arXiv research

We compare the higher analytic torsion of Bismut and Lott of a fibre bundle p: M -> B equipped with a flat vector bundle F -> M and a fibre-wise Morse function h on M with a higher torsion T that is constructed in terms of a families Thom-Smale complex associated to h and F, thereby extending previous joint work with B…

By explicitly comparing constructions, we prove that the higher torsion invariants of smooth bundles defined by Igusa and Klein via Morse theory agree with the higher torsion invariants defined by Badzioch, Dorabiala, Dwyer, Weiss, and Williams using homotopy theoretical methods.

The paper compares two torsion invariants in complex vector bundles.

In this paper we extend Badzioch's, Dorabiala's, and Williams' definition of cohomological higher smooth torsion to a twisted cohomological higher torsion invariant. Additionally, we show that this still satisfies geometric additivity and transfer, and will also satisfy additivity and transfer for coefficients.

This study examines torsion homology in Oeljeklaus-Toma manifolds, extending knot theory concepts.

This paper attempts to investigate the space of various characteristic classes for smooth manifold bundles with local system on the total space inducing a finite holonomy covering. These classes are known as twisted higher torsion classes. We will give a system of axioms that we require these cohomology classes to sati…

We show that J. Lott's equivariant higher analytic torsion for compact group actions depends only on the equivariant Euler characteristic.

We construct a higher Whitehead torsion map, using algebraic K-theory of spaces, and show that it satisfies the usual properties of the classical Whitehead torsion. This is used to describe a "geometric assembly map" defined on stabilized structure spaces in purely homotopy theoretic terms.

We determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible SL(2;C)-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We al…

We study the asymptotics of the higher dimensional Reidemeister torsion for torus knot exteriors, which is related to the results by W. Müller and P. Menal-Ferrer and J. Porti on the asymptotics of the Reidemeister torsion and the hyperbolic volumes for hyperbolic 3-manifolds. We show that the sequence of log |the high…

Let p: M -> B be a family of compact manifolds equipped with a unitarily flat vector bundle F -> M. We generalize Igusa's higher Franz-Reidemeister torsion τ(M/B;F) to the case that the fibre-wise cohomology H^*(M/B;F) -> B carries a parallel metric. If moreover M admits a fibre-wise Morse function, we compute the diff…

Extends Gauduchon's result to higher dimensions, showing balanced metrics.

In this paper we show that the Ray-Singer complex analytic torsion is trivial for even dimensional Calabi-Yau manifolds. Then we define the quaternionic analytic torsion for quaternionic manifolds and prove that they are metric independent. In dimension four, the quaternionic analytic torsion equals to the self-dual an…

We use higher parallel transport -- more precisely, the integration A_{infty}-functor constructed by Block-Smith and Arias Abad-Schaetz -- to define Reidemeister torsion for flat superconnections. We hope that the combinatorial Reidemeister torsion coincides with the analytic torsion defined by Mathai and Wu, thus perm…

We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments.

Motivated by the description of $\mathcal{N}=1$ M-theory compactifications to four-dimensions given by Exceptional Generalized Geometry, we propose a way to geometrize the M-theory fluxes by appropriately relating the compactification space to a higher-dimensional manifold equipped with a torsion-free structure. As a n…

Paper compares higher torsions and removes fiberwise Morse function assumption.

We show that the absolute value at zero of the Ruelle zeta function defined by the geodesic flow coincides with the higher-dimensional Reidemeister torsion for the unit tangent bundle over a 2-dimensional hyperbolic orbifold and a non-unitary representation of the fundamental group. Our proof is based on the integral e…

Using the higher analytic torsion form of Bismut and Lott we construct a characteristic class for smooth sphere bundles. We calculate this class in the case where the sphere bundle comes from a complex vector bundle. Related to these characteristic classes we define nontrivial continuous group cohomology classes of the…

We construct examples in any odd dimension of contact manifolds with finite and non-zero algebraic torsion (in the sense of Latschev-Wendl), which are therefore tight and do not admit strong symplectic fillings. We prove that Giroux torsion implies algebraic $1$-torsion in any odd dimension, which proves a conjecture b…

The abstract aims to generalize classical curve concepts to uniquely define complex curves.

Torsions, curvatures, structure equations and Bianchi identities for locally anisotropic superspaces (containing as particular cases different supersymmetric extensions and prolongations of Riemann, Finsler, Lagrange and Kaluza--Klein spaces) are investigated.

We introduce non-acyclic $PGL_n(\mathbb{C})$-torsion of a 3-manifold with toroidal boundary as an extension of J. Porti's $PGL_2(\mathbb{C})$-torsion, and present an explicit formula of the $PGL_n(\mathbb{C})$-torsion of a mapping torus for a surface with punctures, by using the higher Teichmüler theory due to V. Fock …

Derivative estimates for pluriclosed flow control curvature and torsion.

We define geometric zeta functions for locally symmetric spaces as generalizations of the zeta functions of Ruelle and Selberg. As a special value at zero we obtain the Reidemeister torsion of the manifold. For hermitian spaces these zeta functions have as special value the quotient of the holomorphic torsion of Ray an…

Study fractional structures on bundle gerbe modules using rational homotopy theory.

Given a circle-valued Morse function of a closed oriented manifold, we prove that Reidemeister torsion over a non-commutative formal Laurent polynomial ring equals the product of a certain non-commutative Lefschetz-type zeta function and the algebraic torsion of the Novikov complex over the ring. This paper gives a gen…

Recently twisted and higher order Alexander polynomials were used by Cochran, Harvey, Friedl--Kim and Turaev to give lower bounds on the Thurston norm. We first show how Reidemeister torsion relates to these Alexander polynomials. We then give lower bounds on the Thurston norm in terms of the Reidemeister torsion which…

New theory allows simultaneous block-diagonalization of commuting operator fields.

Develops method to compute Chern-Simons potentials from higher-dimensional Pontryagin densities.

Study tight contact structures on specific 3-manifolds.

New results on homology torsion growth for various groups.

We show that (under mild assumptions) the generating function of log homology torsion of a knot exterior has a meromorphic continuation to the entire complex plane. As corollaries, this gives new proofs of (a) the Silver-Williams asymptotic, (b) Fried's theorem on reconstructing the Alexander polynomial (c) Gordon's th…

Equivalence proven between two torsion invariants for flat vector bundles.

We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of righ…

Torsion invariants for manifolds which are not simply connected were introduced by K. Reidemeister and generalized to higher dimensions by W. Franz. The Reidemeister torsion, was the first invariant of manifolds which was not a homotopy invariant. The analytic counterpart of the combinatorial Reidemeister torsion was i…

Total torsion of 3D lines of curvature is an integer multiple of 2π.

We propose a new, infinite class of brackets generalizing the Frölicher--Nijenhuis bracket. This class can be reduced to a family of generalized Nijenhuis torsions recently introduced. In particular, the Haantjes bracket, the first example of our construction, is relevant in the characterization of Haantjes moduli of o…

Determines higher smooth surgery structure sets of complex projective spaces.

Improved homological dimension for certain subgroups in Lie groups.

Higher index theorem for Dirac operators on finite-volume spaces.

We consider a closed odd-dimensional oriented manifold $M$ together with an acyclic flat hermitean vector bundle $\cF$. We form the trivial fibre bundle with fibre $M$ over the manifold of all Riemannian metrics on $M$. It has a natural flat connection and a vertical Riemannian metric. The higher analytic torsion form …

In this paper we prove that for a fixed neat principal congruence subgroup of a Bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated to the m-th symmetric power of the standard representation of SL_2(C) grows exponentially in m^2. We give upper a…

The Alexander polynomial of a knot has been generalized in three different ways to give twisted invariants. The resulting invariants are usually referred to as twisted Alexander polynomials, higher-order Alexander polynomials and $L^2$-Alexander invariants of knots. We quickly recall the definitions and we summarize an…

The paper proves rigidity for cocycles from higher rank lattices to Out(FN).

Paper proves gluing formula for analytic torsions using Witten deformation for non-Morse functions.

New connections found in higher-dimensional geometries with skew-torsion.

This paper solves Yang-Baxter cohomology for cyclic biquandles.