We show that there is no positive loop inside the component of a fiber in the space of Legendrian embeddings in the contact manifold ST ∗M , provided that the universal cover of M is Rn. We consider some related results in the space of one-jets of functions on a compact manifold. We give an application to the positive isotopies in homogeneous neighborhoods of surfaces in a tight contact 3-manif...

A free action of the direct product of two copies of the symmetric group on 3 elements on the cartesian product of two copies of the 3-sphere is constructed. This nonlinear action is constructed using surgery. The action provides a counterexample to a conjecture of Lewis made in 1968. Results of P. A. Smith [10] and J. Milnor [5] show that a dihedral group cannot act freely on a sphere. This im...

It is known that the second Leibniz homology group HL2(stln(R)) of the Steinberg Leibniz algebra stln(R) is trivial for n ≥ 5. In this paper, we determine HL2(stln(R)) explicitly (which are shown to be not necessarily trivial) for n = 3, 4 without any assumption on the base ring. §

This paper contains a loose collection of remarks on F1-schemes. Etale morphisms and universal coverings are introduced. The relation to toric varieties, at least for integral schemes, is clarified.

The quotient-cusp singularities are isolated complex surface singularities that are double-covered by cusp singularities. We show that the universal abelian cover of such a singularity, branched only at the singular point, is a complete intersection cusp singularity of embedding dimension 4. This supports a general conjecture that we make about the universal abelian cover of a Q-Gorenstein sing...

Let X be a projective manifold, ρ : X̃ → X its universal covering and ρ∗ : V ect(X) → V ect(X̃) the pullback map for the isomorphism classes of vector bundles. This article establishes a connection between the properties of the pullback map ρ∗ and the properties of the function theory on X̃. We prove the following pivotal result: if a universal cover of a projective variety has no nonconstant holo...

Stable, holomorphic vector bundles are constructed on an torus fibered, non-simply connected Calabi-Yau threefold using the method of bundle extensions. Since the manifold is multiply connected, we work with equivariant bundles on the elliptically fibered covering space. The cohomology groups of the vector bundle, which yield the low energy spectrum, are computed using the Leray spectral sequen...

When the Seiberg-Witten curve of a four-dimensional N = 2 supersymmetric gauge theory wraps a Riemann surface as a multi-sheeted cover, a topological constraint requires that in general the curve should develop ramification points. We show that, while some of the branch points of the covering map can be identified with the punctures that appear in the work of Gaiotto, the ramification points gi...

This paper is concerned with the existence of contact structures on (connected, closed, orientable) 5-manifolds with certain finite fundamental groups. As such, it constitutes a sequel to [6] (which gave corresponding existence results for highly connected manifolds of arbitrary (odd) dimension and some ad hoc results for finite fundamental groups) and our joint paper [7], where we showed that ...

Let A be a basic connected finite dimensional algebra over an algebraically closed field k. Assuming that A is monomial and that the ordinary quiver Q of A has no oriented cycle and no multiple arrows, we prove that A admits a universal cover with group the fundamental group of the underlying space of Q.