by the component structure of the various fixed point sets into account.
Abstract: Let G be a countable discrete group and let M be a smooth proper cocompact G-manifold without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant Euler class, in the equivariant KO–homology of M. The universal equivariant Euler characteristic of M, which lives in a group UG(M), counts the equivariant cells of M, taking the component structure of the various fixed point sets into account. We construct a natural homomorphism from UG(M) to the equivariant KO-homology of M. The main result of this paper says that this map sends the universal equivariant Euler characteristic to the equivariant Euler class. In particular this shows that there are no “higher” equivariant Euler characteristics. We show that, rationally, the equivariant Euler class carries the same information as the collection of the orbifold Euler characteristics of the components of the L–fixed point sets ML, where L runs through the finite cyclic subgroups of G. However, we give an example of an action of the symmetric group S3 on the 3–sphere for which the equivariant Euler class has order 2, so there is also some torsion information.
TL;DR: In this article, it was shown that there are algorithms to determine if a 3-manifold contains an essential lamination or a Reebless foliation, and that the algorithms can be used to determine whether the lamination is essential or not.
Abstract: We show that there are algorithms to determine if a 3-manifold contains an essential lamination or a Reebless foliation.
TL;DR: In this article, it was shown that the two quadratic functions coincide in terms of the reduction modulo 1 of the Reidemeister{Turaev torsion of (M;), while the other one can be dened using the intersection pairing of an appropriate compact oriented 4{manifold with boundary M. The proof relies on the comparison of two distinct combinatorial descriptions of Spin c {structures on M :T u- raev's charges vs Chern vectors.
Abstract: Given an oriented rational homology 3{sphere M , it is known how to asso- ciate to any Spin c {structure on M two quadratic functions over the linking pairing. One quadratic function is derived from the reduction modulo 1 of the Reidemeister{Turaev torsion of (M;), while the other one can be dened using the intersection pairing of an appropriate compact oriented 4{manifold with boundary M. In this paper, using surgery presentations of the manifold M ,w e prove that those two quadratic functions coincide. Our proof relies on the comparison be- tween two distinct combinatorial descriptions of Spin c {structures on M :T u- raev's charges vs Chern vectors. AMS Classication numbers Primary: 57M27 Secondary: 57Q10, 57R15
TL;DR: In this paper, it was shown that the classical concordance group of algebraically slice knots has an innite cyclic summand and in particular is not a divisible group.
Abstract: As a corollary of work of Ozsv ath and Szab o [8], it is shown that the classical concordance group of algebraically slice knots has an innite cyclic summand and in particular is not a divisible group.
TL;DR: In this article, the authors give a short proof of a rigidity result by Forester which gives a sufficient condition for a deformation space to contain an Aut(G)-invariant G-tree.
Abstract: The deformation space of a simplicial G-tree T is the set of G-trees which can be obtained from T by some collapse and expansion moves, or equivalently, which have the same elliptic subgroups as T. We give a short proof of a rigidity result by Forester which gives a sufficient condition for a deformation space to contain an Aut(G)-invariant G-tree. This gives a sufficient condition for a JSJ splitting to be invariant under automorphisms of G. More precisely, the theorem claims that a deformation space contains at most one strongly slide-free G-tree, where strongly slide-free means the following: whenever two edges e_1, e_2 incident on a same vertex v are such that G_{e_1} is a subset of G_{e_2}, then e_1 and e_2 are in the same orbit under G_v.
TL;DR: In this article, it was shown that a free period three action on the three-sphere is standard and that the quotient is homeomorphic to a lens space, using a minimax argument involving sweepouts.
Abstract: We show that a free period three action on the three-sphere is standard, ie, the quotient is homeomorphic to a lens space. We use a minimax argument involving sweepouts.
TL;DR: In this paper, a dieomorphism invariant of integral homology 3{spheres is dened using a non-abelian \quaternionic" version of the Seiberg{Witten equations.
Abstract: A new dieomorphism invariant of integral homology 3{spheres is dened using a non-abelian \quaternionic" version of the Seiberg{Witten equations.
TL;DR: In this article, it was shown that every closed oriented 3-manifold admits a hyperbolic cone-shaped manifold structure with a cone-angle arbitrarily close to 2.
Abstract: We prove that every closed oriented 3{manifold admits a hyperbolic cone{ manifold structure with cone{angle arbitrarily close to 2.