719 research outputs found
Compactification, topology change and surgery theory
We study the process of compactification as a topology change. It is shown
how the mediating spacetime topology, or cobordism, may be simplified through
surgery. Within the causal Lorentzian approach to quantum gravity, it is shown
that any topology change in dimensions may be achieved via a causally
continuous cobordism. This extends the known result for 4 dimensions.
Therefore, there is no selection rule for compactification at the level of
causal continuity. Theorems from surgery theory and handle theory are seen to
be very relevant for understanding topology change in higher dimensions.
Compactification via parallelisable cobordisms is particularly amenable to
study with these tools.Comment: 1+19 pages. LaTeX. 9 associated eps files. Discussion of disconnected
case adde
Localized Exotic Smoothness
Gompf's end-sum techniques are used to establish the existence of an infinity
of non-diffeomorphic manifolds, all having the same trivial
topology, but for which the exotic differentiable structure is confined to a
region which is spatially limited. Thus, the smoothness is standard outside of
a region which is topologically (but not smoothly) ,
where is the compact three ball. The exterior of this region is
diffeomorphic to standard . In a
space-time diagram, the confined exoticness sweeps out a world tube which, it
is conjectured, might act as a source for certain non-standard solutions to the
Einstein equations. It is shown that smooth Lorentz signature metrics can be
globally continued from ones given on appropriately defined regions, including
the exterior (standard) region. Similar constructs are provided for the
topology, of the Kruskal form of the Schwarzschild
solution. This leads to conjectures on the existence of Einstein metrics which
are externally identical to standard black hole ones, but none of which can be
globally diffeomorphic to such standard objects. Certain aspects of the Cauchy
problem are also discussed in terms of \models which are
``half-standard'', say for all but for which cannot be globally
smooth.Comment: 8 pages plus 6 figures, available on request, IASSNS-HEP-94/2
Twisted topological structures related to M-branes II: Twisted Wu and Wu^c structures
Studying the topological aspects of M-branes in M-theory leads to various
structures related to Wu classes. First we interpret Wu classes themselves as
twisted classes and then define twisted notions of Wu structures. These
generalize many known structures, including Pin^- structures, twisted Spin
structures in the sense of Distler-Freed-Moore, Wu-twisted differential
cocycles appearing in the work of Belov-Moore, as well as ones introduced by
the author, such as twisted Membrane and twisted String^c structures. In
addition, we introduce Wu^c structures, which generalize Pin^c structures, as
well as their twisted versions. We show how these structures generalize and
encode the usual structures defined via Stiefel-Whitney classes.Comment: 20 page
A handlebody calculus for topology change
We consider certain interesting processes in quantum gravity which involve a
change of spatial topology. We use Morse theory and the machinery of
handlebodies to characterise topology changes as suggested by Sorkin. Our
results support the view that that the pair production of Kaluza-Klein
monopoles and the nucleation of various higher dimensional objects are allowed
transitions with non-zero amplitude.Comment: Latex, 32 pages, 7 figure
Towards a CPT Invariant Quantum Field Theory on Elliptic de Sitter Space
Consequences of Schr\"{o}dinger's antipodal identification on quantum field
theory in de Sitter space are investigated. The elliptic
identification provides observers with complete information. We show that a
suitable confinement on dimension of the elliptic de Sitter space guarantees
the existence of globally defined spinors and orientable
manifold. In Beltrami coordinates, we give exact solutions of scalar and spinor
fields. The CPT invariance of quantum field theory on the elliptic de Sitter
space is presented explicitly.Comment: 16 pages, some references have been added, the structure of paper
have been revised, accepted for publication in Int. J. Mod. Phys.
Invertible Dirac operators and handle attachments on manifolds with boundary
For spin manifolds with boundary we consider Riemannian metrics which are
product near the boundary and are such that the corresponding Dirac operator is
invertible when half-infinite cylinders are attached at the boundary. The main
result of this paper is that these properties of a metric can be preserved when
the metric is extended over a handle of codimension at least two attached at
the boundary. Applications of this result include the construction of
non-isotopic metrics with invertible Dirac operator, and a concordance
existence and classification theorem.Comment: Accepted for publication in Journal of Topology and Analysi
Duality properties of indicatrices of knots
The bridge index and superbridge index of a knot are important invariants in
knot theory. We define the bridge map of a knot conformation, which is closely
related to these two invariants, and interpret it in terms of the tangent
indicatrix of the knot conformation. Using the concepts of dual and derivative
curves of spherical curves as introduced by Arnold, we show that the graph of
the bridge map is the union of the binormal indicatrix, its antipodal curve,
and some number of great circles. Similarly, we define the inflection map of a
knot conformation, interpret it in terms of the binormal indicatrix, and
express its graph in terms of the tangent indicatrix. This duality relationship
is also studied for another dual pair of curves, the normal and Darboux
indicatrices of a knot conformation. The analogous concepts are defined and
results are derived for stick knots.Comment: 22 pages, 9 figure
The multipliers of periodic points in one-dimensional dynamics
It will be shown that the smooth conjugacy class of an unimodal map which
does not have a periodic attractor neither a Cantor attractor is determined by
the multipliers of the periodic orbits. This generalizes a result by M.Shub and
D.Sullivan for smooth expanding maps of the circle
Electrical networks on -simplex fractals
The decimation map for a network of admittances on an
-simplex lattice fractal is studied. The asymptotic behaviour of
for large-size fractals is examined. It is found that in the
vicinity of the isotropic point the eigenspaces of the linearized map are
always three for ; they are given a characterization in terms of
graph theory. A new anisotropy exponent, related to the third eigenspace, is
found, with a value crossing over from to
.Comment: 14 pages, 8 figure
Causal continuity in degenerate spacetimes
A change of spatial topology in a causal, compact spacetime cannot occur when
the metric is globally Lorentzian. One can however construct a causal metric
from a Riemannian metric and a Morse function on the background cobordism
manifold, which is Lorentzian almost everywhere except that it is degenerate at
each critical point of the function. We investigate causal structure in the
neighbourhood of such a degeneracy, when the auxiliary Riemannian metric is
taken to be Cartesian flat in appropriate coordinates. For these geometries, we
verify Borde and Sorkin's conjecture that causal discontinuity occurs if and
only if the Morse index is 1 or n-1.Comment: 34 pages, 11 figures, Latex2e, important references added,
introduction and discussions sections reworded slightl
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