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13
On the Signatures of Even 4Manifolds
, 2000
"... Abstract. In this paper, we prove a number of inequalities between the signature and the Betti numbers of a 4–manifold with even intersection form and prescribed fundamental group. Furthermore, we introduce a new geometric group invariant and discuss some of its properties. 1. ..."
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Cited by 7 (1 self)
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Abstract. In this paper, we prove a number of inequalities between the signature and the Betti numbers of a 4–manifold with even intersection form and prescribed fundamental group. Furthermore, we introduce a new geometric group invariant and discuss some of its properties. 1.
Alexander Numbering of Knotted Surface Diagrams
, 1998
"... Abstract. A formula that relates triple points, branch points, and their distance from infinity is presented. 1. ..."
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Cited by 2 (2 self)
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Abstract. A formula that relates triple points, branch points, and their distance from infinity is presented. 1.
Elimination of singularities of smooth mappings of 4–manifolds into 3– manifolds
 Topology Appl
"... The simplest singularities of smooth mappings are fold singularities. We say that a mapping f is a fold mapping if every singular point of f is of the fold type. We prove 1 that for a closed orientable 4manifold M 4 the following conditions are equivalent: (1) M 4 admits a fold mapping into R 3; (2 ..."
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The simplest singularities of smooth mappings are fold singularities. We say that a mapping f is a fold mapping if every singular point of f is of the fold type. We prove 1 that for a closed orientable 4manifold M 4 the following conditions are equivalent: (1) M 4 admits a fold mapping into R 3; (2) for every orientable 3manifold N 3, every homotopy class of mappings of M 4 into N 3 contains a fold mapping; (3) there exists a cohomology class x ∈ H 2 (M 4; Z) such that x ⌣ x is the first Pontryagin class of M 4. For a simply connected manifold M 4, we show that M 4 admits no fold mappings
Filtrations on instanton homology
"... The Khovanov cohomology Kh.K / of an oriented knot or link is defined in [3] as the cohomology of a cochain complex.C D C.D/; d Kh / associated to a plane diagram D for K: Kh.K / D H C; d Kh: ..."
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Cited by 1 (1 self)
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The Khovanov cohomology Kh.K / of an oriented knot or link is defined in [3] as the cohomology of a cochain complex.C D C.D/; d Kh / associated to a plane diagram D for K: Kh.K / D H C; d Kh:
BETWEEN LOWER AND HIGHER DIMENSIONS (in the work of Terry Lawson)
"... There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had ..."
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There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly welldeveloped theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental
REAL SURFACES IN ELLIPTIC SURFACES
, 2003
"... Abstract. We study the structure of complex points on real surfaces, embedded into complex Elliptic surfaces. We show, for example, that any compact surface has a totally real embedding into a blowup of a K3 surface. We also exhibit smooth disc bundles over compact orientable surfaces that have a S ..."
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Abstract. We study the structure of complex points on real surfaces, embedded into complex Elliptic surfaces. We show, for example, that any compact surface has a totally real embedding into a blowup of a K3 surface. We also exhibit smooth disc bundles over compact orientable surfaces that have a Stein structure as Stein domains inside Elliptic surfaces. 1. Statement of results Let S be a real surface, embedded into a complex surface (X,J). We say that the embedding S ֒ → X is totally real at a point p ∈ S, if TpS+J(TpS) = TpX. If this is not the case, we call p a complex point of the embedding. An embedding is called totally real, if it is totally real at all points. Theorem 1.1. Every compact oriented real surface S has a totally real embedding into any K3 surface. Every compact real surface has a totally real embedding into a blowup of a K3 surface at one point. A blowup of a complex surface is of course not minimal. If we want to have an embedding of all compact surfaces into a minimal surface, we have the following theorem. Theorem 1.2. Every compact real surface has a totally real embedding into any E(3) surface. Let us denote by Σg the compact Riemann surface of genus g ≥ 0, and let n be an integer. We denote by D(g,n) the open unit disc bundle over the surface Σg, with Euler number n. It follows from the adjunction inequality for Stein surfaces that for n> 2g − 2, the smooth manifolds D(g,n) do not have any Stein structure. It is furthermore a consequence of the result of Gompf [8], using a method of Stein surgery developed by Eliashberg in [2], that for n ≤ 2g − 2, the smooth manifolds D(g,n) can be endowed with a Stein structure. We use a method of Stein fattening, introduced by Forstnerič [5], to give a different proof of this result, by explicitly seeing D(n,g) as open strictly pseudoconvex Stein domains in Elliptic surfaces E(n). The definition of Elliptic surfaces E(n) is given latter in the text.
ON THE SIGNATURES OF EVEN 4–MANIFOLDS CHRISTIAN BOHR
, 2000
"... Abstract. In this paper, we prove a number of inequalities between the signature and the Betti numbers of a 4–manifold with even intersection form. Furthermore, we introduce a new geometric group invariant and discuss some of its properties. This paper is a prelimininary version and contains some re ..."
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Abstract. In this paper, we prove a number of inequalities between the signature and the Betti numbers of a 4–manifold with even intersection form. Furthermore, we introduce a new geometric group invariant and discuss some of its properties. This paper is a prelimininary version and contains some results of my Ph.D. thesis. 1.
G G GGG
, 2002
"... 4–manifolds as covers of the 4–sphere branched over nonsingular surfaces ..."
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4–manifolds as covers of the 4–sphere branched over nonsingular surfaces
SINGULAR SEIFERT SURFACES AND SMALE INVARIANTS FOR A FAMILY OF 3SPHERE IMMERSIONS
, 903
"... Abstract. A selftransverse immersion of the 2sphere into 4space with algebraic number of self intersection points equal to n induces an immersion of the circle bundle over the 2sphere of Euler class 2n into 4space. Precomposing the circle bundle immersions with their universal covering maps, we ..."
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Abstract. A selftransverse immersion of the 2sphere into 4space with algebraic number of self intersection points equal to n induces an immersion of the circle bundle over the 2sphere of Euler class 2n into 4space. Precomposing the circle bundle immersions with their universal covering maps, we get for n> 0 immersions gn of the 3sphere into 4space. In this note, we compute the Smale invariants of gn. The computation is carried out by (partially) resolving the singularities of the natural singular map of the punctured complex projective plane which extends gn. As an application, we determine the classes represented by gn in the cobordism group of immersions which is naturally identified with the stable 3stem. It follows in particular that gn represents a generator of the stable 3stem if and only if n is divisible by 3. 1.