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Gauge theory for embedded surfaces
 I, Topology
, 1993
"... (i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional 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 ..."
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Cited by 67 (6 self)
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(i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional 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
4Manifolds with a Symplectic Bias
"... Abstract. This text reviews some state of the art and open questions on (smooth) 4manifolds from the point of view of symplectic geometry. ..."
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Abstract. This text reviews some state of the art and open questions on (smooth) 4manifolds from the point of view of symplectic geometry.
LINKS, TWOHANDLES, AND FOURMANIFOLDS
, 2005
"... Abstract. We show that only finitely many links in a closed 3manifold share the same complement, up to twists along discs and annuli. Using the same techniques, we prove that by adding 2handles on the same link we get only finitely many smooth cobordisms between two given closed 3manifolds. As a ..."
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Abstract. We show that only finitely many links in a closed 3manifold share the same complement, up to twists along discs and annuli. Using the same techniques, we prove that by adding 2handles on the same link we get only finitely many smooth cobordisms between two given closed 3manifolds. As a consequence, there are finitely many closed smooth 4manifolds constructed from some Kirby diagram with bounded number of crossings, discs, and strands, or from some Turaev special shadow with bounded number of vertices. (These are the 4dimensional analogues of Heegaard diagrams and special spines for 3manifolds.) We therefore get two filtrations on the set of all closed smooth 4manifolds with finite sets. The two filtrations are equivalent after linear rescalings, and their cardinality grows at least as n c·n.
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.
An Introduction to Exotic 4manifolds
, 812
"... This article intends to provide an introduction to the ..."
ON A CONCRETE CONSTRUCTION OF A SINGULAR SEIFERT Surface
, 2009
"... First we compute the Smale invariant of a certain immersion of the 3sphere into 4space, originally introduced by Milnor. Our computation is based on a concrete construction of a singular map extending the given immersion, where we utilise unfoldings of a certain complex map germ, regarded as a re ..."
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First we compute the Smale invariant of a certain immersion of the 3sphere into 4space, originally introduced by Milnor. Our computation is based on a concrete construction of a singular map extending the given immersion, where we utilise unfoldings of a certain complex map germ, regarded as a real map germ. The result implies that the immersion has order 8 in the oriented cobordism group of codimension one immersions of 3manifolds, identified with the stable 3stem. This recovers a result due to Ekholm that the immersion has odd Brown invariant. We next apply the same method for another immersion to show that it then represents a generator. Furthermore, we show that this second immersion coincides with Melikhov’s example of an immersion with nontrivial stable Hopf invariant and consequently that his immersion actually represents a generator of the stable 3stem.