A note on reidemeister torsion and period matrix of Riemann surfaces

We consider compact Riemann surfaces Σ _g with genus at least 2. We explain the relation between the Reidemeister torsion of Σ _g and its period matrix.     

Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps

For a non-compact hyperbolic 3-manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of \(\mathrm{SL }_2(\mathbb {C})\) to the corresponding Reidemeister torsion. While the analytic torsion is a spectral invariant of the manifold, the Reidemeister torsion is of combinatorial nature. Our proof rests on an expression of the analytic torsion in terms of special values of Ruelle zeta functions as well as on recent work of Pere Menal-Ferrer and Joan Porti.     

Reidemeister torsion for linear representations and Seifert surgery on knots

We study an invariant of a 3-manifold which consists of Reidemeister torsion for linear representations which pass through a finite group. We show a Dehn surgery formula on this invariant and compute that of a Seifert manifold over S². As a consequence we obtain a necessary condition for a result of Dehn surgery along a knot to be Seifert fibered, which can be applied even in a case where abelian Reidemeister torsion gives no information.     

Analytic and Reidemeister torsion for representations in finite type Hilbert modules

For a closed Riemannian manifold (M, g) we extend the definition of analytic and Reidemeister torsion associated to a unitary representation of ₁ (M) on a finite dimensional vector space to a representation on aA-Hilbert moduleW of finite type whereA is a finite von Neumann algebra. If (M,W) is of determinant class we prove, generalizing the Cheeger-Müller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, theL ₂-analytic andL ₂-Reidemeister torsions are equal.The first three authors were supported by NSF. The first two authors wish to thank the Erwin-Schrödinger-Institute, Vienna, for hospitality and support during the summer of 1993 when part of this work was done.     

An extended Cheeger-Müller theorem for covering spaces

We generalize a theorem of Bismut-Zhang, which extends the Cheeger-Müller theorem on Ray-Singer torsion and Reidemeister torsion, to the case of infinite Galois covering spaces. Our result is stated in the framework of extended cohomology, and generalizes in this case a recent result of Braverman-Carey-Farber-Mathai. It does not use the determinant class condition and thus also (potentially) generalizes several results on L²-torsions due to Burghelea, Friedlander, Kappeler and McDonald. We combine the framework developed by Braverman-Carey-Farber-Mathai on the determinant of extended cohomology with the heat kernel method developed in the original paper of Bismut-Zhang to prove our result.     

On the quantum filtration of the Khovanov-Rozansky cohomology

We prove the quantum filtration on the Khovanov-Rozansky link cohomology Hp with a general degree (n+1) monic potential polynomial p(x) is invariant under Reidemeister moves, and construct a spectral sequence converging to Hp that is invariant under Reidemeister moves, whose E₁ term is isomorphic to the Khovanov-Rozansky sl(n)-cohomology Hn. Then we define a generalization of the Rasmussen invariant, and study some of its properties. We also discuss relations between upper bounds of the self-linking number of transversal links in standard contact S³.     

Absolute torsion and eta-invariant

In a recent joint work with V. Turaev [6], we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion, it has the advantage of having a well-determined sign. Also, the absolute torsion is defined for arbitrary orientable flat vector bundles, and not only for unimodular ones, as is classical Reidemeister torsion. In this paper I show that the sign behavior of the absolute torsion, under a continuous deformation of the flat bundle, is determined by the eta-invariant and the Pontrjagin classes. This result has a twofold significance. Firstly, it justifies the definition of the absolute torsion by establishing a relation to the well-known geometric invariants of manifolds. Viewed differently, the result of this paper allows to express (partially) the eta-invariant, which is defined using analytic tools, in terms of the absolute torsion, having a purely topological definition. The result may find applications in studying the spectral flow by methods of combinatorial topology. Received January 11, 1999; in final form August 16, 1999     

Reidemeister zeta functions of low-dimensional almost-crystallographic groups are rational

We prove that the Reidemeister zeta functions of automorphisms of crystallographic groups with diagonal holonomy ?₂ are rational. As a result, we obtain that Reidemeister zeta functions of automorphisms of almost-crystallographic groups up to dimension 3 are rational.     

Étude d'une forme volume naturelle sur l'espace de représentations du groupe d'un nœud dans SU(2)

For a knot K in S³, we construct in the line of Casson – or more precisely taking into account Lin's (J. Differential Geom. 35 (1992) 337–357) and Heusener's (Topology Appl. 127 (2003) 175–197) further works – a 1-volume form on the SU(2)-representation space of the group of K and we show how to interpret this volume form as a Reidemeister torsion. In the last part of this Note, we give an explicit computation of this volume form for torus knots. To cite this article: J. Dubois, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Symmetric (co)homologies of Lie algebras

Cohomologies of Lie algebras are usually calculated using the Chevalley-Eilenberg cochain complex of skew-symmetric forms. We consider two cochain complexes consisting of forms with some symmetric properties. First, cochains C^*(L) are symmetric in the last 2 arguments, skew-symmetric in the others and satify moreover some kind of Jacobi condition in the last 3 arguments. In characteristic 0, its cohomologies are isomorphic to the cohomologies of the factor-complex C^*(L,L’)/C^*+1(L,K). Second, a symmetric version C_λ^*(A) is defined for an associative algebra A. It is a subcomplex of the cyclic cochain complex. These symmetric cochain complexes are used for the calculation of 3-cohomologies of Cartan Type Lie algebras with trivial coefficients.     

The analytic torsion of a disc

In this article, we study the Reidemeister torsion and the analytic torsion of the m dimensional disc, with the Ray and Singer homology basis (Adv Math 7:145–210, 1971). We prove that the Reidemeister torsion coincides with a power of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger–Müller theorem. We use a formula proved by Brüning and Ma (GAFA 16:767–873, 2006) that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary (Lück, J Diff Geom 37:263–322, 1993). Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the two sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang (Asian J Math 4:695–714, 2000), and we compare the results. The results of these work were announced in the study of Hartmann et al. (BUMI 2:529–533, 2009).     

Elementary divisors of induced transformations on symmetry classes of tensors

Let V_χ(G) denote the symmetry class of tensors over the vector space V associated with the permutation group G and irreducible character χ. Write v₁*v₂*...*v_m for the decomposable symmetrized product of the indicated vectors (m=degG). If T is a linear operator on V, let K(T) denote the associated operator on V_χ(G), i.e., K(T)v₁*v₂*...*v_m=Tv₁*Tv₂*...*Tv_m. Denote by D(T) the derivation operator D(T)v₁*v₂*...*v_m=Tv₁*v₂...*v_m+v₁*Tv₂*v₃* ...*v_m+...+v₁*v₂*...*v_m–1*Tv_m. The article concerns the elementary divisors of K(T) and D(T).     

w* sequential convergence

LetX be an infinite dimensional Banach space, andX* its dual space. Sequences {χ _n ^* } _n=1 ^∞ ?X* which arew* converging to 0 while inf _n ‖x* _n ‖>0, are constructed.     

Universal inequalities in Ehrhart theory

In this paper, we show the existence of universal inequalities for the h*-vector of a lattice polytope P, that is, we show that there are relations among the coefficients of the h*-polynomial that are independent of both the dimension and the degree of P. More precisely, we prove that the coefficients h* ₁ and h* ₂ of the h*-vector (h* ₀, h* ₁,..., h* _d) of a lattice polytope of any degree satisfy Scott’s inequality if h* ₃ = 0.     

Additivity for the parametrized topological Euler characteristic and Reidemeister torsion

Dwyer, Weiss, and Williams have recently defined the notions of the parametrized topological Euler characteristic and the parametrized topological Reidemeister torsion which are invariants of bundles of compact topological manifolds. We show that these invariants satisfy additivity formulas paralleling the additive properties of the classical Euler characteristic and the Reidemeister torsion of CW-complexes.     

The finiteness of the Reidemeister number of morphisms between almost-crystallographic groups

We give a practical criterion to determine whether a given pair of morphisms between almost-crystallographic groups has a finite Reidemeister coincidence number. As an application, we determine all two- and three-dimensional almost-crystallographic groups that have the R _∞ property. We also show that for a pair of continuous maps between oriented infra-nilmanifolds of equal dimension, the Nielsen coincidence number equals the Reidemeister coincidence number when the latter is finite.     

Asymptotic centers and best compact approximation of operators intoC(K)

The existence of best compact approximations for all bounded linear operators fromX intoC(K) is related to the behavior of asymptotic centers inX ^*. IfK is just one convergent sequence, the condition is that everyω ^*-convergent sequence inX ^* will have an asymptotic center. We first study this property, solving some open problems in the theory of asymptotic centers. IfK is more “complex,” the asymptotic centers should behave “continuously.” We use this observation to construct operators fromC[0,1] intoC(ω ²) and from ?₁ intoL ₁ without best compact approximation. We also construct spacesX ₁,X ₂, isomorphic to a Hilbert space, and operatorsT ₁,∶X ₁→C(ω ²),T ₂∶?₁→X ₂ without best compact approximations.