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We prove the following theorems: (1) Every orientable, closed, irreducible 3-manifold that contains an incompressible torus carries a (universally) tight contact structure. (2) Every foliation without Reeb component is a C 0 limit of (universally) tight contact structures.
We construct examples of Lefschetz fibrations with prescribed singular fibers. By taking differences of pairs of such fibrations with the same singular fibers, we obtain new examples of surface bundles over surfaces with nonzero signature. From these we derive new upper bounds for the minimal genus of a surface representing a given element in the second homology of a mapping class group.
The homological systole of a compact Riemann surface X is the minimal length of a simple closed non-separating goedesic curve. Since any homology basis of X must contain curves that intersect any non-separating closed curve, surfaces having small homological systoles cannot have short homology basis. It turns out that this basically the only obstruction to finding short homology basis. We show, in...
Let X be a manifold on which a discrete (pseudo)group of diffeomorphisms Γ acts, and let E be a Γ-equivariant vector bundle on X. We give a construction of cyclic cocycles on the cross product algebra C 0~ (X) Γ representing the equivariant characteristic classes of E. Our formulas can be viewed as a higher-dimensional analogue of Connes' Godbillon-Vey cyclic cocycle. An essential tool...
Let X=G/P be a homogeneous space of a complex semisimple Lie group G equipped with a hermitian metric. We study the action of the Hodge star operator on the space of harmonic differential forms on X. We obtain explicit combinatorial formulas for this action when X is an irreducible hermitian symmetric space of compact type.
Suppose E is a spectrum obtained by the completion of a ring spectrum E at some ideal I of its coefficients E * . We study the homology of the Ω spectrum representing E and show that the techniques of coalgebraic algebra provide a language enabling a simple description of this homology and of the homological view of the process of completion. Results are obtained for a wide range of connective...
In this paper, we prove that a 4-manifold topologically embeds into R 7 if and only if its third normal Stiefel-Whitney class vanishes. In particular, this shows that orientable 4-manifolds topologically embed into R 7 i.e. the hard Whitney embedding theorem holds in dimension 4 in the topological category. The former generalizes the classical Haefliger-Hirsch theorem to dimension...
Let l be an oriented link of d components with nonzero Alexander polynomial Δ(u 1 ,...,u d ). Let Λ be a finite-index subgroup of H 1 (S 3 -l) Z d , and let M Λ be the corresponding abelian cover of S 3 branched along l. The growth rate of the order of the torsion subgroup of H 1 (M Λ ), as a suitable measure of Λ approaches...
Suppose M is a topological m-manifold, X is a generalized n-manifold satisfying the disjoint disks property (DDP), m>n>=5, f:M->X is an approximate fibration, with fiber the shape of a closed topological manifold F, and Y is a closed, 1-LCC, codimension three subset of X. We examine conditions under which f is controlled homeomorphic to an approximate fibration g:M->X such that g|g ...
We express the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. We also construct a canonical representative of the Milnor fibre in a suitable monodromic Grothendieck group.
We compute the mod2 cohomology of Waldhausen's algebraic K-theory spectrum A(*) of the category of finite pointed spaces, as a module over the Steenrod algebra. This also computes the mod2 cohomology of the smooth Whitehead spectrum of a point, denoted Wh Diff (*). Using an Adams spectral sequence we compute the 2-primary homotopy groups of these spectra in dimensions *=<18,...
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