Nine papers published by Geometry & Topology Publications
Geometry and Topology
(1) Algebraic & Geometric Topology 10 (2010) 1609-1625
Continuous interval exchange actions
by Christopher F Novak
URL: http://www.msp.warwick.ac.uk/agt/2010/10-03/p054.xhtml
DOI: 10.2140/agt.2010.10.1609
(2) Algebraic & Geometric Topology 10 (2010) 1627-1664
Fibered orbifolds and crystallographic groups
by John G Ratcliffe and Steven T Tschantz
URL: http://www.msp.warwick.ac.uk/agt/2010/10-03/p055.xhtml
DOI: 10.2140/agt.2010.10.1627
(3) Algebraic & Geometric Topology 10 (2010) 1665-1681
More Cappell-Shaneson spheres are standard
by Robert E Gompf
URL: http://www.msp.warwick.ac.uk/agt/2010/10-03/p056.xhtml
DOI: 10.2140/agt.2010.10.1665
(4) Algebraic & Geometric Topology 10 (2010) 1683-1714
Simplicial models of trace spaces
by Martin Raussen
URL: http://www.msp.warwick.ac.uk/agt/2010/10-03/p057.xhtml
DOI: 10.2140/agt.2010.10.1683
(5) Algebraic & Geometric Topology 10 (2010) 1715-1738
Instanton Floer homology and the Alexander polynomial
by Peter Kronheimer and Tom Mrowka
URL: http://www.msp.warwick.ac.uk/agt/2010/10-03/p058.xhtml
DOI: 10.2140/agt.2010.10.1715
(6) Algebraic & Geometric Topology 10 (2010) 1739-1746
Coalgebraic tensor product and homology operations
by Takuji Kashiwabara
URL: http://www.msp.warwick.ac.uk/agt/2010/10-03/p059.xhtml
DOI: 10.2140/agt.2010.10.1739
Three papers have been published by Geometry & Topology
(7) Geometry & Topology 14 (2010) 1765-1870
Symplectic topology of Mane's critical values
by Kai Cieliebak, Urs Frauenfelder and Gabriel P Paternain
URL: http://www.msp.warwick.ac.uk/gt/2010/14-03/p037.xhtml
DOI: 10.2140/gt.2010.14.1765
(8) Geometry & Topology 14 (2010) 1871-1919
Heegaard surfaces and the distance of amalgamation
by Tao Li
URL: http://www.msp.warwick.ac.uk/gt/2010/14-04/p038.xhtml
DOI: 10.2140/gt.2010.14.1871
(9) Geometry & Topology 14 (2010) 1921-1940
Bounds on exceptional Dehn filling II
by Ian Agol
URL: http://www.msp.warwick.ac.uk/gt/2010/14-04/p039.xhtml
DOI: 10.2140/gt.2010.14.1921
Abstracts follow
(1) Continuous interval exchange actions
by Christopher F Novak
Let E denote the group of all interval exchange transformations on
[0,1). Given a suitable topological group structure on E, it is
possible to classify all one-parameter interval exchange actions
(continuous homomorphisms R --> E). In particular, up to conjugacy in
E, any one-parameter interval exchange action factors through a
rotational torus action.
(2) Fibered orbifolds and crystallographic groups
by John G Ratcliffe and Steven T Tschantz
In this paper, we prove that a normal subgroup N of an n-dimensional
crystallographic group Gamma determines a geometric fibered orbifold
structure on the flat orbifold E^n/Gamma, and conversely every
geometric fibered orbifold structure on E^n/Gamma is determined by a
normal subgroup N of Gamma. In particular, we prove that E^n/Gamma is
a fiber bundle, with totally geodesic fibers, over a
beta_1-dimensional torus, where beta_1 is the first Betti number of
Gamma.
(3) More Cappell-Shaneson spheres are standard
by Robert E Gompf
Akbulut has recently shown that an infinite family of Cappell-Shaneson
homotopy 4-spheres is diffeomorphic to the standard 4-sphere. In the
present paper, a different method shows that a strictly larger family
is standard. This new approach uses no Kirby calculus except through
the relatively simple 1979 paper of Akbulut and Kirby showing that the
simplest example with untwisted framing is standard. Instead, hidden
symmetries of the original Cappell-Shaneson construction are
exploited. In the course of the proof, an example is given showing
that Gluck twists can sometimes be undone using symmetries of fishtail
neighborhoods.
(4) Simplicial models of trace spaces
by Martin Raussen
Directed algebraic topology studies topological spaces in which
certain directed paths (d-paths) are singled out; in most cases of
interest, the reverse path of a d-path is no longer a d-path. We are
mainly concerned with spaces of directed paths between given end
points, and how those vary under variation of the end points. The
original motivation stems from certain models for concurrent
computation. So far, homotopy types of spaces of d-paths and their
topological invariants have only been determined in cases that were
elementary to overlook.
In this paper, we develop a systematic approach describing spaces of
directed paths - up to homotopy equivalence - as finite
prodsimplicial complexes, ie with products of simplices as building
blocks. This method makes use of a certain poset category of binary
matrices related to a given model space. It applies to a class of
directed spaces that arise from a certain class of models of
computation - still restricted but with a fair amount of
generality. In the final section, we outline a generalization to model
spaces known as Higher Dimensional Automata.
In particular, we describe algorithms that allow us to determine not
only the fundamental category of such a model space, but all
homological invariants of spaces of directed paths within it. The
prodsimplical complexes and their associated chain complexes are
finite, but they will, in general, have a huge number of cells and
generators.
(5) Instanton Floer homology and the Alexander polynomial
by Peter Kronheimer and Tom Mrowka
The instanton Floer homology of a knot in the three-sphere is a vector
space with a canonical mod 2 grading. It carries a distinguished
endomorphism of even degree, arising from the 2-dimensional homology
class represented by a Seifert surface. The Floer homology decomposes
as a direct sum of the generalized eigenspaces of this
endomorphism. We show that the Euler characteristics of these
generalized eigenspaces are the coefficients of the Alexander
polynomial of the knot. Among other applications, we deduce that
instanton homology detects fibered knots.
(6) Coalgebraic tensor product and homology operations
by Takuji Kashiwabara
In this paper, we show that Hunton-Turner coalgebraic tensor product
respects various actions of Hopf algebras of homology operations.
(7) Symplectic topology of Mane's critical values
by Kai Cieliebak, Urs Frauenfelder and Gabriel P Paternain
We study the dynamics and symplectic topology of energy hypersurfaces
of mechanical Hamiltonians on twisted cotangent bundles. We pay
particular attention to periodic orbits, displaceability, stability
and the contact type property, and the changes that occur at the Mane
critical value c. Our main tool is Rabinowitz Floer homology. We show
that it is defined for hypersurfaces that are either stable tame or
virtually contact, and that it is invariant under homotopies in these
classes. If the configuration space admits a metric of negative
curvature, then Rabinowitz Floer homology does not vanish for energy
levels k > c and, as a consequence, these level sets are not
displaceable. We provide a large class of examples in which Rabinowitz
Floer homology is nonzero for energy levels k > c but vanishes for k <
c, so levels above and below c cannot be connected by a stable tame
homotopy. Moreover, we show that for strictly 1/4-pinched negative
curvature and nonexact magnetic fields all sufficiently high energy
levels are nonstable, provided that the dimension of the base manifold
is even and different from two.
(8) Heegaard surfaces and the distance of amalgamation
by Tao Li
Let M_1 and M_2 be orientable irreducible 3-manifolds with connected
boundary and suppose the boundary of M_1 is homeomorphic to the
boundary of M_2. Let M be a closed 3-manifold obtained by gluing M_1
to M_2 along the boundary. We show that if the gluing homeomorphism
is sufficiently complicated, then M is not homeomorphic to the
3-sphere and all small-genus Heegaard splittings of M are standard in
a certain sense. In particular, the Heegaard genus of M is the sum of
the Heegaard genuses of M_1 and M_2 minus the Heegaard genuses of the
boundaries of M_1 and M_2. This theorem is also true for certain
manifolds with multiple boundary components.
(9) Bounds on exceptional Dehn filling II
by Ian Agol
We show that there are at most finitely many one cusped orientable hyperbolic
3-manifolds which have more than eight nonhyperbolic Dehn fillings.
Moreover, we show that determining these finitely many manifolds
is decidable.
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