(1) Differential operators and the wheels power series by Andrew Kricker
An earlier work of the author's showed that it was possible to adapt the Alekseev-Meinrenken Chern-Weil proof of the Duflo isomorphism to obtain a completely combinatorial proof of the wheeling isomorphism. That work depended on a certain combinatorial identity, which said that a particular composition of elementary combinatorial operations arising from the proof was precisely the wheeling operation. The identity can be summarized as follows: The wheeling operation is just a graded averaging map in a space enlarging the space of Jacobi diagrams. The purpose of this paper is to present a detailed and self-contained proof of this identity. The proof broadly follows similar calculations in the Alekseev-Meinrenken theory, though the details here are somewhat different, as the algebraic manipulations in the original are replaced with arguments concerning the enumerative combinatorics of formal power series of graphs with graded legs.
(2) Homotopy algebra structures on twisted tensor products and string topology operations by Micah Miller
Given a C-infinity coalgebra C_*, a strict dg Hopf algebra H_* and a twisting cochain tau: C_* -> H_* such that Im(tau) is in Prim(H_*), we describe a procedure for obtaining an A-infinity coalgebra on the tensor product of C_* with H_*. This is an extension of Brown's work on twisted tensor products. We apply this procedure to obtain an A-infinity coalgebra model of the chains on the free loop space LM based on the C-infinity coalgebra structure of H_*(M) induced by the diagonal map M -> M x M and the Hopf algebra model of the based loop space given by T(H_*(M)[-1]). When C_* has cyclic C-infinity coalgebra structure, we describe an A-infinity algebra on the tensor product of C_* with H_*. This is used to give an explicit (nonminimal) A-infinity algebra model of the string topology loop product. Finally, we discuss a representation of the loop product in principal G-bundles.
(3) Meridional destabilizing number of knots by Toshio Saito
We define the meridional destabilizing number of a knot. This together with Heegaard genus (or tunnel number) gives a binary complexity of knots. We study its behavior under connected sum of tunnel number one knots.
(4) Knots which admit a surgery with simple knot Floer homology groups by Eaman Eftekhary
We show that if a positive integral surgery on a knot K inside a homology sphere X results in an induced knot K_n in X_n(K)=Y which has simple Floer homology then n >= 2g(K). Moreover, for X=S^3 the three-manifold Y is an L-space, and the Heegaard Floer homology groups of K are determined by its Alexander polynomial.
(5) Coverings and minimal triangulations of 3-manifolds by William Jaco, J Hyam Rubinstein and Stephan Tillmann
This paper uses results on the classification of minimal triangulations of 3-manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space L(4k,2k-1) and the generalised quaternionic space which is the quotient of the three-sphere by Q_4k have complexity k, where k is at least 2. Moreover, it is shown that their minimal triangulations are unique.
(6) On genus-1 simplified broken Lefschetz fibrations by Kenta Hayano
Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefschetz fibrations in order to describe near-symplectic 4-manifolds. We first study monodromy representations of higher sides of genus-1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism types of such fibrations with connected fibers and with less than six Lefschetz singularities. In these studies, we obtain several families of genus-1 simplified broken Lefschetz fibrations, which we conjecture contain all such fibrations, and determine the diffeomorphism types of the total spaces of these fibrations. Our results are generalizations of Kas' classification theorem of genus-1 Lefschetz fibrations, which states that the total space of a nontrivial genus-1 Lefschetz fibration over the 2-sphere is diffeomorphic to an elliptic surface E(n) for some n at least 1.
(7) Central extensions of smooth 2-groups and a finite-dimensional string 2-group by Christopher J Schommer-Pries
We provide a model of the String group as a central extension of finite-dimensional 2-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 2-category of Lie groupoids, smooth functors and smooth natural transformations. In particular this notion of smooth 2-group subsumes the notion of Lie 2-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 2-group is a special case of such extensions. There is a nerve construction which can be applied to these 2-groups to obtain a simplicial manifold, allowing comparison with the model of Henriques [arXiv:math/0603563]. The geometric realization is an A-infinity-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).
(8) On Gromov-Hausdorff stability in a boundary rigidity problem by Sergei Ivanov
Let M be a compact Riemannian manifold with boundary. We show that M is Gromov-Hausdorff close to a convex Euclidean region D of the same dimension if the boundary distance function of M is C^1-close to that of D. More generally, we prove the same result under the assumptions that the boundary distance function of M is C^0-close to that of D, the volumes of M and D are almost equal, and volumes of metric balls in M have a certain lower bound in terms of radius.
(9) Directed immersions of closed manifolds by Mohammad Ghomi
Given any finite subset X of the n-sphere, n at least 2, which includes no pairs of antipodal points, we explicitly construct smoothly immersed closed orientable hypersurfaces in (n+1)-dimensional Euclidean space whose Gauss map misses X. In particular, this answers a question of M Gromov.
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