Complex-valued Ray-Singer torsion

In the spirit of Ray and Singer we define a complex-valued analytic torsion using non-selfadjoint Laplacians. We establish an anomaly formula which permits to turn this into a topological invariant. Conjecturally this analytically defined invariant computes the complex-valued Reidemeister torsion, including its phase. We establish this conjecture in some non-trivial situations.     

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.     

A refinement of the Ray–Singer torsion

We propose a refinement of the Ray–Singer torsion, which can be viewed as an analytic counterpart of the refined combinatorial torsion introduced by Turaev. Given a closed, oriented manifold of odd dimension with fundamental group Γ, the refined torsion is a complex valued, holomorphic function defined for representations of Γ which are close to the space of unitary representations. When the representation is unitary the absolute value of the refined torsion is equal to the Ray–Singer torsion, while its phase is determined by the η-invariant. As an application we extend and improve a result of Farber about the relationship between the absolute torsion of Farber–Turaev and the η-invariant. To cite this article: M. Braverman, T. Kappeler, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

The gluing formula of the refined analytic torsion for an acyclic Hermitian connection

In the previous study by Huang and Lee (arXiv:1004.1753) we introduced the well-posed boundary conditions ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ and ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the gluing formula of the refined analytic torsion for an acyclic Hermitian connection with respect to the boundary conditions ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ and ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ . In this case the refined analytic torsion consists of the Ray-Singer analytic torsion, the eta invariant and the values of the zeta functions at zero. We first compare the Ray-Singer analytic torsion and eta invariant subject to the boundary condition ${{\mathcal P}_{-, {\mathcal L}_{0}}}$ or ${{\mathcal P}_{+, {\mathcal L}_{1}}}$ with the Ray-Singer analytic torsion subject to the relative (or absolute) boundary condition and eta invariant subject to the APS boundary condition on a compact manifold with boundary. Using these results together with the well known gluing formula of the Ray-Singer analytic torsion subject to the relative and absolute boundary conditions and eta invariant subject to the APS boundary condition, we obtain the main result.     

A Cheeger–Müller theorem for Cappell–Miller torsions on manifolds with boundary

In this paper, we extend the Cappell–Miller analytic torsion to manifolds with boundary under the absolute and relative boundary conditions and using the techniques of Brüning-Ma and Su-Zhang, we get the anomaly formula of it for odd dimensional manifolds. Then by the methods of Brüning-Ma, Cappell–Miller and Su-Zhang, we get the Cheeger–Müller theorem for the Cappell–Miller analytic torsion on odd dimensional manifolds with boundary up to a sign. As a consequence of the main theorem, we get the gluing formula for the Cappell–Miller analytic torsion which generalizes a theorem of Huang.     

Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps

In this paper we derive a relationship of the leading coefficient of the Laurent expansion of the Ruelle zeta function at s=0 and the analytic torsion for hyperbolic manifolds with cusps. Here, the analytic torsion is defined by a certain regularized trace following Melrose [R.B. Melrose, The Atiyah-Patodi-Singer Index Theorem, Res. Notes Math., vol. 4, A.K. Peters, Ltd., Wellesley, MA, 1993]. This extends the result of Fried, which was proved for the compact case in [D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math. 84 (3) (1986) 523-540], to a noncompact case.     

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.     

Analytic torsion of Hirzebruch surfaces

Using different forms of the arithmetic Riemann-Roch theorem and the computations of Bott-Chern secondary classes, we compute the analytic torsion and the height of Hirzebruch surfaces.     

The metric anomaly of analytic torsion at the boundary of an even dimensional cone

The formula for analytic torsion of a cone in even dimensions is composed of three terms. The first two terms are well understood and given by an algebraic combination of the Betti numbers and the analytic torsion of the cone base. The third “singular” contribution is an intricate spectral invariant of the cone base. We identify the third term as the metric anomaly of the analytic torsion coming from the non-product structure of the cone at its regular boundary. Hereby we filter out the actual contribution of the concial singularity and identify the analytic torsion of an even-dimensional cone purely in terms of the Betti numbers and the analytic torsion of the cone base.     

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.     

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.     

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     

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.     

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.     

An explicit formula for the Webster torsion of a pseudo-hermitian manifold and its application to torsion-free hypersurfaces Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

This paper gives an explicit formula for calculating the Webster pseudo torsion for a strictly pseudoconvex pseudo-hermitian hypersurface. As applications, we are able to classify some pseudo torsion-free hypersurfaces, which include real ellipsoids.     

The analytic torsion of a cone over a sphere

We compute the analytic torsion of a cone over a sphere of dimensions 1, 2, and 3, and we conjecture a general formula for the cone over an odd dimensional sphere.     

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.     

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.     

An explicit formula for the Webster torsion of a pseudo-hermitian manifold and its application to torsion-free hypersurfaces

This paper gives an explicit formula for calculating the Webster pseudo torsion for a strictly pseudoconvex pseudo-hermitian hypersurface. As applications, we are able to classify some pseudo torsion-free hypersurfaces, which include real ellipsoids. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Scattering matrix and analytic torsion

We consider a compact manifold with a piece isometric to a (finite-length) cylinder. By making the length of the cylinder tend to infinity, we obtain an asymptotic gluing formula for the zeta determinant of the Hodge Laplacian and an asymptotic expansion of the $L^2$ torsion of the corresponding Mayer–Vietoris exact sequence. As an application, we give a purely analytic proof of the gluing formula for analytic torsion.