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Geometry and Topology

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Registered: 5/24/06

Nine papers published by Geometry & Topology Publications
Posted: May 10, 2011 2:53 PM

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Six papers have been published by Algebraic & Geometric Topology

(1) Algebraic & Geometric Topology 11 (2011) 1107-1162
   Differential operators and the wheels power series
     by Andrew Kricker
   URL: http://www.msp.warwick.ac.uk/agt/2011/11-02/p036.xhtml
   DOI: 10.2140/agt.2011.11.1107

(2) Algebraic & Geometric Topology 11 (2011) 1163-1203
   Homotopy algebra structures on twisted tensor products and string
topology operations
     by Micah Miller
   URL: http://www.msp.warwick.ac.uk/agt/2011/11-02/p037.xhtml
   DOI: 10.2140/agt.2011.11.1163

(3) Algebraic & Geometric Topology 11 (2011) 1205-1242
   Meridional destabilizing number of knots
     by Toshio Saito
   URL: http://www.msp.warwick.ac.uk/agt/2011/11-02/p038.xhtml
   DOI: 10.2140/agt.2011.11.1205

(4) Algebraic & Geometric Topology 11 (2011) 1243-1256
   Knots which admit a surgery with simple knot Floer homology groups
     by Eaman Eftekhary
   URL: http://www.msp.warwick.ac.uk/agt/2011/11-03/p039.xhtml
   DOI: 10.2140/agt.2011.11.1243

(5) Algebraic & Geometric Topology 11 (2011) 1257-1265
   Coverings and minimal triangulations of 3-manifolds
     by William Jaco, J Hyam Rubinstein and Stephan Tillmann
   URL: http://www.msp.warwick.ac.uk/agt/2011/11-03/p040.xhtml
   DOI: 10.2140/agt.2011.11.1257

(6) Algebraic & Geometric Topology 11 (2011) 1267-1322
   On genus-1 simplified broken Lefschetz fibrations
     by Kenta Hayano
   URL: http://www.msp.warwick.ac.uk/agt/2011/11-03/p041.xhtml
   DOI: 10.2140/agt.2011.11.1267

Three papers have been published by Geometry & Topology

(7) Geometry & Topology 15 (2011) 609-676
   Central extensions of smooth 2-groups and a finite-dimensional
string 2-group
     by Christopher J Schommer-Pries
   URL: http://www.msp.warwick.ac.uk/gt/2011/15-02/p017.xhtml
   DOI: 10.2140/gt.2011.15.609

(8) Geometry & Topology 15 (2011) 677-697
   On Gromov-Hausdorff stability in a boundary rigidity problem
     by Sergei Ivanov
   URL: http://www.msp.warwick.ac.uk/gt/2011/15-02/p018.xhtml
   DOI: 10.2140/gt.2011.15.677

(9) Geometry & Topology 15 (2011) 699-705
   Directed immersions of closed manifolds
     by Mohammad Ghomi
   URL: http://www.msp.warwick.ac.uk/gt/2011/15-02/p019.xhtml
   DOI: 10.2140/gt.2011.15.699

Abstracts follow

(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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