The Reidemeister zeta function with applications to Nielsen theory and a connection with Reidemeister torsion

In this paper we study, the Reidemeister zeta function. We prove rationality and functional equations of the Reidemeister zeta function of an endomorphism of finite group. We also obtain these results for eventually commutative endomorphisms. These results are applied to the theory of Reidemeister and Nielsen numbers of self-maps of topological spaces. Our method is to identify the Reidemeister number of a group endomorphism with the number of fixed points in the unitary dual. As a consequence, we show that the Reidemeister torsion of the mapping torus of the unitary dual is a special value of the Reidemeister zeta function. We also prove certain congruences for Reidemeister numbers which are equivalent to a Euler product formula for the Reidemeister zeta function. The congruences are the same as those found by Dold for Lefschetz numbers.     

A Reidemeister trace for fibred maps

A Reidemeister trace for fibred maps is defined as the alternating sum of suitable (elementary) traces for linear morphisms of fibred cellular free modules with local coefficients. This invariant extends in a natural way the classical construction of the generalized Lefschetz number??Reidemeister trace??to the category of fibred CW-complexes.     

Difference of Composition Operators on Some Analytic Function Spaces

For two analytic self-maps φ and ψ defined on the unit disk D, we characterize completely the boundedness and compactness of the difference C_φ-C_ψ of the composition operators C_φ and C_ψ from Bloch space B into Besov space B_ν~∞. Moreover, we also give a complete characterization of the compactness of the difference C_φ-C_ψ on BMOA space.     

Axioms for a local Reidemeister trace in fixed point and coincidence theory on differentiable manifolds

We give axioms which characterize the local Reidemeister trace for orientable differentiable manifolds. The local Reidemeister trace in fixed point theory is already known, and we provide both uniqueness and existence results for the local Reidemeister trace in coincidence theory.     

Limit sets for maps of the circle

A trick which reduces some questions concerning the dynamics of self-maps of the circle to similar questions about self-maps of the interval is suggested and applied to answer two questions of Block and Coppel.     

Images of simple lattice polynomials

Although lattice polynomials, built from the two binary lattice operations and involving constants of the lattice, are mechanical devices to produce isotone self-maps, there is no order-theoretical property common to all lattice polynomial images. This contrasts with the current fact that little is known about isotone self-maps whose images are not, themselves, lattices. It also shuts out an obvious approach to the conjecture that every order polynomial complete lattice is finite.Presented by R. Freese.     

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.     

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.     

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.     

Minimal periods of holomorphic maps on complex tori

We study the set of minimal periods of holomorphic self-maps of one- and two-dimensional complex tori. In particular, we characterize when the set of minimal periods of such maps is finite. In fact, we have an algorithm for doing this characterization for holomorphic self-maps of an arbitrary dimensional complex torus.     

Twisted burnside theory for the discrete Heisenberg group and for wreath products of some groups

For each positive integer N, an automorphism with the Reidemeister number 2N of the discrete Heisenberg group is constructed; an example of determination of points in the unitary dual object being fixed with respect to the mapping induced by the group automorphism is given. For wreath products of finitely generated Abelian groups and the group of integers, it is proved that if the Reidemeister number of an arbitrary automorphism is finite, then it is equal to the number of fixed points of the induced mapping on a finite-dimensional part of the unitary dual object.     

Axioms for the equivariant Lefschetz number and for the Reidemeister trace

We characterize the Reidemeister trace, the equivariant Lefschetz number and the equivariant Reidemeister trace in terms of certain axioms. Dedicated to Albrecht Dold and Edward Fadell     

Unknotting number and number of Reidemeister moves needed for unlinking

Using unknotting number, we introduce a link diagram invariant of type given in Hass and Nowik (2008) [4], which changes at most by 2 under a Reidemeister move. We show that a certain infinite sequence of diagrams of the trivial two-component link need quadratic number of Reidemeister moves for being splitted with respect to the number of crossings.     

Equivariant Lefschetz maps for simplicial complexes and smooth manifolds

Let X be a locally compact space with a continuous proper action of a locally compact group G. Assuming that X satisfies a certain kind of duality in equivariant bivariant Kasparov theory, we can enrich the classical construction of Lefschetz numbers for self-maps to an equivariant K-homology class. We compute the Lefschetz invariants for self-maps of finite-dimensional simplicial complexes and smooth manifolds. The resulting invariants are independent of the extra structure used to compute them. Since smooth manifolds can be triangulated, we get two formulas for the same Lefschetz invariant in this case. The resulting identity is closely related to the equivariant Lefschetz Fixed Point Theorem of Lück and Rosenberg.     

Cohomogeneity One Manifolds and Self-Maps of Nontrivial Degree

We construct natural self-maps of compact cohomogeneity one manifolds and compute their degrees and Lefschetz numbers. On manifolds with simple cohomology rings this yields relations between the order of the Weyl group and the Euler characteristic of a principal orbit. As examples we determine all cohomogeneity one actions on irreducible Riemannian symmetric spaces of compact type that lead to self-maps of degree ≠ −1; 0; 1. We derive explicit formulas for new coordinate polynomial self-maps of the compact matrix groups SU(3), SU(4), and SO(2n). For SU(3) we determine precisely which integers can be realized as degrees of self-maps. Supported by a DFG Heisenberg scholarship and DFG priority program SPP 1154.     

A distance for diagrams of a knot

We introduce a distance for diagrams of an oriented knot by using Reidemeister moves linking the diagrams and we give evaluations of the distance. Furthermore, we apply the distance to construct a knot invariant.     

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.     

Boundary Distortion Estimates for Holomorphic Maps

We establish some estimates of the angular derivatives from below for holomorphic self-maps of the unit disk ${\mathbb {D}}$ at one and two fixed points of the unit circle provided there is no fixed point inside ${\mathbb {D}}$ . The results complement Cowen–Pommerenke and Anderson–Vasil’ev type estimates in the case of univalent functions. We use the method of extremal length and a semigroup approach to deriving inequalities for holomorphic self-maps of the disk which are not necessarily univalent using known inequalities for univalent functions. This approach allowed us to obtain a new Ossermans type estimate as well as inequalities for holomorphic self-maps which images do not separate the origin and the boundary.     

Twisted Conjugacy Classes for Polyfree Groups

Let G be a finitely generated polyfree group. If G has nonzero Euler characteristic then we show that Aut(G) has a finite index subgroup in which every automorphism has infinite Reidemeister number. For certain G of length 2, we show that the number of Reidemeister classes of every automorphism is infinite.