Stiefel Whitney classes for real representations of GL2(𝔽q)

We compute the total Stiefel Whitney class for a real representation [Formula: see text] of [Formula: see text], where [Formula: see text] is odd. The obstruction class of [Formula: see text] is defined to be the Stiefel Whitney class of lowest positive degree that does not vanish. We provide an expression for the obstruction class of [Formula: see text] in terms of its character values if [Formula: see text].

On a Lie Algebraic Approach to Abelian Extensions of Associative Algebras

By using a representation of a Lie algebra on the second Hochschild cohomology group, we construct an obstruction class to extensibility of derivations and a short exact sequence of Wells type for an abelian extension of an associative algebra.

Belyi’s theorem in characteristic two

We prove an analogue of Belyi’s theorem in characteristic two. Our proof consists of the following three steps. We first introduce a new notion called pseudo-tameness for morphisms between curves over an algebraically closed field of characteristic two. Secondly, we prove the existence of a ‘pseudo-tame’ rational function by showing the vanishing of an obstruction class. Finally, we construct a tamely ramified rational function from the ‘pseudo-tame’ rational function.

Finite, Fiber- and Orientation-Preserving Group Actions on Totally Orientable Seifert Manifolds

In this paper we consider the finite groups that act fiber- and orientation-preservingly on closed, compact, and orientable Seifert manifolds that fiber over an orientable base space. We establish a method of constructing such group actions and then show that if an action satisfies a condition on the obstruction class of the Seifert manifold, it can be derived from the given construction. The obstruction condition is refined and the general structure of the finite groups that act via the construction is provided.

Non-Formality of Planar Configuration Spaces in Characteristic 2

We prove that the ordered configuration space of four points or more in the plane has a nonformal singular cochain algebra in characteristic 2. This is proved by constructing an explicit nontrivial obstruction class in the Hochschild cohomology of the cohomology ring of the configuration space, by means of the Barratt–Eccles–Smith simplicial model. We also show that if the number of points does not exceed its dimension then a Euclidean configuration space is intrinsically formal over any ring.

Symplectic Foliations and Generalized Complex Structures

We answer the natural question: when is a transversely holomorphic symplectic foliation induced by a generalized complex structure? The leafwise symplectic form and transverse complex structure determine an obstruction class in a certain cohomology, which vanishes if and only if our question has an affirmative answer. We first study a component of this obstruction, which gives the condition that the leafwise cohomology class of the symplectic form must be transversely pluriharmonic. As a consequence, under certain topological hypotheses, we infer that we actually have a symplectic fibre bundle over a complex base. We then show how to compute the full obstruction via a spectral sequence. We give various concrete necessary and sufficient conditions for the vanishing of the obstruction. Throughout, we give examples to test the sharpness of these conditions, including a symplectic fibre bundle over a complex base that does not come from a generalized complex structure, and a regular generalized complex structure that is very unlike a symplectic fibre bundle, i.e., for which nearby leaves are not symplectomorphic.