Patrick M. Gilmer
Extra structure and the universal construction for the Witten-Reshetikhin-Turaev TQFT.
Wed, 01/11/2012 - 15:01 — SomNambulAuthors: Patrick M. Gilmer, Xuanye Wang
Subjects: Geometric Topology
AbstractA TQFT is a functor from a cobordism category to the category of vector
spaces, satisfying certain properties. An important property is that the vector
spaces should be finite dimensional. For the WRT TQFT, the relevant
2+1-cobordism category is built from manifolds which are equipped with an extra
structure such as a p_1-structure, or an extended manifold structure. The
purpose of this paper is to explain that without this extra structure, one
would not get finite dimensionality.Irreducible factors of modular representations of mapping class groups arising in Integral TQFT.
Thu, 10/27/2011 - 15:01 — SomNambulAuthors: Patrick M. Gilmer, Gregor Masbaum
Subjects: Geometric Topology
AbstractWe find decomposition series of length at most two for modular
representations in positive characteristic of mapping class groups of surfaces
induced by an integral version of the Witten-Reshetikhin-Turaev SO(3)-TQFT at
the p-th root of unity, where p is an odd prime. The dimensions of the
irreducible factors are given by Verlinde-type formulas.- 10 comments
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Maslov index, Lagrangians, Mapping Class Groups and TQFT.
Thu, 12/24/2009 - 16:01 — SomNambulAuthors: Patrick M. Gilmer, Gregor Masbaum
Subjects: Geometric Topology
AbstractGiven a mapping class f of an oriented surface Sigma and a lagrangian lambda
in the first homology of Sigma, we define an integer n_{lambda}(f) modulo 4. We
use n_{lambda}(f) to describe a universal central extension of the mapping
class group of Sigma as an index-four subgroup of the extension constructed
from the Maslov index of triples of lagrangian subspaces in the homology of the
surface. We give two formulas for n_{lambda}(f). One is topological using
surgery, the other is homological and builds on work of Turaev and work of
Walker. Some applications to TQFT are discussed.Heegaard genus, cut number, weak p-congruence, and quantum invariants.
Fri, 08/21/2009 - 19:51 — SomNambulAuthors: Patrick M. Gilmer
Subjects: Geometric Topology
AbstractWe use quantum invariants to define a 3-manifold invariant j_p which lies in
the non-negative integers. We relate j_p to the Heegard genus, and the cut
number. We show that $j_p$ is an invariant of weak p-congruence.Integral TQFT for a one-holed torus.
Thu, 08/20/2009 - 23:06 — SomNambulAuthors: Patrick M. Gilmer, Gregor Masbaum
Subjects: Geometric Topology
AbstractWe give new explicit formulas for the representations of the mapping class
group of a genus one surface with one boundary component which arise from
Integral TQFT. Our formulas allow one to compute the h-adic expansion of the
TQFT-matrix associated to a mapping class in a straightforward way. Truncating
the h-adic expansion gives an approximation of the representation by
representations into finite groups. As a special case, we study the induced
representations over finite fields and identify them up to isomorphism.
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