43 research outputs found
Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory
Let be a smooth, compact, oriented -manifold. Building upon work of
Li-Liu, Ruberman, Nakamura and Konno, we consider a families version of
Seiberg-Witten theory and obtain obstructions to the existence of certain group
actions on by diffeomorphisms. The obstructions show that certain group
actions on preserving the intersection form can not be
lifted an action of the same group on by diffeomorphisms. Using our
obstructions, we construct numerous examples of group actions which can be
realised continuously but can not be realised smoothly for any differentiable
structure. For example, we construct compact simply-connected -manifolds
and involutions such that
can be realised by a continuous involution on or by a diffeomorphism, but
not by an involutive diffeomorphism for any smooth structure on .Comment: 27 page
Topological T-duality for torus bundles with monodromy
We give a simplified definition of topological T-duality that applies to
arbitrary torus bundles. The new definition does not involve Chern classes or
spectral sequences, only gerbes and morphisms between them. All the familiar
topological conditions for T-duals are shown to follow. We determine necessary
and sufficient conditions for existence of a T-dual in the case of affine torus
bundles. This is general enough to include all principal torus bundles as well
as torus bundles with arbitrary monodromy representations. We show that
isomorphisms in twisted cohomology, twisted K-theory and of Courant algebroids
persist in this general setting. We also give an example where twisted K-theory
groups can be computed by iterating T-duality.Comment: 49 page
Arithmetic of singular character varieties and their -polynomials
We calculate the -polynomials of the and
-character varieties of compact oriented surfaces of any
genus and the -polynomials of the and
-character varieties of compact non-orientable surfaces of
any Euler characteristic. Our methods also give a new and significantly simpler
computation of the -polynomials of the -character
varieties of compact orientable surfaces, which were computed by Logares,
Mu\~noz and Newstead for genus and by Martinez and Mu\~noz for . Our technique is based on the arithmetic of character varieties over finite
fields. More specifically, we show how to extend the approach of Hausel and
Rodriguez-Villegas used for non-singular (twisted) character varieties to the
singular (untwisted) case.Comment: 41 page