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The blowup formula for Donaldson invariants
"... Since their introduction in 1984 [D1], the Donaldson invariants of smooth 4manifolds have remained as mysterious as they have been useful. However, in the past year there has been a surge of activity pointed at comprehension of the structure of these invariants [KM, FS]. One key to these advances a ..."
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Cited by 30 (5 self)
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Since their introduction in 1984 [D1], the Donaldson invariants of smooth 4manifolds have remained as mysterious as they have been useful. However, in the past year there has been a surge of activity pointed at comprehension of the structure of these invariants [KM, FS]. One key to these advances and to future insights lies in understanding the relation
Khovanov homology is an unknotdetector
"... Abstract. We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then s ..."
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Cited by 4 (2 self)
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Abstract. We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement: an invariant that is already known to detect the unknot. 1
Surgery and Geometric Topology
, 1996
"... . Ideas from the theory of topological stability of smooth maps are transported into the controlled topological category. For example, the controlled topological equivalence of maps is discussed. These notions are related to the classication of manifold approximate brations and manifold stratied app ..."
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. Ideas from the theory of topological stability of smooth maps are transported into the controlled topological category. For example, the controlled topological equivalence of maps is discussed. These notions are related to the classication of manifold approximate brations and manifold stratied approximate brations. In turn, these maps form a bundle theory which can be used to describe neighborhoods of strata in topologically stratied spaces. 1. Introduction We explore some connections among the theories of topological stability of maps, controlled topology, and stratied spaces. The notions of topological equivalence of maps and locally trivial families of maps play an important role in the theory of topological stability of smooth maps. We formulate the analogues of these notions in the controlled topological category for two reasons. First, the notion of controlled topological equivalence of maps is a starting point for formulating a topological version of Mather's theory of ...
Gauge theory and Rasmussen’s invariant
"... For a knot K S 3, the (smooth) slicegenus g.K / is the smallest genus of any properly embedded, smooth, oriented surface † B4 with boundary K. In [12], Rasmussen used a construction based on Khovanov homology to define a knotinvariant s.K / 2 2Z with the following properties: ..."
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For a knot K S 3, the (smooth) slicegenus g.K / is the smallest genus of any properly embedded, smooth, oriented surface † B4 with boundary K. In [12], Rasmussen used a construction based on Khovanov homology to define a knotinvariant s.K / 2 2Z with the following properties:
Singular Connections on ThreeManifolds and Manifolds with Cylindrical Ends.
, 1998
"... Chapter 1. Fourdimensional Theory. 4 ..."
PROBING MODULI SPACES OF SHEAVES WITH DONALDSON AND SEIBERG WITTEN INVARIANTS
, 2006
"... Abstract. We use Donaldson invariants of regular surfaces with pg> 0 to make quantitative statements about modulispaces of stable rank 2 sheaves. We give two examples: a quantitative existence theorem for stable bundles, and a computation of the rank of the canonical holomorphic two forms on the mod ..."
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Abstract. We use Donaldson invariants of regular surfaces with pg> 0 to make quantitative statements about modulispaces of stable rank 2 sheaves. We give two examples: a quantitative existence theorem for stable bundles, and a computation of the rank of the canonical holomorphic two forms on the moduli space. The results are in some sense dual to the Donaldson and O’Grady non vanishing theorems because they use the Donaldson series of the surface as input. Results in purely algebraic geometric terms can be obtained by using the explicit form of the Donaldson series of the surface. The Donaldson series are easy to compute using the Seiberg Witten invariants and the Witten conjecture which has recently been rigorously proved by Feehan and Leness. 1.