Detecting codimension one manifold factors with topographical techniques

We prove recognition theorems for codimension one manifold factors of dimension $n \geq 4$. In particular, we formalize topographical methods and introduce three ribbons properties: the crinkled ribbons property, the twisted crinkled ribbons property, and the fuzzy ribbons property. We show that $X \times \mathbb{R}$ is a manifold in the cases when $X$ is a resolvable generalized manifold of finite dimension $n \geq 3$ with either: (1) the crinkled ribbons property; (2) the twisted crinkled ribbons property and the disjoint point disk property; or (3) the fuzzy ribbons property

The Bing-Borsuk and the Busemann Conjectures

We present two classical conjectures concerning the characterization of manifolds: the Bing Borsuk Conjecture asserts that every $n$-dimensional homogeneous ANR is a topological $n$-manifold, whereas the Busemann Conjecture asserts that every $n$-dimensional $G$-space is a topological $n$-manifold. The key object in both cases are so-called {\it generalized manifolds}, i.e. ENR homology manifolds. We look at the history, from the early beginnings to the present day. We also list several open problems and related conjectures.Comment: We have corrected three small typos on pages 8 and

Compact Stein surfaces with boundary as branched covers of \B^4

We prove that Stein surfaces with boundary coincide up to orientation preserving diffeomorphisms with simple branched coverings of \B^4 whose branch set is a positive braided surface. As a consequence, we have that a smooth oriented 3-manifold is Stein fillable iff it has a positive open-book decomposition.Comment: 25 pages, 20 postscript figures. LaTeX file. Uses: geom.sty epsf.st

Singular maps on exotic 4-manifold pairs

We show examples of pairs of smooth, compact, homeomorphic 4-manifolds, whose diffeomorphism types are distinguished by the topology of the singular sets of smooth stable maps defined on them. In this distinction we rely on results from Seiberg-Witten theory.Comment: 17 pages, 6 figures, minor modifications following referee's comment