254 research outputs found
Finitness of the basic intersection cohomology of a Killing foliation
We prove that the basic intersection cohomology where is the singular
foliation determined by an isometric action of a Lie group on the compact
manifold , is finite dimensional
The two components of the SO(3)-character space of the fundamental group of a closed surface of genus 2
We use geometric techniques to explicitly find the topological structure of
the space of SO(3)-representations of the fundamental group of a closed surface
of genus 2 quotient by the conjugation action by SO(3). There are two
components of the space. We will describe the topology of both components and
describe the corresponding SU(2)-character spaces by parametrizing them by
spherical triangles. There is the sixteen to one branch-covering for each
component, and the branch locus is a union of 2-spheres or 2-tori. Along the
way, we also describe the topology of both spaces. We will later relate this
result to future work into higher-genus cases and the SL(3,R)-representations
Kinematic Orbits and the Structure of the Internal Space for Systems of Five or More Bodies
The internal space for a molecule, atom, or other n-body system can be
conveniently parameterised by 3n-9 kinematic angles and three kinematic
invariants. For a fixed set of kinematic invariants, the kinematic angles
parameterise a subspace, called a kinematic orbit, of the n-body internal
space. Building on an earlier analysis of the three- and four-body problems, we
derive the form of these kinematic orbits (that is, their topology) for the
general n-body problem. The case n=5 is studied in detail, along with the
previously studied cases n=3,4.Comment: 38 pages, submitted to J. Phys.
Orbits of quantum states and geometry of Bloch vectors for -level systems
Physical constraints such as positivity endow the set of quantum states with
a rich geometry if the system dimension is greater than two. To shed some light
on the complicated structure of the set of quantum states, we consider a
stratification with strata given by unitary orbit manifolds, which can be
identified with flag manifolds. The results are applied to study the geometry
of the coherence vector for n-level quantum systems. It is shown that the
unitary orbits can be naturally identified with spheres in R^{n^2-1} only for
n=2. In higher dimensions the coherence vector only defines a non-surjective
embedding into a closed ball. A detailed analysis of the three-level case is
presented. Finally, a refined stratification in terms of symplectic orbits is
considered.Comment: 15 pages LaTeX, 3 figures, reformatted, slightly modified version,
corrected eq.(3), to appear in J. Physics
Abelian gauge theories on compact manifolds and the Gribov ambiguity
We study the quantization of abelian gauge theories of principal torus
bundles over compact manifolds with and without boundary. It is shown that
these gauge theories suffer from a Gribov ambiguity originating in the
non-triviality of the bundle of connections whose geometrical structure will be
analyzed in detail. Motivated by the stochastic quantization approach we
propose a modified functional integral measure on the space of connections that
takes the Gribov problem into account. This functional integral measure is used
to calculate the partition function, the Greens functions and the field
strength correlating functions in any dimension using the fact that the space
of inequivalent connections itself admits the structure of a bundle over a
finite dimensional torus. The Greens functions are shown to be affected by the
non-trivial topology, giving rise to non-vanishing vacuum expectation values
for the gauge fields.Comment: 33 page
Future asymptotic expansions of Bianchi VIII vacuum metrics
Bianchi VIII vacuum solutions to Einstein's equations are causally
geodesically complete to the future, given an appropriate time orientation, and
the objective of this article is to analyze the asymptotic behaviour of
solutions in this time direction. For the Bianchi class A spacetimes, there is
a formulation of the field equations that was presented in an article by
Wainwright and Hsu, and in a previous article we analyzed the asymptotic
behaviour of solutions in these variables. One objective of this paper is to
give an asymptotic expansion for the metric. Furthermore, we relate this
expansion to the topology of the compactified spatial hypersurfaces of
homogeneity. The compactified spatial hypersurfaces have the topology of
Seifert fibred spaces and we prove that in the case of NUT Bianchi VIII
spacetimes, the length of a circle fibre converges to a positive constant but
that in the case of general Bianchi VIII solutions, the length tends to
infinity at a rate we determine.Comment: 50 pages, no figures. Erronous definition of Seifert fibred spaces
correcte
On the symmetry breaking phenomenon
We investigate the problem of symmetry breaking in the framework of dynamical
systems with symmetry on a smooth manifold. Two cases will be analyzed: general
and Hamiltonian dynamical systems. We give sufficient conditions for symmetry
breaking in both cases
Combinatorial Stokes formulas via minimal resolutions
We describe an explicit chain map from the standard resolution to the minimal
resolution for the finite cyclic group Z_k of order k. We then demonstrate how
such a chain map induces a "Z_k-combinatorial Stokes theorem", which in turn
implies "Dold's theorem" that there is no equivariant map from an n-connected
to an n-dimensional free Z_k-complex.
Thus we build a combinatorial access road to problems in combinatorics and
discrete geometry that have previously been treated with methods from
equivariant topology. The special case k=2 for this is classical; it involves
Tucker's (1949) combinatorial lemma which implies the Borsuk-Ulam theorem, its
proof via chain complexes by Lefschetz (1949), the combinatorial Stokes formula
of Fan (1967), and Meunier's work (2006).Comment: 18 page
3-manifold invariants and periodicity of homology spheres
We show how the periodicity of a homology sphere is reflected in the
Reshetikhin-Turaev-Witten invariants of the manifold. These yield a criterion
for the periodicity of a homology sphere.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-34.abs.htm
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