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We provide a model of the String group as a central extension of finite-dimensional –groups in the bicategory of Lie groupoids, left-principal bibundles, and bibundle maps. This bicategory is a geometric incarnation of the bicategory of smooth stacks and generalizes the more naive –category of Lie groupoids, smooth functors, and smooth natural transformations. In particular this notion of smooth –group subsumes the notion of Lie –group introduced by Baez and Lauda [Theory Appl. Categ. 12 (2004) 423–491]. More precisely we classify a large family of these central extensions in terms of the topological group cohomology introduced by Segal [Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), Academic Press, London (1970) 377–387], and our String –group is a special case of such extensions. There is a nerve construction which can be applied to these –groups to obtain a simplicial manifold, allowing comparison with the model of Henriques [arXiv:math/0603563]. The geometric realization is an –space, and in the case of our model, has the correct homotopy type of String(n). Unlike all previous models, our construction takes place entirely within the framework of finite-dimensional manifolds and Lie groupoids. Moreover within this context our model is characterized by a strong uniqueness result. It is a canonical central extension of Spin(n).
Let be a compact Riemannian manifold with boundary. We show that is Gromov–Hausdorff close to a convex Euclidean region of the same dimension if the boundary distance function of is –close to that of . More generally, we prove the same result under the assumptions that the boundary distance function of is –close to that of , the volumes of and are almost equal, and volumes of metric balls in have a certain lower bound in terms of radius.
Given any finite subset of the sphere , , which includes no pairs of antipodal points, we explicitly construct smoothly immersed closed orientable hypersurfaces in Euclidean space whose Gauss map misses . In particular, this answers a question of M Gromov.
Given a measured lamination on a finite area hyperbolic surface we consider a natural measure on the real line obtained by taking the push-forward of the volume measure of the unit tangent bundle of the surface under an intersection function associated with the lamination. We show that the measure gives summation identities for the Rogers dilogarithm function on the moduli space of a surface.
In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that one may determine the existence of a such a metric by analyzing algebraic properties of the Lie algebra and infinitesimal deformations of any initial metric.
Our second main result concerns the isometry groups of such distinguished metrics. Among the completely solvable unimodular Lie groups (this includes nilpotent groups), if the Lie group admits such a metric, we show that the isometry group of this special metric is maximal among all isometry groups of left-invariant metrics.
We prove a version of Gromov’s compactness theorem for pseudoholomorphic curves which holds locally in the target symplectic manifold. This result applies to sequences of curves with an unbounded number of free boundary components, and in families of degenerating target manifolds which have unbounded geometry (eg no uniform energy threshold). Core elements of the proof regard curves as submanifolds (rather than maps) and then adapt methods from the theory of minimal surfaces.
We shall show that for a given homeomorphism type and a set of end invariants (including the parabolic locus) with necessary topological conditions which a topologically tame Kleinian group with that homeomorphism type must satisfy, there is an algebraic limit of minimally parabolic, geometrically finite Kleinian groups which has exactly that homeomorphism type and the given end invariants. This shows that the Bers–Sullivan–Thurston density conjecture follows from Marden’s conjecture proved by Agol and Calegari–Gabai combined with Thurston’s uniformisation theorem and the ending lamination conjecture proved by Minsky, partially collaborating with Masur, Brock and Canary.
Alan Weinstein showed that certain characteristic numbers of any Riemannian submersion with totally geodesic fibers and positive vertizontal curvatures are nonzero. In this paper we explicitly compute these invariants in terms of Chern and Pontrjagin numbers of the bundle. This allows us to show that many bundles do not admit such metrics.
We show that an orientable –dimensional manifold admits a complete riemannian metric of bounded geometry and uniformly positive scalar curvature if and only if there exists a finite collection of spherical space-forms such that is a (possibly infinite) connected sum where each summand is diffeomorphic to or to some member of . This result generalises G Perelman’s classification theorem for compact –manifolds of positive scalar curvature. The main tool is a variant of Perelman’s surgery construction for Ricci flow.
Let be a tree with an action of a finitely generated group . Given a suitable equivalence relation on the set of edge stabilizers of (such as commensurability, coelementarity in a relatively hyperbolic group, or commutation in a commutative transitive group), we define a tree of cylinders . This tree only depends on the deformation space of ; in particular, it is invariant under automorphisms of if is a JSJ splitting. We thus obtain –invariant cyclic or abelian JSJ splittings. Furthermore, has very strong compatibility properties (two trees are compatible if they have a common refinement).
We prove a Milnor–Wood inequality for representations of the fundamental group of a compact complex hyperbolic manifold in the group of isometries of quaternionic hyperbolic space. Of special interest is the case of equality, and its application to rigidity. We show that equality can only be achieved for totally geodesic representations, thereby establishing a global rigidity theorem for totally geodesic representations.
We give an algorithmic proof of the theorem that a closed orientable irreducible and atoroidal –manifold has only finitely many Heegaard splittings in each genus, up to isotopy. The proof gives an algorithm to determine the Heegaard genus of an atoroidal –manifold.
Thanks to recent work of Stipsicz, Szabó and the author and of Bhupal and Stipsicz, one has a complete list of resolution graphs of weighted homogeneous complex surface singularities admitting a rational homology disk (“”) smoothing, that is, one with Milnor number . They fall into several classes, the most interesting of which are the classes whose resolution dual graph has central vertex with valency . We give a uniform “quotient construction” of the smoothings for those classes; it is an explicit –Gorenstein smoothing, yielding a precise description of the Milnor fibre and its non-abelian fundamental group. This had already been done for two of these classes; what is new here is the construction of the third class, which is far more difficult. In addition, we explain the existence of two different smoothings for the first class.
We also prove a general formula for the dimension of a smoothing component for a rational surface singularity. A corollary is that for the valency cases, such a component has dimension and is smooth. Another corollary is that “most” –shaped resolution graphs cannot be the graph of a singularity with a smoothing. This result, plus recent work of Bhupal and Stipsicz, is evidence for a general conjecture:
Conjecture The only complex surface singularities admitting a smoothing are the (known) weighted homogeneous examples.
Let be a nontrivial knot in , and let and be two distinct rational numbers of same sign. We prove that there is no orientation-preserving homeomorphism between the manifolds and . We further generalize this uniqueness result to knots in arbitrary L–space homology spheres.
We prove that the deformation space of marked hyperbolic –manifolds homotopy equivalent to a fixed compact –manifold with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian surface groups are locally connected at quasiconformally rigid points. Similar results are obtained for deformation spaces of acylindrical –manifolds and Bers slices.