15,865 research outputs found
The 2-Hilbert Space of a Prequantum Bundle Gerbe
We construct a prequantum 2-Hilbert space for any line bundle gerbe whose
Dixmier-Douady class is torsion. Analogously to usual prequantisation, this
2-Hilbert space has the category of sections of the line bundle gerbe as its
underlying 2-vector space. These sections are obtained as certain morphism
categories in Waldorf's version of the 2-category of line bundle gerbes. We
show that these morphism categories carry a monoidal structure under which they
are semisimple and abelian. We introduce a dual functor on the sections, which
yields a closed structure on the morphisms between bundle gerbes and turns the
category of sections into a 2-Hilbert space. We discuss how these 2-Hilbert
spaces fit various expectations from higher prequantisation. We then extend the
transgression functor to the full 2-category of bundle gerbes and demonstrate
its compatibility with the additional structures introduced. We discuss various
aspects of Kostant-Souriau prequantisation in this setting, including its
dimensional reduction to ordinary prequantisation.Comment: 97 pages; v2: minor changes; Final version to be published in Reviews
in Mathematical Physic
Categories, norms and weights
The well-known Lawvere category R of extended real positive numbers comes
with a monoidal closed structure where the tensor product is the sum. But R has
another such structure, given by multiplication, which is *-autonomous.
Normed sets, with a norm in R, inherit thus two symmetric monoidal closed
structures, and categories enriched on one of them have a 'subadditive' or
'submultiplicative' norm, respectively. Typically, the first case occurs when
the norm expresses a cost, the second with Lipschitz norms.
This paper is a preparation for a sequel, devoted to 'weighted algebraic
topology', an enrichment of directed algebraic topology. The structure of R,
and its extension to the complex projective line, might be a first step in
abstracting a notion of algebra of weights, linked with physical measures.Comment: Revised version, 16 pages. Some minor correction
Positively curved Killing foliations via deformations
We show that a compact manifold admitting a Killing foliation with positive
transverse curvature fibers over finite quotients of spheres or weighted
complex projective spaces, provided that the singular foliation defined by the
closures of the leaves has maximal dimension. This result is obtained by
deforming the foliation into a closed one while maintaining transverse
geometric properties, which allows us to apply results from the Riemannian
geometry of orbifolds to the space of leaves. We also show that the basic Euler
characteristic is preserved by such deformations. Using this fact we prove that
a Riemannian foliation of a compact manifold with finite fundamental group and
nonvanishing Euler characteristic is closed. As another application we obtain
that, for a positively curved Killing foliation of a compact manifold, if the
structural algebra has sufficiently large dimension then the basic Euler
characteristic is positive.Comment: 23 pages, several corrections, results have change
Equivariant Euler characteristics and K-homology Euler classes for proper cocompact G-manifolds
Let G be a countable discrete group and let M be a smooth proper cocompact
G-manifold without boundary. The Euler operator defines via Kasparov theory an
element, called the equivariant Euler class, in the equivariant K-homology of
M. The universal equivariant Euler characteristic of M, which lives in a group
U^G(M), counts the equivariant cells of M, taking the component structure of
the various fixed point sets into account. We construct a natural homomorphism
from U^G(M) to the equivariant KO-homology of M. The main result of this paper
says that this map sends the universal equivariant Euler characteristic to the
equivariant Euler class. In particular this shows that there are no `higher'
equivariant Euler characteristics. We show that, rationally, the equivariant
Euler class carries the same information as the collection of the orbifold
Euler characteristics of the components of the L-fixed point sets M^L, where L
runs through the finite cyclic subgroups of G. However, we give an example of
an action of the symmetric group S_3 on the 3-sphere for which the equivariant
Euler class has order 2, so there is also some torsion information.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper16.abs.htm
On modular homology in projective space
AbstractFor a vector space V over GF(q) let Lk be the collection of subspaces of dimension k. When R is a field let Mk be the vector space over it with basis Lk. The inclusion map ∂:Mk→Mk−1 then is the linear map defined on this basis via ∂(X)≔∑Y where the sum runs over all subspaces of co-dimension 1 in X. This gives rise to a sequenceM:0←M0←M1←⋯←Mk−1←Mk←⋯which has interesting homological properties if R has characteristic p>0 not dividing q. Following on from earlier papers we introduce the notion of π-homological, π-exact and almost π-exact sequences where π=π(p,q) is some elementary function of the two characteristics. We show that M and many other sequences derived from it are almost π-exact. From this one also obtains an explicit formula for the Brauer character on the homology modules derived from M. For infinite-dimensional spaces we give a general construction which yields π-exact sequences for finitary ideals in the group ring RPΓL(V)
From Topology to Noncommutative Geometry: -theory
We associate to each unital -algebra a geometric object---a diagram
of topological spaces representing quotient spaces of the noncommutative space
underlying ---meant to serve the role of a generalized Gel'fand spectrum.
After showing that any functor from compact Hausdorff spaces to a suitable
target category can be applied directly to these geometric objects to
automatically yield an extension which acts on all unital
-algebras, we compare a novel formulation of the operator functor to
the extension of the topological -functor.Comment: 14 page
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