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33
An invariant of integral homology 3spheres which is universal for all finite type invariants
, 1996
"... In [LMO] a 3manifold invariant Ω(M) is constructed using a modification of the Kontsevich integral and the Kirby calculus. The invariant Ω takes values in a graded Hopf algebra of Feynman 3valent graphs. Here we show that for homology 3spheres the invariant Ω is universal for all finite type inv ..."
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Cited by 64 (4 self)
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In [LMO] a 3manifold invariant Ω(M) is constructed using a modification of the Kontsevich integral and the Kirby calculus. The invariant Ω takes values in a graded Hopf algebra of Feynman 3valent graphs. Here we show that for homology 3spheres the invariant Ω is universal for all finite type invariants, i.e. Ωn is an invariant order 3n which dominates all other invariants of the same order. Some corollaries are discussed.
Finite Type 3Manifold Invariants, The Mapping Class Group And Blinks
, 1996
"... The goal of the present paper is to find higher genus surgery formulas for the set of finite type invariants of integral homology 3spheres, and to develop a theory of finite type invariants which will be applied in a subsequent publication [GL3] in the study of subgroups of the mapping class ..."
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Cited by 26 (5 self)
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The goal of the present paper is to find higher genus surgery formulas for the set of finite type invariants of integral homology 3spheres, and to develop a theory of finite type invariants which will be applied in a subsequent publication [GL3] in the study of subgroups of the mapping class group. The main result is to show that six filtrations on the vector space generated by oriented integral homology 3spheres (three coming from surgery on special classes of links and three coming from subgroups of the mapping class group) are equal. En route we introduce the notion of blink (a special case of a link) and of a new subgroup of the mapping class group.
On finite type 3manifold invariants III: manifold weight systems, Topology, in press. M.N. Gusarov, On nequivalence of knots and invariants of finite degrees, Topology of manifolds and varieties, edited by
, 1994
"... Abstract. The present paper is a continuation of [Oh2] and [GL] devoted to the study of finite type invariants of integral homology 3spheres. We introduce the notion of manifold weight systems, and show that type m invariants of integral homology 3spheres are determined (modulo invariants of type ..."
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Cited by 25 (10 self)
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Abstract. The present paper is a continuation of [Oh2] and [GL] devoted to the study of finite type invariants of integral homology 3spheres. We introduce the notion of manifold weight systems, and show that type m invariants of integral homology 3spheres are determined (modulo invariants of type m − 1) by their associated manifold weight systems. In particular we deduce a vanishing theorem for finite type invariants. We show that the space of manifold weight systems forms a commutative, cocommutative Hopf algebra and that the map from finite type invariants to manifold weight systems is an algebra map. We conclude with better bounds for the graded space of finite type invariants of integral homology
The SU(3) Casson Invariant For Integral Homology 3Spheres
, 1998
"... We derive a gauge theoretic invariant of integral homology 3spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, p ..."
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Cited by 21 (7 self)
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We derive a gauge theoretic invariant of integral homology 3spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, perturbed flat orbits. Our formula for the correction term generalizes that given by Walker in his extension of Casson’s SU(2) invariant to rational homology 3spheres.
The Alexander polynomial and finite type 3manifold invariants
 Math. Ann
"... Abstract. Using elementary counting methods, we calculate the universal perturbative invariant (also known as the LMO invariant) of a 3manifold M, satisfying H1(M, Z) = Z, in terms of the Alexander polynomial of M. We show that +1 surgery on a knot in the 3sphere induces an injective map from fin ..."
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Cited by 20 (7 self)
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Abstract. Using elementary counting methods, we calculate the universal perturbative invariant (also known as the LMO invariant) of a 3manifold M, satisfying H1(M, Z) = Z, in terms of the Alexander polynomial of M. We show that +1 surgery on a knot in the 3sphere induces an injective map from finite type invariants of integral homology 3spheres to finite type invariants of knots. We also show that weight systems of degree 2m on knots, obtained by applying finite type 3m invariants of integral homology 3spheres, lie in the algebra of AlexanderConway weight systems, thus answering the questions raised in [Ga].
Integration of singular braid invariants and graph cohomology
, 1995
"... Abstract. We prove necessary and sucient conditions for an arbitrary invariant of braids with m double points to be the \mth derivative " of a braid invariant. We show that the \primary obstruction to integration " is the only obstruction. This gives a slight generalization of the existen ..."
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Cited by 14 (0 self)
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Abstract. We prove necessary and sucient conditions for an arbitrary invariant of braids with m double points to be the \mth derivative " of a braid invariant. We show that the \primary obstruction to integration " is the only obstruction. This gives a slight generalization of the existence theorem for Vassiliev invariants of braids. We give a direct proof by induction on m which works for invariants with values in any abelian group. We nd that to prove our theorem, we must show that every relation among fourterm relations satises a certain geometric condition. To nd the relations among relations we show that H1 of a variant of Kontsevich’s graph complex vanishes. We discuss related open questions for invariants of links and other things. 1.
Calculus of clovers and finite type invariants . . .
, 2001
"... A clover is a framed trivalent graph with some additional structure, embedded in a 3manifold. We define surgery on clovers, generalizing surgery on Y{graphs used earlier by the second author to define a new theory of finitetype invariants of 3manifolds. We give a systematic exposition of a topolo ..."
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Cited by 13 (2 self)
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A clover is a framed trivalent graph with some additional structure, embedded in a 3manifold. We define surgery on clovers, generalizing surgery on Y{graphs used earlier by the second author to define a new theory of finitetype invariants of 3manifolds. We give a systematic exposition of a topological calculus of clovers and use it to deduce some important results about the corresponding theory of nite type invariants. In particular, we give a description of the weight systems in terms of unitrivalent graphs modulo the AS and IHX relations, reminiscent of the similar results for links. We then compare several definitions of finite type invariants of homology spheres (based on surgery on Ygraphs, blinks, algebraically split links, and boundary links) and prove in a selfcontained way their equivalence.
FINITE TYPE 3MANIFOLD INVARIANTS AND THE STRUCTURE OF THE TORELLI GROUP I
"... Abstract. Using the recently developed theory of nite type invariants of integral homology 3spheres we study the structure of the Torelli group of a closed surface. Explicitly,we construct (a) natural cocycles of the Torelli group (with coe cients in a space of trivalent graphs) and cohomology clas ..."
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Cited by 10 (0 self)
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Abstract. Using the recently developed theory of nite type invariants of integral homology 3spheres we study the structure of the Torelli group of a closed surface. Explicitly,we construct (a) natural cocycles of the Torelli group (with coe cients in a space of trivalent graphs) and cohomology classes of the abelianized Torelli group; (b) group homomorphisms that detect (rationally) the nontriviality of the lower central series of the Torelli group. Our results are motivated by the appearance of trivalent graphs in topology and in representation theory and the dual role played by the Casson invariant in the theory of nite type invariants of integral homology 3spheres and in Morita's study [Mo2, Mo3] of the structure of the Torelli group Our results generalize those of S. Morita [Mo2, Mo3] and complement the recent calculation, due to R. Hain [Ha2], of the Iadic completion of the rational group ring of the Torelli group. We also give analogous results for two other subgroups of the mapping class group.