On the geography of symplectic 6-manifolds

The article investigates the geography of closed, connected and simply connected, six-dimensional manifolds. It is proved that any triple of integers satisfying some necessary arithmetical restrictions occurs as the Chern triple of such a manifold. The main tools used for producing the examples are the symplectic connected sum and the symplectic blow-up. Received: 28 May 1998 / Revised version: 22 January 1999     

Singhof fillings of closed 3-manifolds

We give a complete classification of all closed, connected 3-manifolds which admit a Singhof filling with any number of solid tori. Received: 15 March 2001 / Revised version: 17 September 2001     

The topological types of some irregular Kähler surfaces

The topological type of generalized Kummer surfaces is described in terms of sphere bundles over Riemann surfaces and the complex projective plane. Explicit examples of sets of pairwise non-diffeomorphic K?hler surfaces of the same topological type are given. Received: 5 January 2000     

On abelian quantum invariants of links in 3-manifolds

From a finite abelian group G, a quadratic form onG and an element in , we define a topological invariant of a pair(M,L) where is a closed oriented 3-manifold and L an oriented, framedn-component link inM. The main result consists in an explicit formula for this invariant, based on a reciprocity formula for Gauss sums, which features a special linking pairing. This pairing depends on both the quadratic form q and the linking pairing of M. A necessary and sufficient condition for the invariant to vanish is described in terms of a characteristic class for this pairing. We also discuss torsion spin-structures and related structures which appear in this context. Received May 13, 1998 / Accepted November 11, 1999 / Published online February 5, 2001     

On symplectic cobordisms

In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3-manifolds are “symplectic cobordism equivalent”. Received: 26 March 2001 / Revised version: 1 May 2001 / Published online: 28 February 2002     

The LMO-invariant of 3-manifolds of rank one and the Alexander polynomial

We prove that the LMO-invariant of a 3-manifold of rank one is determined by the Alexander polynomial of the manifold, and conversely, that the Alexander polynomial is determined by the LMO-invariant. Furthermore, we show that the Alexander polynomial of a null-homologous knot in a rational homology 3-sphere can be obtained by composing the weight system of the Alexander polynomial with the ?rhus invariant of knots. Received February 14, 2000 / Published online October 11, 2000     

Handle attaching in symplectic homology and the Chord Conjecture

Arnold conjectured that every Legendrian knot in the standard contact structure on the 3-sphere possesses a characteristic chord with respect to any contact form. I confirm this conjecture if the know has Thurston-Bennequin invariant −1. More generally, existence of chords is proved for a standard Legendrian unknot on the boundary of a subcritical Stein manifold of any dimension. There is also a multiplicity result which implies in some situations existence of infinitely many chords.?The proof relies on the behaviour of symplectic homology under handle attaching. The main observation is that symplectic homology only changes in the presence of chords. Received July 14, 2000 / final version received June 1, 2001?Published online August 1, 2001     

Bordisme de S1-fibrés vectoriels munis¶de structures supplémentaires

Bordism of S ¹-vector bundles with additional structures We give isomorphisms between equivariant bordism groups of certain S ¹-vector bundles and bordism groups of suitable “classifying” spaces determined by certain caracterestic classes. In the spinorial case, we detect the even or odd type of the S ¹-action and give a relationship with elleptic homology. Furthermore, we define a new type of $S^1$-actions, depending on the actions and the given slice type. This new type differs, in certain cases, from the classical odd or even type of S ¹-actions on spinorial manifolds. Received: 7 July 2000 / Revised version: 10 February 2001     

Nonisomorphic Lefschetz fibrations on knot surgery 4-manifolds

In this article we construct an infinite family of simply connected minimal symplectic 4-manifolds, each of which admits at least two nonisomorphic Lefschetz fibration structures with the same generic fiber. We obtain such examples by performing knot surgery on an elliptic surface E(n) using a special type of 2-bridge knots. This work was supported by grant No. R01-2005-000-10625-0 from the KOSEF and by the Korea Research Foundation Grant funded by the Korean Government (KRF-2007-314-C00024).     

Contact structures, equivariant spin bordism, and periodic fundamental groups

Using contact surgery and equivariant bordism theory, we prove the existence of contact structures on all 5-dimensional spin manifolds with certain finite fundamental groups. Received September 13, 1999 / Revised version September 13, 2000 / Published online April 12, 2001     

Seifert unions of solid tori

We obtain a list of all 3-manifolds that can be obtained by gluing 3-balls and solid tori along mutually disjoint surfaces in their boundaries. Received: 22 February 2001; in final form: 18 October 2001 / Published online: 4 April 2002     

Conjugations on 6-manifolds

Conjugation spaces are spaces with an involution such that the fixed point set of the involution has \({\mathbb{Z} _2}\)-cohomology ring isomorphic to the \({\mathbb{Z} _2}\)-cohomology of the space itself, with the difference that all degrees are divided by two (e.g. \({\mathbb{C} {\rm P}^n}\) with the complex conjugation has \({\mathbb{R} {\rm P}^n}\) as fixed point set). One also requires that a certain conjugation equation is fulfilled. We give a new characterisation of conjugation spaces and apply it to the following realization problem: given M, a closed orientable 3-manifold, does there exist a simply connected 6-manifold X and a conjugation on X with fixed point set M? We give an affirmative answer.     

Cobordism of immersions of surfaces¶in non-orientable 3-manifolds

Some properties of non-orientable 3-manifolds are shown. In particular, for a connected, non-orientable 3-manifold M, the group of cobordism clases of immersions of surfaces in M is isomorphic to a group structure on the set H ₂(M,Z/2Z)×H ₁(M,Z/2Z)×Z/2Z. Received: 8 June 2000 / Revised version: 2 October 2000     

Constructing infinitely many smooth structures on $3\mathbb{CP}^2 \# n\overline{\mathbb{CP}}^2$

Using Seiberg-Witten theory and rational blow-down procedures of R. Fintushel and R.J. Stern, we construct infinitely many irreducible smooth structures, both symplectic and non-symplectic, on the four-manifold for each integer n lying in the interval . Received: 17 January 2000 / Published online: 18 January 2002     

Vanishing structure set of Haken 3-manifolds

In this article we prove that the surgery groups of the fundamental group of a certain class of Haken 3-manifolds can be computed in terms of a generalized homology theory even if the manifolds do not support any nonpositively curved Riemannian metric. A consequence of this result is that the integral Novikov conjecture is true for the fundamental group of this class of manifolds. Received October 2, 1998 / in revised form February 10, 2000 / Published online July 20, 2000     

Kauffman bracket skein module of a connected sum of 3-manifolds

We show that for the Kauffman bracket skein module over the field of rational functions in variable A, the module of a connected sum of 3-manifolds is the tensor product of modules of the individual manifolds. Received: 12 January 1998 / Revised version: 15 September 1999     

A primary obstruction to topological embeddings¶and its applications

For a proper continuous map f:M→N between topological manifolds M and N with m≡ dimM < dimN≡m+k, a primary obstruction to topological embeddings θ(f) ∈H ^c _{m − k} (M; Z ₂) has been defined and studied by the authors in {9, 8, 2, 3], where H ^c _* denotes the singular homology with closed support. In this paper, we study the obstruction from the viewpoint of differential topology and give various applications. We first give some characterizations of embeddings among generic differentiable maps, which are refinements of the results in [9, 10]. Then we give a result concerning the number of connected components of the complement of the image of a codimension-1 continuous map with a normal crossing point, which generalizes the results in [6, 4, 5, 9]. Finally we give a simple proof of a theorem of Li and Peterson [20] about immersions of m-manifolds into (2m-1)-manifolds. Received: 3 December 1999 / Revised version: 10 October 2000