The homology of cyclic branched covers ofS 3

Partially supported by NSF grant DMS-9208052 and the MSRI NSF grant DMS-9022140. The author held an MSRI Research Professorship while the paper was being written     

The homology of irregular dihedral branched covers ofS 3

Partially supported by NSF grant DMS-9208052 and the MSRI NSF grant DMS-9022140. The author held an MSRI Research Professorship while the paper was being written     

Fundamental groups of algebraic fiber spaces

Let be a dominant morphism, where E and B are smooth irreducible complex quasi-projective varieties. Suppose that the general fiber of $f$ is connected. We present an algebro-geometric condition under which the boundary homomorphism is well-defined, and makes the sequence exact. As an application, we calculate the fundamental group of the complement to the dual hypersurface of a smooth projective curve. Received: October 3, 2001     

Fundamental groups of spaces of generalized perfect splines

Efficient versions of the Cauchy criterion for the convergence in L _p of greedy approximants of trigonometric Fourier series are discussed.     

Fundamental groups of spaces of generalized perfect splines

The fundamental groups π ₁(Ω _n ) of the spaces Ω _n of generalized perfect splines are calculated. For n ≥ 2, the groups are trivial; π ₁(Ω₁) is a free group with three generators.     

Semigroup actions, covering spaces and Schützenberger groups

We associate a 2-complex to the following data: a presentation of a semigroup S and a transitive action of S on a set V by partial transformations. The automorphism group of the action acts properly discontinuously on this 2-complex. A sufficient condition is given for the 2-complex to be simply connected. As a consequence we obtain simple topological proofs of results on presentations of Schützenberger groups. We also give a geometric proof that a finitely generated regular semigroup with finitely many idempotents has polynomial growth if and only if all its maximal subgroups are virtually nilpotent.     

Compactifying sufficiently regular covering spaces of compact 3-manifolds

In this paper it is proven that if the group of covering translations of the covering space of a compact, connected, -irreducible 3-manifold corresponding to a non-trivial, finitely-generated subgroup of its fundamental group is infinite, then either the covering space is almost compact or the subgroup is infinite cyclic and has normalizer a non-finitely-generated subgroup of the rational numbers. In the first case additional information is obtained which is then used to relate Thurston's hyperbolization and virtual bundle conjectures to some algebraic conjectures about certain 3-manifold groups.     

Arc-smooth generalized universal covering spaces

Let Y be a path-connected subset of a CAT(0) space Z, allowing for a map to a 1-dimensional separable metric space X, such that the nontrivial point preimages of f form a null sequence of convex subsets of Z. Such Y need not be homotopy equivalent to a 1-dimensional space.

We prove that Y admits a generalized universal covering space, which we equip with an arc-smooth structure by consistently and continuously selecting one tight representative from each path homotopy class of Y. It follows that all homotopy groups of Y vanish in dimensions greater than 1.     

Soap films and covering spaces

A new mathematical model of soap films is proposed, called the “covering space model.” The two sides of a film are modeled as currents on different sheets of a covering space branching along the film boundary. Hence a film may be seen as the minimal cut separating one sheet of the covering space from the others. The film is thus the oriented boundary of one sheet, which represents the exterior of the film. As oriented boundaries, films may be calibrated with differential forms on the covering space, a version of the min-cut, max-flow duality of network theory. This model applies to unoriented films, films with singularities, films touching only part of a knotted curve, films that deformation retract to their boundaries, and other examples that have proved troublesome for previous soap film models. Communicated by Frederick Almgren     

Selective covering properties of product spaces

We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals.     

Cyclic branched coverings of lens spaces

Some infinite family is constructed of orientable three-dimensional closed manifoldsM _n (p, q), where n ≥ 2, p ≥ 3, 0 < q < p, and (p, q) = 1, such that M _n (p, q) is an n-fold cyclic covering of the lens space L(p, q) branched over a two-component link.