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     

A constructive proof of Ky Fan's generalization of Tucker's lemma

We present a proof of Ky Fan's combinatorial lemma on labellings of triangulated spheres that differs from earlier proofs in that it is constructive. We slightly generalize the hypotheses of Fan's lemma to allow for triangulations of Sⁿ that contain a flag of hemispheres. As a consequence, we can obtain a constructive proof of Tucker's lemma that holds for a more general class of triangulations than the usual version.     

Obstruction theory and coincidences of maps between nilmanifolds

Let f, g : M N be two maps between two compact nilmanifolds with dim M dim N = n. In this paper, we show that either the Nielsen coincidence number N(f, g) = 0 or N(f, g) = R(f, g) where R(f, g) denotes the Reidemeister number of f and g. Furthermore, we show that if N(f, g) > 0 then the primary obstruction o_n(f, g) to deforming f and g to be coincidence free on the n-th skeleton of M is non-trivial.Received: 30 April 2004; revised: 20 July 2004     

Euler’s fixed point theorem: The axis of a rotation

We give an elementary proof of what is perhaps the earliest fixed point theorem; namely Leonhard Euler’s theorem of 1775 on the existence of an axis v for any three-dimensional rotation R. The proof is constructive and shows that no multiplications are required to compute v. Dedicated to the memory of Leonhard Euler, “The Master of us all”, on the occasion of the 300th anniversary of his birth     

A simple proof of Bernstein-Lunts equivalence

We give a new proof of the Bernstein-Lunts equivalence of ordinary and equivariant derived categories of Harish-Chandra modules. This proof requires no boundedness assumptions. It uses K-projective resolutions of equivariant complexes, which are shown to exist and constructed explicitly when the group under consideration is reductive. The general case can be obtained from the reductive case by techniques of [MP].     

On a property of functions on the sphere and its application

Given a continuous function f:S^m+n−2→R^m, and n points u₁,u₂,…,u_n∈S^m+n−2; does there exist a rotation r∈SO(m+n−1) such that f(ru₁)=f(ru₂)=?=f(ru_n)? In this paper, we study the property of a continuous map from a sphere to a Euclidean space by using the theory of Smith periodic transformation and Brouwer degree of map theorem. The conjecture is proved under the case of n=2 and m being even. Furthermore, this conjecture is proved for the case when u_j⋅u_j+1=λ and the dimension of the sphere is not less than m+n−2. This paper generalizes the Borsuk-Ulam theorem and then presents its application.     

On fixed points of stratified maps

Nielsen fixed point theory deals with the fixed point sets of self maps on compact polyhedra. In this note, we shall extend it to stratified maps, to consider fixed points on (noncompact) strata. The extension was motivated by our recent work on the braid forcing problem in which the deleted symmetric products are indispensable. The stratified viewpoint is theoretically as natural as the equivariant Nielsen fixed point theory, while it can be more tractable computationally and more flexible in applications. This work was partially supported by an NSFC grant and a BMEC grant.     

Is topologically simple simple?

In this note we determine when topological simplicity of a subsemigroup implies its simplicity. A measure theoretic approach is used. The main result is: In a locally compact completely simple semigroup whose maximal subgroups are compact, every subsemigroup which is either locally compact or of non-empty interior is completely simple. As a corollary, we have: In a compact semigroup, any topologically simple subsemigroup which is either locally compact or of non-empty interior is simple.     

A fixed point theorem for a class of Sobolev maps

We prove an analog of the Brouwer fixed point theorem for a map whose differential and adjoint are integrable with exponents n−1 and n/(n−1) respectively. Here Ω is a convex bounded open subset of Rⁿ.

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The least number, of n-periodic points of a self-map of a solvmanifold, can be realised by a smooth map

Shub and Sullivan noticed that there is a large gap between the least number of periodic points, taken over the family of continuous maps homotopic to a given self-map f and the analogous number taken over the family of smooth maps homotopic to f. In this paper we show that in the case of self-maps of solvmanifolds the minimal numbers of n-periodic points, in both categories, are equal.     

Computation of Nielsen numbers for maps of compact surfaces with boundary

In this paper we study the Nielsen number of a self-map f:M→M of a compact connected surface with boundary. Let G=π₁(M) be the fundamental group of M which is a finitely generated free group. We introduce a new algebraic condition called “bounded solution length” on the induced endomorphism φ:G→G of f and show that many maps which have no remnant satisfy this condition. For a map f that has bounded solution length, we describe an algorithm for computing the Nielsen number N(f).     

A very simple proof of the eta transformation formula

We give a very simple proof of a classical transformation formula for the Dedekind eta function. This proof is a simplified version of an approach suggested by H. Petersson.     

A generalized Banach contraction principle that characterizes metric completeness

We prove a fixed point theorem that is a very simple generalization of the Banach contraction principle and characterizes the metric completeness of the underlying space. We also discuss the Meir-Keeler fixed point theorem.