A base-point-free definition of the Lefschetz invariant
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In classical Lefschetz-Nielsen theory, one defines the Lefschetz invariant L( f ) of an endomorphism f of a manifold M. The definition depends on the fundamental group of M, and hence on choosing a base point ∗ ∈ M and a base path from ∗ to f (∗). At times, it is inconvenient or impossible to make these choices. In this paper, we use the fundamental groupoid to define a base-point-free version of the Lefschetz invariant. Copyright © 2006 Vesta Coufal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In classical Lefschetz fixed point theory [3], one considers an endomorphism f : M → M of a compact, connected polyhedron M. Lefschetz used an elementary trace construction to define the Lefschetz invariant L( f ) ∈ Z. The Hopf-Lefschetz theorem states that if L( f ) = 0, then every map homotopic to f has a fixed point. The converse is false. However, a converse can be achieved by strengthening the invariant. To begin, one chooses a base point ∗ of M and a base path τ from ∗ to f (∗). Then, using the fundamental group and an advanced trace construction one defines a Lefschetz-Nielsen invariant L( f , ∗,τ), which is an element of a zero-dimensional Hochschild homology group [4]. Wecken proved that when M is a compact manifold of dimension n > 2, L( f , ∗,τ) = 0 if and only if f is homotopic to a map with no fixed points. We wish to extend Lefschetz-Nielsen theory to a family of manifolds and endomorphisms, that is, a smooth fiber bundle p : E → B together with a map f : E → E such that p = p ◦ f . One problem with extending the definitions comes from choosing base points in the fibers, that is, a section s of p, and the fact that f is not necessarily fiber homotopic to a map which fixes the base points (as is the case for a single path connected space and a single endomorphism.) To avoid this difficulty, we reformulate the classical definitions of the Lefschetz-Nielsen invariant by employing a trace construction over the fundamental groupoid, rather than the fundamental group. Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 34143, Pages 1–20 DOI 10.1155/FPTA/2006/34143
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A base-point-free definition of the Lefschetz invariant
In Section 2, we describe the classical (strengthened) Lefschetz-Nielsen invariant following the treatment given by Geoghegan [4] (see also Jiang [6], Brown [3] and L¨uck [8]). We also introduce the Hattori-Stallings trace, which will replace the usual trace in the construction of the algebraic invariant. In Section 3, we develop the background necessary to explain our base-point-free definitions. This includes the general theory of groupoids and modules over ringoids, as well as our version of the Hattori-Stallings trace. In Section 4, we present our base-point-free definitions of the Lefschetz-Nielsen invariant, and show that they are equivalent to the classical definitions
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