(10) Geometry & Topology 15 (2011) 1169-1224 Local topology in deformation spaces of hyperbolic 3-manifolds by Jeffrey F Brock, Kenneth W Bromberg, Richard D Canary and Yair N Minsky URL: http://www.msp.warwick.ac.uk/gt/2011/15-02/p032.xhtml DOI: 10.2140/gt.2011.15.1169
(1) Poincare duality and periodicity by John R Klein and William Richter
We construct periodic families of Poincare complexes, partially solving a question of Hodgson, and infinite families of Poincare complexes whose top cell falls off after one suspension but which fail to embed in a sphere of codimension one. We give a homotopy theoretic description of the four-fold periodicity in knot cobordism.
(2) Flipping bridge surfaces and bounds on the stable bridge number by Jesse Johnson and Maggy Tomova
We show that if K is a knot in the 3-sphere and Sigma is a bridge sphere for K with high distance and 2n punctures, the number of perturbations of K required to interchange the two balls bounded by Sigma via an isotopy is n. We also construct a knot with two different bridge spheres with 2n and 2n-1 bridges respectively for which any common perturbation has at least 3n-4 bridges. We generalize both of these results to bridge surfaces for knots in any 3-manifold.
(3) On the Chabauty space of locally compact abelian groups by Yves Cornulier
This paper contains several results about the Chabauty space of a general locally compact abelian group. Notably, we determine its topological dimension, we characterize when it is totally disconnected or connected; we characterize isolated points.
(4) Generic deformations of the colored sl(N)-homology for links by Hao Wu
We generalize the works of Lee [Adv. Math. 197 (2005) 554--586] and Gornik [arXiv math.QA/0402266] to construct a basis for generic deformations of the colored sl(N)-homology defined in [arXiv 1002.2662v2]. As applications, we construct nondegenerate pairings and co-pairings which lead to dualities of generic deformations of the colored sl(N)-homology. We also define and study colored sl(N)-Rasmussen invariants. Among other things, we observe that these invariants vanish on amphicheiral knots and discuss some implications of this observation.
(5) Rational Z_p-equivariant spectra by David Barnes
We find a simple algebraic model for rational Z_p-equivariant spectra, via a series of Quillen equivalences. This model, along with an Adams short exact sequence, will allow us to easily perform constructions and calculations.
(6) SO(3) homology of graphs and links by Benjamin Cooper, Matt Hogancamp and Vyacheslav Krushkal
The SO(3) Kauffman polynomial and the chromatic polynomial of planar graphs are categorified by a unique extension of the Khovanov homology framework. Many structural observations and computations of homologies of knots and spin networks are included.
(7) On macroscopic dimension of rationally essential manifolds by Alexander Dranishnikov
We construct a counterexamples in dimensions n>3 to Gromov's conjecture that the macroscopic dimension of rationally essential n-dimensional manifolds equals n.
(8) On rational homology disk smoothings of valency 4 surface singularities by Jonathan Wahl
Thanks to recent work of Stipsicz, Szabo 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 ("QHD") smoothing, that is, one with Milnor number 0. They fall into several classes, the most interesting of which are the 3 classes whose resolution dual graph has central vertex with valency 4. We give a uniform "quotient construction" of the QHD smoothings for those classes; it is an explicit Q-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 QHD smoothings for the first class.
We also prove a general formula for the dimension of a QHD smoothing component for a rational surface singularity. A corollary is that for the valency 4 cases, such a component has dimension 1 and is smooth. Another corollary is that "most" H-shaped resolution graphs cannot be the graph of a singularity with a QHD smoothing. This result, plus recent work of Bhupal and Stipsicz, is evidence for a general conjecture:
Conjecture: The only complex surface singularities admitting a QHD smoothing are the (known) weighted homogeneous examples.
(9) Cosmetic surgery in L-space homology spheres by Zhongtao Wu
Let K be a nontrivial knot in the 3-sphere, and let r and r' be two distinct rational numbers of same sign. We prove that there is no orientation-preserving homeomorphism between the manifolds given by Dehn surgery along K with slopes r and r'. We further generalize this uniqueness result to knots in arbitrary L-space homology spheres.
(10) Local topology in deformation spaces of hyperbolic 3-manifolds by Jeffrey F Brock, Kenneth W Bromberg, Richard D Canary and Yair N Minsky
We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M 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 3-manifolds and Bers slices.
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