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     

Natural maps on the iterated jet prolongation of a fibred manifold

Summary Using an analytic procedure by the first author [5, 6], we first determine all natural transformations of the second iterated jet prolongation J¹J¹Y of a fibred manifold YX into itself depending on a linear connection on the base manifold X. We obtain two 3-parameter families and we interpret them geometrically. Our results clarify the distinguished role of the involution on J¹ J¹ Y depending on introduced by the second author [11]. Then we discuss the role of our transformations in the theory of the natural operators transforming a connection on a fibred manifold YX and a linear connection on X into a connection on the first jet prolongation J¹YX of Y. In the final remark we determine all natural transformations of the second sesquiholonomic and holomonic jet prolongations of Y into themselves. Our attention to second order jet spaces is due to the role they play in fundamental geometric and physical theories (cf. curvature of connections and lagrangian theories).     

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.     

Periodic polynomial of trace maps

Let σ be an endomorphism of the free group on two generators and Φσ the trace map associated with σ. A polynomial P is said to be periodic for σ if, for some positive integer n, it is invariant under , i.e., . In this note we study the structure of the ring of periodic polynomials for σ.     

Reidemeister and Nielsen zeta-functions

In this paper we formulate a rationality theorem for the Reidemeister and Nielsen zeta-functions modulo a normal subgroup of the fundamental group. We give conditions under which these zeta-functions coincide. We formulate a conjecture aboutentropy for the Reidemeister numbers. We show that the radius of convergence of the Nielsen zeta-function for an orientation-preserving homeomorphism f of a compact surface is an invariant of a three-dimensional manifold, the torus of the map f, and a special flow on it. In special cases we derive a functional equation for the Nielsen zeta-function. We give an example of a transcendental Nielsen zeta function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 164–168, 1988.In conclusion the author expresses thanks to V. B. Piloginaya, V. G. Turaev, Boju Jiang, N. V. Ivanov for stimulating discussions and to D. Fried for sending preprints.     

Flatness and fibred powers

We prove that a complex map is flat if and only if none of its fibred powers has vertical (isolated or embedded) components. A commutative algebraic analogue of this statement follows: flatness is equivalent to torsion freeness of all tensor powers. Received: 21 August 1997 / Revised version: 6 April 1998     

In memoriam Kurt Reidemeister

Ohne Zusammenfassung     

On the application of fibred mapping spaces to exponential laws for bundles,ex-spaces and other categories of maps

A previous paper constructed exponential laws in the category Top_B of spaces over B. The present paper relates these laws to constructions known for locally trivial maps, and constructs also new exponential laws for ex-spaces, fibred section spaces and fibred relative lifting spaces. Versions of these laws for homotopy classes of maps are discussed.     

A lower bound for the number of Reidemeister moves of type III

We study the number of Reidemeister type III moves using Fox n-colorings of knot diagrams.     

On the surjectivity of some trace maps

LetK be a commutative ring with a unit element 1. Let Γ be a finite group acting onK via a mapt: Γ→Aut(K). For every subgroupH≤Γ define tr _H :K→K ^H by tr _h (x)=Σ_σ∈H σ(x). We proveTheorem: tr_Γ is surjective onto K ^Γ if and only if tr _P is surjective onto K ^P for every (cyclic) prime order subgroup P of Γ. This is false for certain non-commutative ringsK.     

Bounded G-theory with fibred control

We use filtered modules over a Noetherian ring and fibred bounded control on homomorphisms to construct a new kind of controlled algebra with applications in geometric topology. The resulting theory can be thought of as a “pushout” of bounded K-theory with fibred control and bounded G-theory constructed and used by the authors. Bounded G-theory was geared toward constructing a G-theoretic version of assembly maps and proving the Novikov injectivity conjecture for them. The G-theory with fibred control is needed in the study of surjectivity of the assembly map. The relation between the K- and G-theories is the classical one: K-theory is meaningful, however G-theory is easier to compute, and the relationship is expressed via a Cartan map. This map turns out to be an equivalence under very mild constraints in terms of metric geometry such as finite decomposition complexity. The fibred theory is certainly more complicated than the absolute theory. This paper contains the non-equivariant theory including fibred controlled excision theorems known to be crucial for computations.     

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.