Universal Lefschetz fibrations and Lefschetz cobordisms

We construct universal Lefschetz fibrations, defined in analogy with classical universal bundles. We also introduce the cobordism groups of Lefschetz fibrations, and we see how these groups are quotients of the singular bordism groups via the universal Lefschetz fibrations.Comment: 14 pages; minor revision to match the published versio

Representing Dehn twists with branched coverings

We show that any homologically non-trivial Dehn twist of a compact surface F with boundary is the lifting of a half-twist in the braid group B_n, with respect to a suitable branched covering p : F -> B^2. In particular, we allow the surface to have disconnected boundary. As a consequence, any allowable Lefschetz fibration on B^2 is a branched covering of B^2 x B^2.Comment: Major revision. It has been added Corollary 3 about the lifting homomorphism. Are been added also some remarks and are given some other definitions. There are now 19 figures and 23 page

Branched coverings of CP2CP^2 and other basic 4-manifolds

We give necessary and sufficient conditions for a 4-manifold to be a branched covering of $CP^2$, $S^2\times S^2$, $S^2 \mathbin{\tilde\times} S^2$ and $S^3 \times S^1$, which are expressed in terms of the Betti numbers and the intersection form of the 4-manifold.Comment: 16 pages, 1 figure, 19 reference

A concave holomorphic filling of an overtwisted contact 33-sphere

In this paper we prove that the closed $4$-ball admits non-K\"ahler complex structures with strictly pseudoconcave boundary. Moreover, the induced contact structure on the boundary $3$-sphere is overtwisted.Comment: 11 pages, 0 figur

A universal ribbon surface in B^4

We construct an orientable ribbon surface F in B^4, which is universal in the following sense: any compact orientable pl 4-manifold having a handle decomposition with 0-, 1- and 2-handles can be represented as a cover of B^4 branched over F.Comment: 19 pages, 28 figures, 28 references. LaTeX 2.09 file. Uses: amstext.sty amscd.sty geom.sty epsf.st

On branched covering representation of 4-manifolds

We provide new branched covering representations for bounded and/or non-compact 4-manifolds, which extend the known ones for closed 4-manifolds. Assuming $M$ to be a connected oriented PL 4-manifold, our main results are the following: (1) if $M$ is compact with (possibly empty) boundary, there exists a simple branched cover $p:M \to S^4 - \mathop{\mathrm{Int}}(B^4_1 \cup \dots \cup B^4_n)$, where the $B^4_i$'s are disjoint PL 4-balls, $n \geq 0$ is the number of boundary components of $M$; (2) if $M$ is open, there exists a simple branched cover $p : M \to S^4 - \mathop{\mathrm{End}} M$, where $\mathop{\mathrm{End}} M$ is the end space of $M$ tamely embedded in $S^4$. In both cases, the degree $d(p)$ and the branching set $B_p$ of $p$ can be assumed to satisfy one of these conditions: (1) $d(p)=4$ and $B_p$ is a properly self-transversally immersed locally flat PL surface; (2) $d(p)=5$ and $B_p$ is a properly embedded locally flat PL surface. In the compact (resp. open) case, by relaxing the assumption on the degree we can have $B^4$ (resp. $R^4$) as the base of the covering. We also define the notion of branched covering between topological manifolds, which extends the usual one in the PL category. In this setting, as an interesting consequence of the above results, we prove that any closed oriented topological 4-manifold is a 4-fold branched covering of $S^4$. According to almost-smoothability of 4-manifolds, this branched cover could be wild at a single point.Comment: 16 pages, 9 figure

On smooth functions with two critical values

We prove that every smooth closed manifold admits a smooth real-valued function with only two critical values. We call a function of this type a \emph{Reeb function}. We prove that for a Reeb function we can prescribe the set of minima (or maxima), as soon as this set is a PL subcomplex of the manifold. In analogy with Reeb's Sphere Theorem, we use such functions to study the topology of the underlying manifold. In dimension $3$, we give a characterization of manifolds having a Heegaard splitting of genus $g$ in terms of the existence of certain Reeb functions. Similar results are proved in dimension $n\geq 5$

On codimension-1 submanifolds of the real and complex projective space

Inspired by the analogous result in the algebraic setting (Theorem1) we show (Theorem2) that the product M 7 RP^n of a closed and orientable topological manifold M with the n-dimensional real projective space cannot be embedded into RP^(m+n+1) for all even n > m