Search

We develop a method to construct representations of the homotopy 2-groupoid of a manifold as a 2-category by means of Chen’s formal homology connections. As an application we describe 2-holonomy maps for hyperplane arrangements and discuss representations of the category of braid cobordisms.

We review various aspects of representations of the holonomy Lie algebras and the associated monodromy representations of the fundamental groups for the complement of hyperplane arrangements. In particular, we describe the relation between monodromy representations of the KZ equation and homological representations of the braid groups by means of hypergeometric integrals.

We describe a relationship between homological representations of the braid groups due to R. Lawrence, D. Krammer, and S. Bigelow and the monodromy representations of the Knizhnik-Zamolodchikov (KZ) connection. First, we describe the comparison theorem in the case of generic parameters. Then, we discuss the nongeneric case from the viewpoint of conformal field theory.

Let X be a finite CW-complex, denote its fundamental group by G. Let R be an n-dimensional complex repesentation of G. Any element A of the first cohomology group of X with complex coefficients gives rise to the exponential deformation of the representation R, which can be considered as a curve in the space of representations. We show that the cohomology of X with local coefficients corresponding to the generic point of this curve is computable from a spectral sequence starting from the cohomology of X with R-twisted coefficients. We compute the differentials of the spectral sequence in terms of Massey products. We show that the spectral sequence degenerates in case when X is a Kaehler manifold, and the representation R is semi-simple. If A is a real cohomology class, one associates to the triple (X,R,A) the twisted Novikov homology (a module over the Novikov ring). We show that the twisted Novikov Betti numbers equal the Betti numbers of X with coefficients in the above local system. We investigate the dependance of these numbers on A and prove that they are constant in the complement to a finite number of integral hyperplanes in the first cohomology group., Final version, references added

We consider subgroups of the braid groups which are generated by $$n$$ th powers of the standard generators and prove that any infinite intersection (with even $$n$$ ) is trivial. This is motivated by some conjectures of Squier concerning the kernels of Burau’s representations of the braid groups at roots of unity. Furthermore, we show that the image of the braid group on 3 strands by these representations is either a finite group, for a few roots of unity, or a finite extension of a triangle group, by using geometric methods.

Let A be an arrangement of complex hyperplanes and M A the complement of the union of hyperplanes in A . The Orlik–Solomon algebra of A determines a subcomplex of the de Rham complex of the loop space of M A , which is called the bar complex of the Orlik–Solomon algebra. The dual of this complex is isomorphic to the tensor algebra of the homology of M A equipped with a derivation arising from the product structure of the Orlik–Solomon algebra. Based on this construction we give an explicit description of Chen's iterated integrals of logarithmic forms depending only on the homotopy class of a loop.

We prove that, except for a few explicit roots of unity, the quantum image of any Johnson subgroup of the mapping class group contains an explicit free non-abelian subgroup., Comment: revised version 14p., 4 figures. We removed the sections containing applications of the largeness results presented here to finite index subgroups of mapping class groups, as the proofs in the former version have a gap. arXiv admin note: substantial text overlap with arXiv:0907.0568

We study finite type invariants for links in the product of the torus and the unit interval by means of the monodromy of the elliptic KZ equation. We show that the weight system for the chord diagrams on the torus defined for any representation of the Lie algebra sl(N, C ) and a certain projectively flat connection on the torus can be integrated to a link invariant by using the elliptic KZ system.

We construct a representation of the homotopy 2-groupoid of a manifold by means of Chen’s formal homology connections. By using the idea of this 2-holonomy map, we describe a method to obtain a representation of the category of braid cobordisms.

We review some aspects of the homology of a local system on the complement of a hyperplane arrangement. We describe a relationship between linear representations of the braid groups due to R. Lawrence, D. Krammer and S. Bigelow and the holonomy representations of the KZ connection. We explain a method to describe such representations by the iterated integrals of logarithmic 1-forms.

By means of a description of the solutions of the KZ equation using hypergeometric integrals we show that the homological representations of the braid groups studied by Lawrence, Krammer and Bigelow are equivalent at generic complex values to the monodromy of the KZ equation with values in the space of null vectors in the tensor product of Verma modules of sl2(C).

We give a relation between the homological representations of the braid groups studied by Lawrence, Krammer and Bigelow and the monodromy of the KZ equation with values in the space of null vectors in the tensor product of Verma modules of sl2(C) when the parameters are generic. We also discuss the case of special parameters in relation with integral representations of the space of conformal blocks by hypergeometric integrals.

Let A be an essential complex hyperplane arrangement in an n-dimensional complex vector space V. Let H denote the union of the hyperplanes, and M denote the complement to H in V. We develop the real-valued and circle-valued Morse theory for M and prove, in particular, that M has the homotopy type of a space obtained from a manifold fibered over a circle, by attaching cells of dimension n. We compute the Novikov homology of M for a large class of homomorphisms of the fundamental group of M to R., Comment: 15 pages, revised version

The purpose of this paper is to give a brief review on the author's approach to define 3-manifold invariants using representations of modular groups appearing in conformal field theory on Riemann surfaces. After the Witten's discovery of 3-manifold invariants based on Chern-Simons gauge theory. Reshetikhin and Turaev gave a Dehn surgery formula for Witten invariant, which was studied extensively by Kirby and Melvin and others. In this paper the authors describe the nature of these representations. The representations of the mapping class groups discussed in this paper are projectively linear and we write down the associated 2-cocyle in a combinatorial way. The authors also show that these representations are unitary. Combining this unitarity with the comparison of our approach and the Dehn surgery formula, the authors give a lower estimate of the Heegaard genus of a closed orientable 3-manifold by means of the Witten invariant. In this paper the authors discuss the case of SU(2) WZW model at level K.

The total homology of the loop space of the configuration space of ordered distinct n points in R^m has a structure of a Hopf algebra defined by the 4-term relations if m>2. We describe a relation of between the cohomology of this loop space and the set of finite type invariants for the pure braid group with n strands. Based on this we give expressions of certain link invariants as integrals over cycles of the above loop space., Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon4/paper10.abs.html Version 2: corrections to superscripts on pages 148 and 149

Motive par la theorie des champs conforme en dimension deux avec la symetrie de jauge, on etudie la monodromie des connexions integrables associees aux algebres de Lie simples. Ceci donne une serie de representations lineaires du groupe de tresses dont la forme explicite est exprimee par des solutions de l'equation de Yang-Baxter

Motive par la theorie des champs conforme en dimension deux avec la symetrie de jauge, on etudie la monodromie des connexions integrables associees aux algebres de Lie simples. Ceci donne une serie de representations lineaires du groupe de tresses dont la forme explicite est exprimee par des solutions de l'equation de Yang-Baxter

The purpose of this paper is to establish the following isomorphism of Lie algebras.Main Theorem. Let X be the complement of a hypersurface S in the complex projective space PN. Then the tower of nilpotent complex Lie algebras associated with the fundamental group π1(X, *) and the holonomy Lie algebraattached to S are isomorphic.

LET A4 be a closed orientable 3-manifold. The purpose of this paper is to define topological invariants 4,(M) parametrized by K = 1,2,. . . using representations of the mapping class group of an orientable surface. In [27], Witten introduced new topological invariants for 3manifolds based on quantum field theory and motivated by Witten’s Dehn surgery formula, Reshetikhin and Turaev [I93 gave a formula expressed by the Jones polynomial and its relatives, by means of representations of the quantized universal enveloping algebra in the case q is a root of unity. Our approach using a Heegaard decomposition of a 3-manifold is difTerent from theirs. Let Xr denote a closed oricntablc surface of genus y. First, WC consider a pants decomposition of C, and WC dcnotc by Y its dual graph, which is a trivalent graph. Given a positive intcgcr K called a Icvel, WC introduce a finite dimensional complex vector space Z,(y) whose basis is in one-to-one correspondence with the set of admissible weights /: edge(y) + (0, l/2, * . . , K/2} satisfying the Clebsch-Gordan condition and the algebraic constraintj(c,) +/(cL) +/(cJ) IS K for any edges c,, c2 and c, meeting at a vertex. This vector space appears as a combinatorial description of the space of conformal blocks for SU(2) Wess-Zumino-Witten model at level K (see [23]). Let yr and Y2 be trivalent graphs associated with markings of C,. By means of the fusing matrices describing the holonomy of the Knizhnik-Zamolodchikov equation, we obtain a canonical isomorphism Z,(Yr) z Z,(y2). A detailed study of the monodromy of the Knizhnik-Zamolodchikov equation was pursued by Tsuchiya and Kanie [22]. It is known that the representations of the braid groups appearing as the monodromy of the Knizhnik-Zamolodchikov equation provide the Jones polynomial and its relatives (see [13], [IS] and [22]). More recently, Drinfel’d established a relation between the monodromy and the quantized universal enveloping algebras in a general situation (see [6]). using the notion of quasi-Hopf algebras. In [is], Moore and Seiberg wrote down a series of polynomial equations among fusing matrices, braiding matrices and so-called switching operators expected from the viewpoint of the consistency conditions in conformal field theory. The holonomy of the Knizhnik-Zamolodchikov equation gives solutions to these polynomial equations, and as a consequence we obtain projectively linear representations of the mapping class group

We show that the bar complex of the configuration space of ordered distinct points in the complex plane is acyclic. The 0-dimensional cohomology of this bar complex is identified with the space of finite type invariants for braids. We construct a universal holonomy homomorphism from the braid group to the space of horizontal chord diagrams over Q, which provides finite type invariants for braids with values in Q.