International Journal of Mathematics

We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, on some Seifert fibered 3-manifolds we describe open books which are horizontal with respect to their plumbing description. As another application we describe horizontal open books isomorphic to Milnor open books for some complex surface singularities. Moreover we give examples of tight contact 3-manifolds supported by planar open books. As a consequence, the Weinstein conjecture holds for these tight contact structures [1].

We study the number of equivalence classes of proper holomorphic embeddings of a Stein space X into ℂⁿ. In this paper we prove that if the automorphism group of X is a Lie group and there exists a proper holomorphic embedding of X into ℂⁿ, 0 < dim X < n, then for any k ≥ 0 there exist uncountably many non-equivalent proper holomorphic embeddings Φ: X × ℂ^k ↪ ℂⁿ × ℂ^k. For k = 0 all embeddings will be proved to satisfy the additional property of ℂⁿ\Φ(X) being (n - dim X)-Eisenman hyperbolic. As a corollary we conclude that there are uncountably many non-equivalent proper holomorphic embeddings of ℂ^k into ℂⁿ whenever 0 < k < n.

This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let a₁G₁,…,a_kG_k be left cosets in a group G such that covers each element of G at least m times but none of its proper subsystems does. We show that if G is cyclic, or G is finite and G₁,…,G_k are normal Hall subgroups of G, then the inequality holds, where if p₁,…,p_r are distinct primes and α₁,…,α_r are nonnegative integers. When all the a_i are the identity element of G and all the G_i are subnormal in G, we prove that there is a composition series from to G whose factors are of prime orders. The paper also includes some other results and two challenging conjectures.

In this paper we study the spectral asymmetry of (possibly nonselfadjoint) elliptic ΨDO's in terms of the difference of zeta functions coming from different cuttings. Refining previous formulas of Wodzicki in the case of odd class elliptic ΨDO's, our main results have several consequence concerning the local independence with respect to the cutting, the regularity at integer points of eta functions and a geometric expression for the spectral asymmetry of Dirac operators which, in particular, yields a new spectral interpretation of the Einstein–Hilbert action in gravity.

We show that the multiplier algebra of a simple stable nuclear C*-algebra has a property similar to that of an AFD or hyperfinite von Neumann algebra. Specifically, we prove the following:

Theorem 0.1. Let be a unital simple separable C*-algebra. Let be the multiplier algebra of the stabilization of . Then is nuclear if and only if has the AFD-type property.

We also study a stronger property called the "strong AFD-type property". We show that if is a unital simple real rank zero AT-algebra with unique trace, then the multiplier algebra of the stabilization of has the strong AFD-type property, and we raise the question of whether this is true more generally.

We are concerned with the equivalence problem of third-order ordinary differential equations (ODEs) under bundle diffeomorphisms of three-dimensional real space. For those ODEs, we define canonical structure equations of a bundle on the second jet space, and introduce four torsion parts. We characterize linear third-order ODEs and projective linear third-order ODEs defined on tangent scrolls of space curves by vanishing of some torsion parts.