58 research outputs found
Orbit configuration spaces associated to discrete subgroups of PSL(2,R)
No full textAbstractThe purpose of this article is to analyze several Lie algebras associated to “orbit configuration spaces” obtained from a group G acting freely, and properly discontinuously on the upper half-plane H2. The Lie algebra obtained from the descending central series for the associated fundamental group is shown to be isomorphic, up to a regrading, to 1.the Lie algebra obtained from the higher homotopy groups of analogous constructions associated to H2×Cq modulo torsion, as well as2.the Lie algebra obtained from horizontal chord diagrams for surfaces. The resulting Lie algebras are similar to those studied in [T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Contemp. Math. 78 (1988) 339–363; T. Kohno, Vassiliev invariants and de Rham complex on the space of knots, Contemp. Math. 179 (1994) 123–138; T. Kohno, Elliptic KZ system, braid groups of the torus and Vassiliev invariants, Topology and its Applications 78 (1997) 79–94; D.C. Cohen, Monodromy of fiber-type arrangements and orbit configuration spaces, Forum Math. 13 (2001) 505–530; F.R. Cohen, M. Xicoténcatl, On orbit configuration spaces associated to the Gaussian integers: homology and homotopy groups, Topology Appl. 118 (2002) 17–29; E. Fadell and S. Husseini, The space of loops on configuration spaces and the Majer-Terracini index, Topol. Methods Nonlinear Anal. J. Julius Schauder Center 11 (1998), 249–271; E. Fadell and S. Husseini, Geometry and Topology of Configuration Spaces, in: Springer Monographs in Mathematics, Springer-Verlag, 2001; F.R. Cohen and T. Sato, On groups of morphisms of coalgebras, (submitted for publication)]. The structure of a related graded Poisson algebra defined below and obtained from an analogue of the infinitesimal braid relations parametrized by G is also addressed
Conformal field theory and topology
No full textThe aim of this book is to provide the reader with an introduction to conformal field theory and its applications to topology. The author starts with a description of geometric aspects of conformal field theory based on loop groups. By means of the holonomy of conformal field theory he defines topological invariants for knots and 3-manifolds. He also gives a brief treatment of Chern-Simons perturbation theory
Bar complex of the Orlik–Solomon algebra
No full textAbstractLet A be an arrangement of complex hyperplanes and MA 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 MA, 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 MA 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
Rank 2 bundles with meromorphic connections with poles of Poincaré rank 1 :
No full textHolomorphic vector bundles on ℂ × , a complex manifold, with meromorphic connections with poles of Poincaré rank 1 along {0} × , arise naturally in algebraic geometry. They are called ()-structures here. This paper takes an abstract point of view. It gives a complete classification of all ()-structures of rank 2 over germs (, ⁰) of manifolds. In the case of , they separate into four types. Those of the three types have universal unfoldings; those of the fourth type (the logarithmic type) do not. The classification of unfoldings of ()-structures of the fourth type is rich and interesting. The paper finds and lists all ()-structures which are basic in the following sense: Together they induce all rank 2 ()-structures, and each of them is not induced by any other ()-structure in the list. Their base spaces turn out to be 2-dimensional F-manifolds with Euler fields. The paper also provides a classification of all rank 2 ()-structures over it. Also, this classification is surprisingly rich. The backbone of the paper is normal forms. Though also the monodromy and the geometry of the induced Higgs fields and of the base spaces are important and are considered.This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 242588615. I would like to thank Liana David for a lot of joint work on (TE)-structures
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