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Statesum construction of twodimensional openclosed TQFTs
 In preparation
"... We present a state sum construction of twodimensional extended Topological Quantum Field Theories (TQFTs), socalled openclosed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional twodimensional cobordisms to those of openclosed cobordisms, i.e. smoo ..."
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Cited by 11 (5 self)
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We present a state sum construction of twodimensional extended Topological Quantum Field Theories (TQFTs), socalled openclosed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional twodimensional cobordisms to those of openclosed cobordisms, i.e. smooth compact oriented 2manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a twodimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an openclosed TQFT with a finite set of Dbranes using the example of the groupoid algebra of a finite groupoid.
Bottom tangles and universal invariants
, 2006
"... A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite ..."
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Cited by 5 (2 self)
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A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of B, which provides an especially convenient way to generate all the bottom tangles, and hence all the framed, oriented links, via closure. We also define a kind of “braided Hopf algebra action ” on the set of bottom tangles. Using the universal invariant of bottom tangles associated to each ribbon Hopf algebra H, we define a braided functor J from B to the category ModH of left H–modules. The functor J, together with the set of generators of B, provides an algebraic method to study the range of quantum invariants of links. The braided Hopf algebra action on bottom tangles is mapped by J to the standard braided Hopf algebra structure for H in ModH. Several notions in knot theory, such as genus, unknotting number, ribbon knots, boundary links, local moves, etc are given algebraic interpretations in the setting involving the category B. The functor J provides a convenient way to study the relationships between these notions and quantum invariants.
A FUNCTORIAL LMO INVARIANT FOR LAGRANGIAN COBORDISMS
, 2007
"... Abstract. Lagrangian cobordisms are threedimensional compact oriented cobordisms between oncepunctured surfaces, subject to some homological conditions. We extend the Le–Murakami–Ohtsuki invariant of homology threespheres to a functor ..."
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Cited by 4 (2 self)
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Abstract. Lagrangian cobordisms are threedimensional compact oriented cobordisms between oncepunctured surfaces, subject to some homological conditions. We extend the Le–Murakami–Ohtsuki invariant of homology threespheres to a functor
INFINITESIMAL MORITA HOMOMORPHISMS AND THE TREELEVEL OF THE LMO INVARIANT
, 809
"... Abstract. Let Σ be a compact connected oriented surface with one boundary component, and let π be the fundamental group of Σ. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of Σ, whose kth term consists of the selfhomeomorphisms of Σ that act trivially at the leve ..."
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Cited by 3 (0 self)
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Abstract. Let Σ be a compact connected oriented surface with one boundary component, and let π be the fundamental group of Σ. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of Σ, whose kth term consists of the selfhomeomorphisms of Σ that act trivially at the level of the kth nilpotent quotient of π. Morita defined a homomorphism from the kth term of the Johnson filtration to the third homology group of the kth nilpotent quotient of π. In this paper, we replace groups by their Malcev Lie algebras and we study the “infinitesimal ” version of the kth Morita homomorphism, which corresponds to the original version by a canonical isomorphism. We provide a diagrammatic description of the kth infinitesimal Morita homomorphism and, given an expansion of the free group π that is “symplectic ” in some sense, we derive it from Kawazumi’s “total Johnson map ” (which is a way of recording the action of the Torelli group of Σ on π). We also consider the diagrammatic representation of the Torelli group that we obtained from the Le–Murakami–Ohtsuki invariant of 3manifolds in a previous joint work with Cheptea and Habiro, and which we call the “LMO homomorphism. ” We show that the LMO invariant induces a symplectic expansion of π for which the total Johnson map tantamounts to the treereduction of the LMO homomorphism. We deduce that the kth infinitesimal Morita homomorphism coincides with the degree [k, 2k [ part of the treereduction of the LMO homomorphism. Our results also apply to the monoid of homology cylinders over Σ.
COVERING MOVES AND KIRBY CALCULUS
, 2004
"... We show that simple coverings of B 4 branched over ribbon surfaces up to certain local ribbon moves coincide with orientable 4dimensional 2handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S 3 b ..."
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Cited by 1 (0 self)
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We show that simple coverings of B 4 branched over ribbon surfaces up to certain local ribbon moves coincide with orientable 4dimensional 2handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S 3 branched over links, in terms of local moves. This result generalizes to coverings of any degree the ones by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of our equivalence theorem to possibly nonsimple coverings of S 3 branched over embedded graphs.
Multiplicative Structures of 2dimensional Topological Quantum Field Theory
, 2003
"... Category theory provides a uniform method of encoding mathematical structures and universal constructions with them. In this article we apply the method of additional structures on the objects of a category to deform a comonoid structure, used implicitly in all categories. To deform this comultiplic ..."
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Category theory provides a uniform method of encoding mathematical structures and universal constructions with them. In this article we apply the method of additional structures on the objects of a category to deform a comonoid structure, used implicitly in all categories. To deform this comultiplication we consider internal categories in a monoidal category with some special properties. Then we consider structures over comonoids and show that deformed internal categories form a 2category. This provides the possibility to study, in a uniform way, different types of generalized multiplicative and comultiplicative structures of 2dimensional Topological Quantum Field Theory.