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HOLOMORPHIC DISKS AND THREEMANIFOLD INVARIANTS: PROPERTIES AND APPLICATIONS
, 2001
"... ... and HFred(Y, s) associated to oriented rational homology 3spheres Y and Spin c structures s ∈ Spin c (Y). In the first part of this paper we extend these constructions to all closed, oriented 3manifolds. In the second part, we study the properties of these invariants. The properties include a ..."
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Cited by 201 (31 self)
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... and HFred(Y, s) associated to oriented rational homology 3spheres Y and Spin c structures s ∈ Spin c (Y). In the first part of this paper we extend these constructions to all closed, oriented 3manifolds. In the second part, we study the properties of these invariants. The properties include a relationship between the Euler characteristics of HF ± and Turaev’s torsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to threemanifold topology.
Hopf algebras, cyclic cohomology and the transverse index theorem
 Comm. Math. Phys
, 1998
"... In this paper we present the solution of a longstanding internal problem of noncommutative geometry, namely the computation of the index of transversally elliptic operators on foliations. ..."
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Cited by 192 (20 self)
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In this paper we present the solution of a longstanding internal problem of noncommutative geometry, namely the computation of the index of transversally elliptic operators on foliations.
Monstrous moonshine and monstrous Lie superalgebras
 INVENT. MATH
, 1992
"... We prove Conway and Norton’s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by Frenkel, Lepowsky and Meurman. To do this we use the noghost theorem from string theory to construct a family of generalized KacMoody superalgebras of rank 2, w ..."
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Cited by 166 (0 self)
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We prove Conway and Norton’s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by Frenkel, Lepowsky and Meurman. To do this we use the noghost theorem from string theory to construct a family of generalized KacMoody superalgebras of rank 2, which are closely related to the monster and several of the other sporadic simple groups. The denominator formulas of these superalgebras imply relations between the Thompson functions of elements of the monster (i.e. the traces of elements of the monster on Frenkel, Lepowsky, and Meurman’s representation), which are the replication formulas conjectured by Conway and Norton. These replication formulas are strong enough to verify that the Thompson functions have most of the “moonshine ” properties conjectured by Conway and Norton, and in particular they are modular functions of genus 0. We also construct a second family of KacMoody superalgebras related to elements of Conway’s sporadic simple group Co1. These superalgebras have even rank between 2 and 26; for example two of the Lie algebras we get have ranks 26 and 18, and one of the superalgebras has rank 10. The denominator formulas of these algebras give some new infinite product identities, in the same way that the denominator
Noncommutative FiniteDimensional Manifolds  I. SPHERICAL MANIFOLDS AND RELATED EXAMPLES
, 2001
"... We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 d ..."
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Cited by 125 (15 self)
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We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 defined by basic Ktheoretic equations. We find a 3parameter family of deformations of the standard 3sphere S 3 and a corresponding 3parameter deformation of the 4dimensional Euclidean space R 4. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R 4 u are isomorphic to the algebras introduced by Sklyanin in connection with the YangBaxter equation. Special values of the deformation parameters do not give rise to Sklyanin algebras and we extract a subclass, the θdeformations, which we generalize in any dimension and various contexts, and study in some details. Here, and
Holomorphic triangles and invariants for smooth fourmanifolds
"... Abstract. The aim of this article is to introduce invariants of oriented, smooth, closed fourmanifolds, built using the Floer homology theories defined in [8] and [12]. This fourdimensional theory also endows the corresponding threedimensional theories with additional structure: an absolute gradi ..."
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Cited by 124 (24 self)
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Abstract. The aim of this article is to introduce invariants of oriented, smooth, closed fourmanifolds, built using the Floer homology theories defined in [8] and [12]. This fourdimensional theory also endows the corresponding threedimensional theories with additional structure: an absolute grading of certain of its Floer homology groups. The cornerstone of these constructions is the study of holomorphic disks in the symmetric products of Riemann surfaces. 1.
Deriving Dg Categories
, 1993
"... We investigate the (unbounded) derived category of a differential Zgraded category (=DG category). As a first application, we deduce a 'triangulated analogue` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories among abelian categori ..."
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Cited by 123 (9 self)
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We investigate the (unbounded) derived category of a differential Zgraded category (=DG category). As a first application, we deduce a 'triangulated analogue` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a 'Morita theorem` (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories. Summary We give an account of the contents of this paper for the special case of DG algebras. Let k be a commutative ring and A a DG (k)algebra, i.e. a Zgraded kalgebra A = a p2Z A p endowed with a differential d of degree 1 such that d(ab) = (da)b + (\Gamma1) p a(db) for all a 2 A p , b 2 A. A DG (right) Amodule is a Zgraded Amodule M = ` p2Z M p endowed with a differential d of degree 1 such that d(ma) = (dm)a + (\Gamma1) p m(da) for all m 2 M p , a 2 A. A morphism of DG Amodules is a homogeneous morphism of degree 0 of the underlying graded Amodules commuting with the differentials. The DG Amodules form an abelian category CA. A morphism f : M ! N of CA is nullhomotopic if f = dr + rd for some homogeneous morphism r : M ! N of degree1 of the underlying graded Amodules.
On the nonexistence of elements of Hopf invariant one
 ANN. OF MATH
, 1960
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