International Journal of Mathematics

For smooth embeddings of an integral homology 3-sphere in the 6-sphere, we define an integer invariant in terms of their Seifert surfaces. Our invariant gives a bijection between the set of smooth isotopy classes of such embeddings and the integers. It also gives rise to a complete invariant for homology bordism classes of all embeddings of homology 3-spheres in the 6-sphere. As a consequence, we show that two embeddings of an oriented integral homology 3-sphere in the 6-sphere are isotopic if and only if they are homology bordant. We also relate our invariant to the Rohlin invariant and accordingly characterize those embeddings which are compressible into the 5-sphere.

There are two main goals in this article, one of them is to minimize the number of invariants needed to obtain Whitney equisingular one parameter families of finitely determined holomorphic map germs f_t:(ℂⁿ,0) → (ℂ^p,0), with n < p. The other is to show how to compute the local Euler obstruction of the stable types which appear in a finitely determined map germ in these dimensions.

The polar multiplicities of all stable types and the 0-stable singularities are the invariants that guarantee the Whitney equisingularity of such families and the polar multiplicities are the numbers that also allow us to compute the local Euler obstruction. Therefore our first step is to describe all stable types which appear when n < p and show the relationship between the polar multiplicities in each stable type. Using the fact that these polar multiplicities are upper semi-continuous we minimize the number of invariants that guarantee Whitney equisingularity of such a family. We also apply the relationship between the polar multiplicities in each stable type and a result of Lê and Teissier to show how to compute the local Euler obstruction of the stable types which appear in these dimensions.

We extend the Ornstein–Weiss theorem on paving of amenable discrete groups to amenable discrete Kac algebras.

We characterize the special Lagrangian submanifolds of a generalized Calabi–Yau manifold, with vanishing Maslov class. Then, we carefully describe several examples, including a non-Kähler generalized Calabi–Yau manifold foliated by special Lagrangian submanifolds.

Given a fibration f : S → ℙ¹, and the associated foliation , the problem of bounding the genus of the general fiber of f in terms of the sheaf is studied. Using unipotent reduction of f, several bounds are obtained, under positivity assumptions on . In Sec.4, the Poincaré problem is solved, for non-degenerate , assuming that all the eigenvalues of the linear part of near singularities are greater than 3.

For certain bordered submanifolds M ⊂ ℂ² we show that M can be embedded properly and holomorphically into ℂ². An application is that any subset of a torus with two boundary components can be embedded properly into ℂ².

We describe a logarithmic tensor product theory for certain module categories for a "conformal vertex algebra". In this theory, which is a natural, although intricate, generalization of earlier work of Huang and Lepowsky, we do not require the module categories to be semisimple, and we accommodate modules with generalized weight spaces. The corresponding intertwining operators contain logarithms of the variables.