48 research outputs found
On open 3-manifolds proper homotopy equivalent to geometrically-simply connected polyhedra
We prove that an open 3-manifold proper homotopy equivalent to a
geometrically simply connected polyhedron is simply connected at infinity,
generalizing a theorem of V.Poenaru.Comment: revised version, 11 pages, Topology Appl. (to appear
An infinite genus mapping class group and stable cohomology
We exhibit a finitely generated group \M whose rational homology is
isomorphic to the rational stable homology of the mapping class group. It is
defined as a mapping class group associated to a surface \su of infinite
genus, and contains all the pure mapping class groups of compact surfaces of
genus with boundary components, for any and . We
construct a representation of \M into the restricted symplectic group of the real Hilbert space generated by the homology
classes of non-separating circles on \su, which generalizes the classical
symplectic representation of the mapping class groups. Moreover, we show that
the first universal Chern class in H^2(\M,\Z) is the pull-back of the
Pressley-Segal class on the restricted linear group
via the inclusion .Comment: 14p., 8 figures, to appear in Commun.Math.Phy
On Bohr-Sommerfeld bases
This paper combines algebraic and Lagrangian geometry to construct a special
basis in every space of conformal blocks, the Bohr-Sommerfeld (BS) basis. We
use the method of [D. Borthwick, T. Paul and A. Uribe, Legendrian distributions
with applications to the non-vanishing of Poincar\'e series of large weight,
Invent. math, 122 (1995), 359-402, preprint hep-th/9406036], whereby every
vector of a BS basis is defined by some half-weighted Legendrian distribution
coming from a Bohr-Sommerfeld fibre of a real polarization of the underlying
symplectic manifold. The advantage of BS bases (compared to bases of theta
functions in [A. Tyurin, Quantization and ``theta functions'', Jussieu preprint
216 (Apr 1999), e-print math.AG/9904046, 32pp.]) is that we can use information
from the skillful analysis of the asymptotics of quantum states. This gives
that Bohr-Sommerfeld bases are unitary quasi-classically. Thus we can apply
these bases to compare the Hitchin connection with the KZ connection defined by
the monodromy of the Knizhnik-Zamolodchikov equation in combinatorial theory
(see, for example, [T. Kohno, Topological invariants for 3-manifolds using
representations of mapping class group I, Topology 31 (1992), 203-230; II,
Contemp. math 175} (1994), 193-217]).Comment: 43 pages, uses: latex2e with amsmath,amsfonts,theore
Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t = -1
We prove that the hyperelliptic Torelli group is generated by Dehn twists about
separating curves that are preserved by the hyperelliptic involution. This verifies a
conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel
of the Burau representation evaluated at t = −1 and also the fundamental group of
the branch locus of the period mapping, and so we obtain analogous generating sets
for those. One application is that each component in Torelli space of the locus of
hyperelliptic curves becomes simply connected when curves of compact type are added
Profinite rigidity for Seifert fibre spaces
An interesting question is whether two 3-manifolds can be distinguished by
computing and comparing their collections of finite covers; more precisely, by
the profinite completions of their fundamental groups. In this paper, we solve
this question completely for closed orientable Seifert fibre spaces. In
particular, all Seifert fibre spaces are distinguished from each other by their
profinite completions apart from some previously-known examples due to Hempel.
We also characterize when bounded Seifert fibre space groups have isomorphic
profinite completions, given some conditions on the boundary
Boundary Conformal Field Theories, Limit Sets of Kleinian Groups and Holography
In this paper,based on the available mathematical works on geometry and
topology of hyperbolic manifolds and discrete groups, some results of Freedman
et al (hep-th/9804058) are reproduced and broadly generalized. Among many new
results the possibility of extension of work of Belavin, Polyakov and
Zamolodchikov to higher dimensions is investigated. Known in physical
literature objections against such extension are removed and the possibility of
an extension is convincingly demonstrated.Comment: 62 pages, 5 figure
Non-Abelian Anyons and Topological Quantum Computation
Topological quantum computation has recently emerged as one of the most
exciting approaches to constructing a fault-tolerant quantum computer. The
proposal relies on the existence of topological states of matter whose
quasiparticle excitations are neither bosons nor fermions, but are particles
known as {\it Non-Abelian anyons}, meaning that they obey {\it non-Abelian
braiding statistics}. Quantum information is stored in states with multiple
quasiparticles, which have a topological degeneracy. The unitary gate
operations which are necessary for quantum computation are carried out by
braiding quasiparticles, and then measuring the multi-quasiparticle states. The
fault-tolerance of a topological quantum computer arises from the non-local
encoding of the states of the quasiparticles, which makes them immune to errors
caused by local perturbations. To date, the only such topological states
thought to have been found in nature are fractional quantum Hall states, most
prominently the \nu=5/2 state, although several other prospective candidates
have been proposed in systems as disparate as ultra-cold atoms in optical
lattices and thin film superconductors. In this review article, we describe
current research in this field, focusing on the general theoretical concepts of
non-Abelian statistics as it relates to topological quantum computation, on
understanding non-Abelian quantum Hall states, on proposed experiments to
detect non-Abelian anyons, and on proposed architectures for a topological
quantum computer. We address both the mathematical underpinnings of topological
quantum computation and the physics of the subject using the \nu=5/2 fractional
quantum Hall state as the archetype of a non-Abelian topological state enabling
fault-tolerant quantum computation.Comment: Final Accepted form for RM
Surgery Equivalence And Finite Type Invariants For Homology 3-Spheres
. One considers two equivalence relations on 3-manifolds related to finite type invariants. The first one requires to have matching invariants in degree less than k + 1 and it is based on a filtration introduced by Garoufalidis and Levine ([4, 1]). The other one allows manifolds to be cut open along embedded surfaces and twist by an element of the k + 1-th term of the lower central series of the group of BSCC maps. The main result states the two relations coincide on the level of homology 3-spheres. The analogous result for the Torelli group was announced by Habiro ([6]), using claspers theory. Similar results for knots and Vassiliev invariants were obtained by Gusarov, Habiro and Stanford ([10]). 1. Finite type invariants and H 1 -bordism classes 1.1. The subgroup of BSCC maps. Let \Sigma g (respectively \Sigma g;1 ) be a closed oriented surface of genus g 2 (with one hole). The mapping class group M g;1 is the group of orientation preserving diffeomorphisms of the surface \Sigma g;..