Twisted Cohomology Groups Research Articles

Topological theory for Selberg type integral associated with rigid Fuchsian systems

Using the analytic realization of middle convolution due to Dettweiler and Reiter, we show that any rigid Fuchsian system can be obtained as a subsystem of some generating system which has an integral representation of solutions of Selberg type. Twisted homology groups and twisted cohomology groups associated with such integrals are studied. In particular, contiguity relations and twisted cycles which realize local exponents are obtained.

Tautness for riemannian foliations on non-compact manifolds

For a riemannian foliation \({\mathcal{F}}\) on a closed manifold M, it is known that \({\mathcal{F}}\) is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form \(\kappa_\mu\) (relatively to a suitable riemannian metric μ) is zero (cf. Alvarez in Ann Global Anal Geom 10:179–194, 1992). In the transversally orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group \(H^{n}\,(M\,/\,{\mathcal{F}})\) , where \(n = {\rm codim}\,{\mathcal{F}}\) (cf. Masa in Comment Math Helv 67:17–27, 1992). By the Poincare Duality (cf. Kamber et and Tondeur in Asterisque 18:458–471, 1984) this last condition is equivalent to the non-vanishing of the basic twisted cohomology group \(H^{0}_{\kappa_\mu}(M\,/\,{\mathcal{F}})\) , when M is oriented. When M is not compact, the tautness class is not even defined in general. In this work, we recover the previous study and results for a particular case of riemannian foliations on non compact manifolds: the regular part of a singular riemannian foliation on a compact manifold (CERF).

Twisted homology and cohomology groups associated to the Wirtinger integral

The first half of this paper (Section 1) deals with the structure of the twisted homology group associated to the Wirtinger integral. A basis of the first homology group is given, and the vanishing of the homology groups of the other dimensions is proved. The second half (Section 2) deals with the structure of the twisted cohomology groups associated to the Wirtinger integral. The isomorphism between the twisted cohomology groups and the cohomology groups associated to a certain subcomplex of the de Rham complex is established, and a basis of the first cohomology group of this subcomplex (therefore, of the first twisted cohomology group) is given. The vanishing of the cohomology groups of the other dimensions is also proved.

Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups

Etant donné un noeud K dans la sphère tridimensionnelle et une représentation régulière de son groupe G K dans SU(2), on construit une forme torsion de Reidemeister non abélienne sur le premier groupe de cohomologie tordue de l’extérieur de K. Cette forme torsion de Reidemeister permet de définir une forme volume sur l’espace de représentations de G k dans SU(2). D’un autre point de vue, en s’inspirant de la construction originale de l’invariant de Casson, on construit une forme volume naturelle sur l’espace de représentations de G K dans SU(2). On établit enfin que ces deux points de vue en apparence distincts produisent en fait le même invariant topologique de noeuds.

Discrete Torsion and Twisted Orbifold Cohomology

One of the remarkable insights of orbifold string theory is an indication of the existence of a new cohomology theory of orbifolds containing the so-called twisted sectors as the contribution of singularities. Mathematically, such an orbifold cohomology theory has been constructed by Chen-Ruan [CR]. The author believes that there is a stringy geometry and topology of orbifolds whose core is orbifold cohomology. One aspect of this new geometry and topology is the twisted orbifold cohomology and its relation to discrete torsion. Again, the twisting process has its roots in physics. Physicists usually work over a global quotient X = Y/G only, where G is a finite group acting smoothly on Y. A discrete torsion is a cohomology class α ∈ H2(G, U(1)). Physically, a discrete torsion counts the freedom to choose a phase factor to weight the path integral over each twisted sector without destroying the consistency of string theory. For each α, Vafa-Witten [VW] constructed the twisted orbifold cohomology group H*orb,α(X/G,ℂ).

On the Instanton Complex of Holomorphic Morse Theory

Consider a holomorphic torus action on vector bundles over a complex manifold which lifts to a holomorphic vector bundle. When the connected components of the fixed-point set are partially ordered, we construct, using sheaf-theoretical techniques, two spectral sequences that converges to the twisted Dolbeault cohomology groups and those with compact support, respectively. These spectral sequences are the holomorphic counterparts of the instanton complex in standard Morse theory. The results proved imply holomorphic Morse inequalities and fixed-point formulas on a possibly non-compact manifold. Finally, a number of examples and applications are given.

Twisted quandle homology theory and cocycle knot invariants

The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to group extension theories for group cocycles. Explicit formulas for cocycles corresponding to extensions are given, and used to prove non-triviality of cohomology groups for some quandles. The corresponding generalization of the quandle cocycle knot invariants is given, by using the Alexander numbering of regions in the definition of state-sums. The invariants are used to derive information on twisted cohomology groups.

Semi-localization of a one pointed Kan complex

First we give a course in semi-simplicial homotopy theory. Then we construct the non-simple Postnikov tower for a one pointed Kan complex. This is done by means of twisted coefficient transgression theory. Then we develope a twisted coefficient obstruction theory for non-simple complexes. Finally we extend the notion of P-localization to an arbitrary one pointed Kan complex, where P is a collection of prime integers. We show that if X is a one pointed Kan complex there exists a map f: X $\to$ X$\sb{p}$ of one pointed Kan complexes such that $f\sb*$: $\pi\sb1$(X) $\to$ $\pi\sb1$(X$\sb{p}$) is an isomorphism, and $f\sb{k}$: $\pi\sb{k}$(X) $\to$ $\pi\sb{k}$(X$\sb{p}$) is P-localization for all k $>$ 1. The Appendix contains a theorem computing the twisted cohomology groups for the models L$\sb\pi$(G, n).