Stefan Haller

Complex valued Ray–Singer torsion II

In this paper we extend Witten–Helffer–Sjostrand theory from selfadjoint Laplacians based on fiber wise Hermitian metrics to nonselfadjoint Laplacians based on fiber wise non-degenerate symmetric bilinear forms. As an application we show that results about the asymptotics of the Ray–Singer torsion of self-adjoint Witten deformation, as well as the strategy proposed by Burghelea–Friedlander–Kappeler to derive the comparison of Ray–Singer and Reidemeister torsion, can be extended to nonself-adjoint Witten deformation. This is then used to conclude the equality of complex analytic and Milnor–Turaev torsion, at least for odd dimensional manifolds, up to sign (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On the Group of Diffeomorphisms Preserving a Locally Conformal Symplectic Structure

The automorphism group of a locally conformal symplectic structure is studied. It is shown that this group possesses essential features of the symplectomorphism group. By using a special type of cohomology the flux and Calabi homomorphisms are introduced. The main theorem states that the kernels of these homomorphisms are simple groups (for the precise statement, see Section 7). Some of the methods used may also be interesting in the symplectic case.

Torsion, as a Function on the Space of Representations

Riemannian Geometry, Topology and Dynamics permit to introduce partially defined holomorphic functions on the variety of representations of the fundamental group of a manifold. The functions we consider are the complex-valued Ray-Singer torsion, the Milnor-Turaev torsion, and the dynamical torsion. They are associated essentially to a closed smooth manifold equipped with a (co)Euler structure and a Riemannian metric in the first case, a smooth triangulation in the second case, and a smooth flow of type described in Section 2 in the third case. In this paper we define these functions, describe some of their properties and calculate them in some case. We conjecture that they are essentially equal and have analytic continuation to rational functions on the variety of representations. We discuss what we know to be true. As particular cases of our torsion, we recognize familiar rational functions in topology such as the Lefschetz zeta function of a diffeomorphism, the dynamical zeta function of closed trajectories, and the Alexander polynomial of a knot. A numerical invariant derived from Ray-Singer torsion and associated to two homotopic acyclic representations is discussed in the last section.

Euler structures, the variety of representations and the Milnor-Turaev torsion

e characteristic, to arbitrary manifolds. We use the Poincar´ e dual concept, co-Euler structures, to remove all geometric ambiguities from the Ray‐Singer torsion by providing a slightly modified object which is a topological invariant. We show that when the co-Euler structure is integral then the modified Ray‐Singer torsion when regarded as a function on the variety of generically acyclic complex representations of the fundamental group of the manifold is the absolute value of a rational function which we call in this paper the Milnor‐Turaev torsion. 57R20; 58J52

Reduction for locally conformal symplectic manifolds

Abstract It is shown how one can do symplectic reduction for locally conformal symplectic manifolds, especially with an action of a Lie group. This generalizes well-known procedures for symplectic manifolds to the slightly larger class of locally conformal symplectic manifolds. The whole setting is very conformally invariant.

Dynamics, Laplace transform and spectral geometry

In this paper, we consider vector fields on a closed manifold whose instantons and closed trajectories can be ‘counted’. Vector fields which admit Lyapunov closed one forms belong to this class. We show that under an additional hypothesis, ‘the exponential growth property’, the counting functions of instantons and closed trajectories have Laplace transforms which can be related to the topology and the geometry of the underlying manifold. The purpose of this paper is to introduce and explore the concept ‘exponential growth property’, and to describe these Laplace transforms.

Topology of angle valued maps, bar codes and Jordan blocks

In this paper one presents a collection of results about the “bar codes” and “Jordan blocks” introduced in Burghelea and Dey (Discret Comput Geom 50: 69–98 2013) as computer friendly invariants of a tame angle-valued map and one relates these invariants to the Betti numbers, Novikov–Betti numbers and the monodromy of the underlying space and map. Among others, one organizes the bar codes as two configurations of points in

Totally geodesic subgroups of diffeomorphisms

\mathbb C{\setminus } 0