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We construct finitely generated groups with strong fixed point properties. Let be the class of Hausdorff spaces of finite covering dimension which are mod– acyclic for at least one prime . We produce the first examples of infinite finitely generated groups with the property that for any action of on any , there is a global fixed point. Moreover, may be chosen to be simple and to have Kazhdan’s property (T). We construct a finitely presented infinite group that admits no nontrivial action on any manifold in . In building , we exhibit new families of hyperbolic groups: for each and each prime , we construct a nonelementary hyperbolic group which has a generating set of size , any proper subset of which generates a finite –group.
We study the rigidity of polyhedral surfaces using variational principles. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a nontriangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first discovered and used by Fenchel and Nielsen. By studying the derivative of the cosine laws, we discover a uniform approach to several variational principles on polyhedral surfaces with or without boundary. As a consequence, the work of Penner, Bobenko and Springborn and Thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context.
Let where is a compact, connected, oriented surface with and nonempty boundary.
(1) The projective class of the chain intersects the interior of a codimension one face of the unit ball in the stable commutator length norm on .
(2) The unique homogeneous quasimorphism on dual to (up to scale and elements of ) is the rotation quasimorphism associated to the action of on the ideal boundary of the hyperbolic plane, coming from a hyperbolic structure on .
These facts follow from the fact that every homologically trivial –chain in rationally cobounds an immersed surface with a sufficiently large multiple of . This is true even if has no boundary.
Let denote a compact, orientable 3–dimensional manifold and let a denote a contact 1–form on ; thus a da is nowhere zero. This article explains how the Seiberg–Witten Floer homology groups as defined for any given structure on give closed, integral curves of the vector field that generates the kernel of .
In 1997, T Cochran, K Orr, and P Teichner [Ann. of Math. (2) 157 (2003) 433-519] defined a filtration of the classical knot concordance group ,
The filtration is important because of its strong connection to the classification of topological –manifolds. Here we introduce new techniques for studying and use them to prove that, for each , the group has infinite rank. We establish the same result for the corresponding filtration of the smooth concordance group. We also resolve a long-standing question as to whether certain natural families of knots, first considered by Casson–Gordon and Gilmer, contain slice knots.
As a continuation of the recent results of Y Lee and the second author [Invent. Math. 170 (2007) 483-505] and the authors [Geom. Topol. 13 (2009) 743-767], we construct a simply connected minimal complex surface of general type with and by using a rational blow-down surgery and –Gorenstein smoothing theory.
We describe a modification of Khovanov homology [Duke Math. J. 101 (2000) 359-426], in the spirit of Bar-Natan [Geom. Topol. 9 (2005) 1443-1499], which makes the theory properly functorial with respect to link cobordisms.
This requires introducing "disorientations" in the category of smoothings and abstract cobordisms between them used in Bar-Natan’s definition. Disorientations have "seams" separating oppositely oriented regions, coming with a preferred normal direction. The seams satisfy certain relations (just as the underlying cobordisms satisfy relations such as the neck cutting relation).
We construct explicit chain maps for the various Reidemeister moves, then prove that the compositions of chain maps associated to each side of each of Carter, Reiger and Saito’s movie moves [J. Knot Theory Ramifications 2 (1993) 251-284; Adv. Math. 127 (1997) 1-51] always agree. These calculations are greatly simplified by following arguments due to Bar-Natan and Khovanov, which ensure that the two compositions must agree, up to a sign. We set up this argument in our context by proving a result about duality in Khovanov homology, generalising previous results about mirror images of knots to a "local" result about tangles. Along the way, we reproduce Jacobsson’s sign table [Algebr. Geom. Topol. 4 (2004) 1211-1251] for the original "unoriented theory", with a few disagreements.
We present a method to desingularize a compact manifold with isolated conical singularities by cutting out a neighbourhood of each singular point and gluing in an asymptotically conical manifold . Controlling the error on the overlap gluing region enables us to use a result of Joyce to conclude that the resulting compact smooth –manifold admits a torsion-free structure, with full holonomy.
There are topological obstructions for this procedure to work, which arise from the degree and degree cohomology of the asymptotically conical manifolds which are glued in at each conical singularity. When a certain necessary topological condition on the manifold with isolated conical singularities is satisfied, we can introduce correction terms to the gluing procedure to ensure that it still works. In the case of degree obstructions, these correction terms are trivial to construct, but in the case of degree obstructions we need to solve an elliptic equation on a noncompact manifold. For this we use the Lockhart–McOwen theory of weighted Sobolev spaces on manifolds with ends. This theory is also used to obtain a good asymptotic expansion of the structure on an asymptotically conical manifold under an appropriate gauge-fixing condition, which is required to make the gluing procedure work.
We classify abelian subgroups of up to finite index in an algorithmic and computationally friendly way. A process called disintegration is used to canonically decompose a single rotationless element into a composition of finitely many elements and then these elements are used to generate an abelian subgroup that contains . The main theorem is that up to finite index every abelian subgroup is realized by this construction. As an application we give an explicit description, in terms of relative train track maps and up to finite index, of all maximal rank abelian subgroups of and of .
We give a complete solution for the reduced Gromov–Witten theory of resolved surface singularities of type , for any genus, with arbitrary descendent insertions. We also present a partial evaluation of the –equivariant relative Gromov–Witten theory of the threefold which, under a nondegeneracy hypothesis, yields a complete solution for the theory. The results given here allow comparison of this theory with the quantum cohomology of the Hilbert scheme of points on the surfaces. We discuss generalizations to linear Hodge insertions and to surface resolutions of type . As a corollary, we present a new derivation of the stationary Gromov–Witten theory of .
Greg Friedman has pointed out that there are sign errors in our paper ‘On the chain-level intersection pairing for PL manifolds’, linked above, and in particular Lemma 10.5(b) (which is a key step in the proof of the main theorem) is not correct as stated.
The purpose of this note is to provide a correction.
We construct virtual fundamental classes for dg–manifolds whose tangent sheaves have cohomology only in degrees 0 and 1. This condition is analogous to the existence of a perfect obstruction theory in the approach of Behrend and Fantechi [Invent. Math 128 (1997) 45-88] or Li and Tian [J. Amer. Math. Soc. 11 (1998) 119-174]. Our class is initially defined in –theory as the class of the structure sheaf of the dg–manifold. We compare our construction with that of Behrend and Fantechi as well as with the original proposal of Kontsevich. We prove a Riemann–Roch type result for dg–manifolds which involves integration over the virtual class. We prove a localization theorem for our virtual classes. We also associate to any dg–manifold of our type a cobordism class of almost complex (smooth) manifolds. This supports the intuition that working with dg–manifolds is the correct algebro-geometric replacement of the analytic technique of“deforming to transversal intersection".
For the free group of finite rank we construct a canonical Bonahon-type, continuous and –invariant geometric intersection form
Here is the closure of unprojectivized Culler–Vogtmann Outer space in the equivariant Gromov–Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that consists of all very small minimal isometric actions of on –trees. The projectivization of provides a free group analogue of Thurston’s compactification of Teichmüller space.
As an application, using the intersection graph determined by the intersection form, we show that several natural analogues of the curve complex in the free group context have infinite diameter.