(i) Topology of embedded surfaces. Let X be a smooth, simply-connected 4-manifold, and ξ a 2-dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly

Abstract not found

Abstract. In this paper we show that the number of deformation types of complex structures on a fixed smooth oriented four-manifold can be arbitrarily large. The examples that we consider in this paper are locally simple abelian covers of rational surfaces. The proof involves the algebraic description of rational blow down, classical Brill-Noether theory and deformation theory of normal flat abelian covers. One of the main problems concerning the differential topology of algebraic surfaces leaving unsolved by the “Seiberg-Witten revolution ” was to determine whether the differential type of a compact complex surface determines the deformation type. Two compact complex manifolds have the same deformation type if they are fibres of a proper smooth family over a connected

Abstract. We show that if a closed manifold M admits an F-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S 1-action. As a corollary we obtain that the simplicial volume of a manifold admitting an F-structure is zero. We also show that if M admits an F-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative. We show that F-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold. We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S 4, CP 2, CP 2, S 2 × S 2 and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 4, CP 2, S 2 × S 2, CP 2 #CP 2 or CP 2 #CP 2. Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 5, S 3 ×S 2, the nontrivial S 3-bundle over S 2 or the Wu-manifold SU(3)/SO(3). 1.

We prove that a complete noncompact Kähler manifold M n of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of C n and we show that the manifold is topologically R 2n. In particular, when M n is a Kähler surface of positive bisectional curvature satisfying certain natural geometric growth conditions, it is biholomorphic to C 2. 1.

Abstract: A construction of convex flag triangulations of five and higher dimensional spheres, whose h-polynomials fail to have only real roots, is given. We show that there is no such example in dimensions lower than five. A condition weaker than realrootedness is conjectured and some evidence is provided. Let the f-polynomial fX of a simplicial complex X be defined by the formula fX(t): = ∑ t #σ. There is a classical problem: what can be said in general about the f-polynomials of (a certain class of) simplicial complexes? In particular, it is well known what polynomials appear as f-polynomials of • general simplicial complexes, or • triangulations of spheres that are the boundary complexes of convex polytopes (the reader may consult [St1] for ample discussion). The question concerning all triangulations of spheres remains still open. However the answer is conjecturally the same. What we are interested in is the special case of the latter. Namely, what can be said in general about

The Poincaré Conjecture was posed ninetynine years ago and may possibly have been proved in the last few months. This note will be an account of some of the major results over the past hundred years which have paved the way towards a proof and towards the even more ambitious project of classifying all compact 3-dimensional manifolds. The final paragraph provides a brief description of the latest developments, due to Grigory Perelman. A more serious discussion of Perelman’s work will be provided in a subsequent note by Michael Anderson.

Abstract not found

Abstract. It is known that if any prime power branched cyclic cover of a knot in S 3 is a homology sphere, then the knot has vanishing Casson-Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in S 3 whose prime power branched cyclic covers are homology spheres. We show that these knots generate an infinite rank subgroup of F(1.0)/F(1.5) for which Casson-Gordon invariants vanish in Cochran-Orr-Teichner’s filtration of the classical knot concordance group. As a corollary, it follows that Casson-Gordon invariants are not a complete set of obstructions to a second layer of Whitney disks. 1.

Dedicated to Ronald J. Stern on the occasion of his sixtieth birthday Abstract. Let M be either CP 2 #3CP 2 or 3CP 2 #5CP 2. We construct the first example of a simply-connected irreducible symplectic 4-manifold that is homeomorphic but not diffeomorphic to M. 1.