Anomaly formulas for the complex-valued analytic torsion on compact bordisms

We extend the complex-valued analytic torsion, introduced by Burghelea and Haller on closed manifolds, to compact Riemannian bordisms. We do so by considering a flat complex vector bundle over a compact Riemannian manifold, endowed with a fiberwise nondegenerate symmetric bilinear form. The Riemmanian metric and the bilinear form are used to define non-selfadjoint Laplacians acting on vector-valued smooth forms under absolute and relative boundary conditions. In order to define the complex-valued analytic torsion in this situation, we study spectral properties of these generalized Laplacians. Then, as main results, we obtain so-called anomaly formulas for this torsion. Our reasoning takes into account that the coefficients in the heat trace asymptotic expansion associated to the boundary value problem under consideration, are locally computable. The anomaly formulas for the complex-valued Ray–Singer torsion are derived first by using the corresponding ones for the Ray–Singer metric, obtained by Brüning and Ma on manifolds with boundary, and then an argument of analytic continuation. In odd dimensions, our anomaly formulas are in accord with the corresponding results of Su, without requiring the variations of the Riemannian metric and bilinear structures to be supported in the interior of the manifold.     

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.     

Smooth parametrized torsion: A manifold approach

We present a construction of a torsion invariant of bundles of smooth manifolds which is based on the work of Dwyer, Weiss and Williams on smooth structures on fibrations.     

Ray-Singer type theorem for the refined analytic torsion

We show that the refined analytic torsion is a holomorphic section of the determinant line bundle over the space of complex representations of the fundamental group of a closed oriented odd-dimensional manifold. Further, we calculate the ratio of the refined analytic torsion and the Farber-Turaev combinatorial torsion. As an application, we establish a formula relating the eta-invariant and the phase of the Farber-Turaev torsion, which extends a theorem of Farber and earlier results of ours. This formula allows to study the spectral flow using methods of combinatorial topology.     

On the geometry of point correspondences of three conformal spaces

Point correspondences of three conformal spaces are studied on the base of G. F. Laptev??s invariant methods. We establish the basic equations and geometrical objects of the point correspondences in question. We construct invariant normalizations of the spaces, single out the basic tensors of the correspondences, establish a connection of the correspondences with the theory of multidimensional 3-webs, and find the torsion and the curvature tensors of a point correspondence. For a series of particular cases, we prove existence theorems.     

Equivalence of higher torsion invariants

We show that the smooth torsion of bundles of manifolds constructed by Dwyer, Weiss, and Williams satisfies the axioms for higher torsion developed by Igusa. As a consequence we obtain that the smooth Dwyer–Weiss–Williams torsion is proportional to the higher torsion of Igusa and Klein.     

Reidemeister torsion and the K-theory of von Neumann algebras

We introduce a topological-type invariant for a cocompact properly discontinuous action of a discrete group on a Riemannian manifold generalizing classical notions of Reidemeister torsion. It takes values in the weak algebraic K-theory of the von Neumann algebra of . We give basic tools for its computation like sum and product formulas and calculate it in several cases. It encompasses, for instance, the Alexander polynomial and is related to analytic torsion.     

A Cheeger-M¨uller Theorem for Symmetric Bilinear Torsions

The authors establish a Cheeger-Müller type theorem for the complex valued analytic torsion introduced by Burghelea and Hailer for fiat vector bundles carrying nondegenerate symmetric bilinear forms. As a consequence, they prove the Burghelea-Haller conjecture in full generality, which gives an analytic interpretation of （the square of） the Turaev torsion.     

Asymptotic behavior of stochastic Schrödinger lattice systems driven by nonlinear noise

Abstract

We study the random dynamics of the N-dimensional stochastic Schrödinger lattice systems with locally Lipschitz diffusion terms driven by locally Lipschitz nonlinear noise. We first prove the existence and uniqueness of solutions and define a mean random dynamical system associated with the solution operators. We then establish the existence and uniqueness of weak pullback random attractors in a Bochner space. We finally prove the existence of invariant measures of the stochastic equation in the space of complex-valued square-summable sequences. The tightness of a family of probability distributions of solutions is derived by the uniform estimates on the tails of the solutions at far field.     

Maximally movable Riemannian spaces with torsion

We prove that, among all Riemannian spaces of constant curvature, only three-dimensional spaces have torsion which is invariant under the group of motions. The torsion tensor in these spaces is covariantly constant and determines the torsion form. The ratio of the integral of this form over a bounded domain to its volume is a constant determining the torsion of the space. We introduce the notions of volume torsion and scalar torsion.     

Turaev torsion invariants of 3-orbifolds

We construct a combinatorial invariant of 3-orbifolds with singular set a link that generalizes the Turaev torsion invariant of 3-manifolds. We give several gluing formulas from which we derive two consequences. The first is an understanding of how the components of the invariant change when we remove a curve from the singular set. The second is a formula relating the invariant of the 3-orbifold to the Turaev torsion invariant of the underlying 3-manifold in the case when the singular set is a nullhomologous knot.     

A Cheeger-Müller Theorem for Symmetric Bilinear Torsions

The authors establish a Cheeger-Müller type theorem for the complex valued analytic torsion introduced by Burghelea and Hailer for fiat vector bundles carrying non- degenerate symmetric bilinear forms. As a consequence, they prove the Burghelea-Haller conjecture in full generality, which gives an analytic interpretation of (the square of) the Turaev torsion.     

格序群的极小扭类,准素扭类和极扭类

本文给出了格序群的扭类是准素扭类的一个充分必要条件，证明了任一非零扭类至少包含一个极小扭类，从而将文〔１〕和〔２〕中在有限制条件情况下的有关结论推广到无任何限制条件的情形。本文还给出了一个扭类是极扭类的几个充分必要条件，并得到了格序群的极扭类、准素扭类和极小扭类之间的关系。     

Invariant manifolds for stochastic wave equations

In this paper, we consider a class of stochastic wave equations with nonlinear multiplicative noise. We first show that these stochastic wave equations generate random dynamical systems (or stochastic flows) by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism. Then, we establish the existence of random invariant manifolds for the random wave equations. Due to the temperedness of the nonlinearity, we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases. Based on these random dynamical systems, we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium. Finally, we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations.     

Intermediately fully invariant subgroups of abelian groups

Describing intermediately fully invariant subgroups of divisible and torsion groups, we show that the intermediately fully invariant subgroups are direct summands in a completely decomposable group whose every homogeneous component is decomposable. For torsion groups, we find out when all their fully invariant subgroups are intermediately fully invariant; and for torsion-free groups, this question comes down to the reduced case. Also, in a torsion group that is the sum of cyclic subgroups, its subgroup is shown to be intermediately inert if and only if it is commensurable with some intermediately fully invariant subgroup.     

Some compactifications of a semitopological semigroup

Let S be a semitopological semgroup and let C_b (S) denote the B^*-algebra of all continuous bounded complex-valued functions on S. In this paper, we consider a left m-introverted and translation invariant B^*-subalgebra F of C_b(S) containing the constant functions. Our concern is with ΔF, the maximal ideal space of F up to an isomorphic homeomorphism, when it is made into a compact right topological semigroup containing a dense continous homomorphic image of S under the Gelfand topology and a suitably chosen binary operation. We establish a representation of the closed left ideals of ΔF and study its centre and ideal structure in some special cases.     

紧群作用下的扭化解析挠率

在紧群中的非单位元的作用只有孤立的不动点的情形下, 我们研究了扭化de Rham 复形上的解析挠率的等变情形. 在一些条件下, 我们也研究了Z₂- 分次的椭圆复形上的离域的(delocalized) L²-解析挠率.     

On asymptotic dimension of countable Abelian groups

We compute the asymptotic dimension of the rationals given with an invariant proper metric. We also show that a countable torsion Abelian group taken with an invariant proper metric has asymptotic dimension zero.     

Torsions analytiques équivariantes en théorie de de Rham

We announce a comparison formula for two natural definitions of equivariant analytic torsion in de Rham theory. In this formula, a new invariant of equivariant fibrations with odd dimensional compact fibres appears, whose main properties are described. Our results are formally very close to corresponding results which we obtained for holomorphic torsion.     

Total torsion of curves in three-dimensional manifolds

A classical result in differential geometry assures that the total torsion of a closed spherical curve in the three-dimensional space vanishes. Besides, if a surface is such that the total torsion vanishes for all closed curves, it is part of a sphere or a plane. Here we extend these results to closed curves in three dimensional Riemannian manifolds with constant curvature. We also extend an interesting companion for the total torsion theorem, which was proved for surfaces in by L. A. Santaló, and some results involving the total torsion of lines of curvature. Dedicated to Professor Manfredo P. do Carmo on his 80th birthday.