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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 68 (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
Higher order intersection numbers of 2spheres in 4manifolds
 ALGEBRAIC & GEOMETRIC TOPOLOGY
, 2000
"... This is the beginning of an obstruction theory for deciding whether a map f: S2 → X is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres in a topological 4manifold X. The first obstruction is Wall’s well known selfintersection nu ..."
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Cited by 16 (9 self)
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This is the beginning of an obstruction theory for deciding whether a map f: S2 → X is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres in a topological 4manifold X. The first obstruction is Wall’s well known selfintersection number µ(f) which tells the whole story in higher dimensions. Our second order obstruction τ(f) is defined if µ(f) vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of π1X modulo S3symmetry (rather then just one copy modulo S2symmetry). It generalizes to the nonsimply connected setting the KervaireMilnor invariant defined in [2] and [12] which corresponds to the Arfinvariant of knots in 3space. We also give necessary and sufficient conditions for homotoping three maps f1, f2, f3: S2 → X to a position in which they have disjoint images. The obstruction λ(f1, f2, f3) generalizes Wall’s intersection number λ(f1, f2) which answers the same question for two spheres but is not sufficient (in dimension 4) for three spheres. In the same way as intersection numbers correspond to linking numbers in 3space, our new invariant corresponds to the Milnor invariant µ(1, 2, 3), generalizing the Matsumoto triple to the non simplyconnected setting. Finally, we explain some simple algebraic properties of these new cubic forms on π2(X) in Theorem 3. These are straightforward generalizations of the properties of quadratic forms as defined by Wall [14, §5]. A particularly attractive formula is λ(f, f, f) = ∑ τ(f) σ σ∈S3 which generalizes the well known fact that Wall’s invariants satisfy λ(f, f) = µ(f) + µ(f) = ∑ µ(f) σ for an immersion f with trivial normal bundle. σ∈S2 1.
Generalized Seifert surfaces and signatures of colored links
, 2005
"... In this paper, we use ‘generalized Seifert surfaces’ to extend the LevineTristram signature to colored links in S³. This yields an integral valued function on the µdimensional torus, where µ is the number of colors of the link. The case µ = 1 corresponds to the LevineTristram signature. We show t ..."
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Cited by 6 (2 self)
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In this paper, we use ‘generalized Seifert surfaces’ to extend the LevineTristram signature to colored links in S³. This yields an integral valued function on the µdimensional torus, where µ is the number of colors of the link. The case µ = 1 corresponds to the LevineTristram signature. We show that many remarkable properties of the latter invariant extend to this µvariable generalization: it vanishes for achiral colored links, it is ‘piecewise continuous’, and the places of the jumps are determined by the Alexander invariants of the colored link. Using a 4dimensional interpretation and the AtiyahSinger Gsignature theorem, we also prove that this signature is invariant by colored concordance, and that it provides a lower bound for the ‘slice genus’ of the colored link.
BOUNDS ON GENUS AND GEOMETRIC INTERSECTIONS FROM CYLINDRICAL END MODULI SPACES
, 2003
"... Abstract. In this paper we present a way of computing a lower bound for genus of any smooth representative of a homology class of positive selfintersection in a smooth fourmanifold X with second positive Betti number b + 2 (X) = 1. We study the solutions of the SeibergWitten equations on the cyl ..."
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Abstract. In this paper we present a way of computing a lower bound for genus of any smooth representative of a homology class of positive selfintersection in a smooth fourmanifold X with second positive Betti number b + 2 (X) = 1. We study the solutions of the SeibergWitten equations on the cylindrical end manifold which is the complement of the surface representing the class. The result can be formulated as a form of generalized adjunction inequality. The bounds obtained depend only on the rational homology type of the manifold, and include the Thom conjecture as a special case. We generalize this approach to derive lower bounds on the number of intersection points of n algebraically disjoint surfaces of positive selfintersection in manifolds with b + 2 (X) = n.
BOUNDS ON GENUS AND GEOMETRIC INTERSECTIONS FROM CYLINDRICAL END MODULI SPACES
, 2002
"... Abstract. In this paper we present a way of computing a lower bound for genus of any smooth representative of a homology class of positive selfintersection in a smooth fourmanifold X with second positive Betti (X) = 1. We study the solutions of the SeibergWitten equations on the cylindrical end ..."
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Abstract. In this paper we present a way of computing a lower bound for genus of any smooth representative of a homology class of positive selfintersection in a smooth fourmanifold X with second positive Betti (X) = 1. We study the solutions of the SeibergWitten equations on the cylindrical end manifold which is the complement of the surface representing the class. The result can be formulated as a form of generalized adjunction inequality. The bounds obtained depend only on the rational homology type of the manifold, and include the Thom conjecture as a special case. We generalize this approach to derive lower bounds on the number of intersection points of n algebraically disjoint number b + 2 surfaces of positive selfintersection in manifolds with b + 2
BOUNDS ON GENUS AND GEOMETRIC INTERSECTIONS FROM CYLINDRICAL END MODULI SPACES
, 2002
"... Abstract. In this paper we present a way of computing a lower bound for genus of any smooth representative of a homology class of positive selfintersection in a smooth fourmanifold X with second positive Betti (X) = 1. We study the solutions of the SeibergWitten equations on the cylindrical end ..."
Abstract
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Abstract. In this paper we present a way of computing a lower bound for genus of any smooth representative of a homology class of positive selfintersection in a smooth fourmanifold X with second positive Betti (X) = 1. We study the solutions of the SeibergWitten equations on the cylindrical end manifold which is the complement of the surface representing the class. The result can be formulated as a form of generalized adjunction inequality. The bounds obtained depend only on the rational homology type of the manifold, and include the Thom conjecture as a special case. We generalize this approach to derive lower bounds on the number of intersection points of n algebraically disjoint number b + 2 surfaces of positive selfintersection in manifolds with b + 2