No abstract

Given a group G and a class of manifolds C (e.g. symplectic, contact, Kähler, etc.), it is an old problem to find a manifold M G ∈ C whose fundamental group is G. This article refines it: for a group G and a positive integer r find M G ∈ C such that π 1 (M G ) = G and π i (M G ) = 0 for 1 < i < r. We thus provide a unified point of view systematizing known and new results in this direction for various different classes of manifolds. The largest r for which such an M G ∈ C can be found is called the homotopical height ht C (G). Homotopical height provides a dimensional obstruction to finding a K(G, 1) space within the given class C, leading to a hierarchy of these classes in terms of "softness" or "hardness"à la Gromov. We show that the classes of closed contact, CR, and almost complex manifolds as well as the class of (open) Stein manifolds are soft. The classes SP and CA of closed symplectic and complex manifolds exhibit intermediate "softness" in the sense that every finitely presented group G can be realized as the fundamental group of a manifold in SP and a manifold in CA. For these classes, ht C (G) provides a numerical invariant for finitely presented groups. We give explicit computations of these invariants for some standard finitely presented groups. We use the notion of homotopical height within the "hard" category of Kähler groups to obtain partial answers to questions of Toledo regarding second cohomology and second group cohomology of Kähler groups. We also modify and generalize a construction due to Dimca, Papadima and Suciu to give a potentially large class of projective groups violating property FP.1450123-1 I. Biswas, M. Mj & D. Pancholi

In this article we show that every closed orientable smooth 4-manifold admits a smooth embedding in the complex projective 3-space.

In this note, we discuss embeddings of 3-manifolds via open books. First we show that every open book of every closed orientable 3-manifold admits an open book embedding in any open book decompistion of S 2 × S 3 and S 2 ×S 3 with the page a disk bundle over S 2 and monodromy the identity. We then use open book embeddings to reprove that every closed orientable 3-manifold embeds in S 5 .

ABSTRACT. We give a possible generalization of a Lutz twist to all dimensions. This reproves the fact that every contact manifold can be given a non-fillable contact structure and also shows great flexibility in the manifolds that can be realized as cores of overtwisted families. We moreover show that R 2n+1 has at least three distinct contact structures.This version of the paper contains both the texts of the published version of the paper together with an Erratum to the published version appended to the end.