Six papers published by Geometry & Topology Publications
Geometry and Topology
(1) Algebraic & Geometric Topology 8 (2008) 1523-1565
Organizing volumes of right-angled hyperbolic polyhedra
by Taiyo Inoue
URL: http://www.msp.warwick.ac.uk/agt/2008/08-03/p054.xhtml
DOI: 10.2140/agt.2008.8.1523
(2) Algebraic & Geometric Topology 8 (2008) 1567-1579
The curvature of contact structures on 3-manifolds
by Vladimir Krouglov
URL: http://www.msp.warwick.ac.uk/agt/2008/08-03/p055.xhtml
DOI: 10.2140/agt.2008.8.1567
(3) Algebraic & Geometric Topology 8 (2008) 1581-1592
Sign refinement for combinatorial link Floer homology
by Etienne Gallais
URL: http://www.msp.warwick.ac.uk/agt/2008/08-03/p056.xhtml
DOI: 10.2140/agt.2008.8.1581
(4) Algebraic & Geometric Topology 8 (2008) 1593-1646
Link concordance and generalized doubling operators
by Tim Cochran, Shelly Harvey and Constance Leidy
URL: http://www.msp.warwick.ac.uk/agt/2008/08-03/p057.xhtml
DOI: 10.2140/agt.2008.8.1593
Two papers have been published by Geometry & Topology
(5) Geometry & Topology 12 (2008) 2327-2378
Connected sums of unstabilized Heegaard splittings are unstabilized
by David Bachman
URL: http://www.msp.warwick.ac.uk/gt/2008/12-04/p053.xhtml
DOI: 10.2140/gt.2008.12.2327
(6) Geometry & Topology 12 (2008) 2379-2452
Cobordism of singular maps
by Andras Szucs
URL: http://www.msp.warwick.ac.uk/gt/2008/12-04/p054.xhtml
DOI: 10.2140/gt.2008.12.2379
Abstracts follow
(1) Organizing volumes of right-angled hyperbolic polyhedra
by Taiyo Inoue
This article defines a pair of combinatorial operations on the
combinatorial structure of compact right-angled hyperbolic polyhedra
in dimension three called decomposition and edge surgery. It is shown
that these operations simplify the combinatorics of such a polyhedron,
while keeping it within the class of right-angled objects, until it is
a disjoint union of Loebell polyhedra, a class of polyhedra which
generalizes the dodecahedron. Furthermore, these combinatorial
operations are shown to have geometric realizations which are volume
decreasing. This allows for an organization of the volumes of
right-angled hyperbolic polyhedra and allows, in particular, the
determination of the polyhedra with smallest and second smallest
volumes.
(2) The curvature of contact structures on 3-manifolds
by Vladimir Krouglov
We study the sectional curvature of plane distributions on
3-manifolds. We show that if a distribution is a contact structure it
is easy to manipulate its curvature. As a corollary we obtain that
for every transversally oriented contact structure on a closed
3-dimensional manifold, there is a metric such that the sectional
curvature of the contact distribution is equal to -1. We also
introduce the notion of Gaussian curvature of the plane distribution.
For this notion of curvature we get similar results.
(3) Sign refinement for combinatorial link Floer homology
by Etienne Gallais
Link Floer homology is an invariant for links which has recently been
described entirely in a combinatorial way. Originally constructed
with mod 2 coefficients, it was generalized to integer coefficients
thanks to a sign refinement. In this paper, thanks to the spin
extension of the permutation group we give an alternative construction
of the combinatorial link Floer chain complex associated to a grid
diagram with integer coefficients. In particular we prove that the
sign refinement comes from a 2-cohomological class corresponding to
the spin extension of the permutation group.
(4) Link concordance and generalized doubling operators
by Tim Cochran, Shelly Harvey and Constance Leidy
We introduce a technique for showing classical knots and links are not
slice. As one application we show that the iterated Bing doubles of
many algebraically slice knots are not topologically slice. Some of
the proofs do not use the existence of the Cheeger-Gromov bound, a
deep analytical tool used by Cochran-Teichner. We define generalized
doubling operators, of which Bing doubling is an instance, and prove
our nontriviality results in this more general context. Our main
examples are boundary links that cannot be detected in the algebraic
boundary link concordance group.
(5) Connected sums of unstabilized Heegaard splittings are unstabilized
by David Bachman
Let M_1 and M_2 be closed, orientable 3-manifolds. Let H_i denote a
Heegaard surface in M_i. We prove that if H_1 # H_2 comes from
stabilizing a lower genus splitting of M_1 # M_2 then one of H_1 or
H_2 comes from stabilizing a lower genus splitting. This answers a
question of C Gordon (Problem 3.91 from Kirby's problem list). We also
show that every unstabilized Heegaard splitting has a unique
expression as the connected sum of Heegaard splittings of prime
3-manifolds.
(6) Cobordism of singular maps
by Andras Szucs
Throughout this paper we consider smooth maps of positive
codimensions, having only stable singularities (see Section 1.4 of
Arnold, Gusein-Zade and Varchenko [Monographs in Math. 83, Birkhauser,
Boston (1988)]. We prove a conjecture, due to M Kazarian, connecting
two classifying spaces in singularity theory for this type of singular
maps. These spaces are: 1) Kazarian's space (generalising Vassiliev's
algebraic complex and) showing which cohomology classes are
represented by singularity strata. 2) The space X_tau giving homotopy
representation of cobordisms of singular maps with a given list of
allowed singularities as in work of Rimanyi and the author [Topology
37 (1998) 1177--1191; Mat. Sb. (N.S.) 108 (150) (1979) 433--456, 478;
Lecture Notes in Math. 788, Springer, Berlin (1980) 223--244].
We obtain that the ranks of cobordism groups of singular maps with a
given list of allowed stable singularities, and also their p-torsion
parts for big primes p coincide with those of the homology groups of
the corresponding Kazarian space. (A prime p is ``big'' if it is
greater than half of the dimension of the source manifold.) For all
types of Morin maps (ie when the list of allowed singularities
contains only corank 1 maps) we compute these ranks explicitly.
We give a very transparent homotopical description of the classifying
space X_tau as a fibration. Using this fibration we solve the problem
of elimination of singularities by cobordisms. (This is a modification
of a question posed by Arnold [Itogi Nauki i Tekniki, Moscow (1988)
5--257].)