Philip R. Heath

Nielsen-type numbers for periodic points II

Abstract This paper continues [4], and discusses the Nielsen-type number NΦ n (ƒ) of periodic points of all periods m | n of a self-map ƒ of a compact connected ANR. Various lower bounds are given for this number, and conditions are given for these lower bounds to be optimal; conditions for it to be equal to ∑ m | n NP m (ƒ) is the Nielsen-type number of periodic point classes of period exactly m defined in [4, 5]; and conditions under which it is equal to N (ƒ n ) the ordinary Nielsen number of ƒ n . Many illustrative examples are given.

Addition formulae for Nielsen numbers and for Nielsen type numbers of fibre preserving maps

Abstract In this paper we generalize well known product formulae for the Nielsen number of a fibre preserving map, to give addition formulae for such maps. We give necessary and sufficient conditions for when a naive addition formula expressing the Nielsen number of the fibre map as a simple sum of Nielsen numbers on the fibres is valid. In the second part of the paper we extend to the nonorientable situation the definition and properties of a Nielsen type number of a fibre preserving map introduced by the first author.

Coglueing homotopy equivalences

in which the front square is a pull-back. Then P is often called the fibre-product o f f and p, and it is also said that ~: P ~ X is induced by f from p. The map ~: Q~ P is determined by ~01 and q)2. Our object is to give conditions on the front and back squares which ensure that if q~l, q~2 and ~o are homotopy equivalences, then so also is ~. First of all we shall assume throughout that the back square of (1.1) as well as the front square, is a pull-back. Second recall that a map q: E --* B has the W C H P (weak covering homotopy property) if it has the covering property for all homotopies Z x I ---, B which are stationary on Z x [0, 89 This property has been shown by Dold [3] and Weinzweig [7] to be convenient for studying fiber homotopy equivalences, and our results will extend some of theirs. For the rest of this section we will assume that in (1.1) p and q have the WCHP. Then our main object is the following theorem which will be proved in Sections 2 to 5.

Coincidence theory on the complement

Abstract In this work we generalize two aspects of Nielsen fixed point theory on the complement to Nielsen coincidence theory. The first aspect concerns the location (under relative homotopies) of coincidence points. It prepares the way for equivariant coincidence theory and the for second part. A minimum theorem is forthcoming under the condition that the subspace can be by-passed. The second aspect (the study of surplus periodic points on the complement) gives a parallel (but quite different) theory when the subspace cannot be by-passed. Other features of this work include a modified fundamental group approach which simplifies the exposition. Secondly in addition to the usual Jiang condition it includes an analogue of it which ensures that the Reidemeister and Nielsen numbers are the same when the Lefschetz number is nonzero.

A Nielsen type number for fibre preserving maps

Abstract In this paper, we introduce a Nielsen type number N F (ƒ, p) for a fibre preserving map ƒ of a fibration p; we show that it is a lower bound for the least number of fixed points within the fibre homotopy class of ƒ. The number N F (ƒ, p), which can be thought of as the dual of the relative Nielsen number due to Schirmer, is often much bigger than the ordinary Nielsen number, N(ƒ), of ƒ. It shares with N(ƒ) such properties as homotopy invariance and commutativity. The definition of N F (ƒ, p) is reminiscent of the so-called naive product formula due to Brown. In this paper, we also exhibit and exploit a connection between the relative Nielsen number and N F (ƒ, p); we compare N F (ƒ, p) and N(ƒ); give necessary and sufficient conditions for N F (ƒ, p) and N(ƒ) to coincide, and show, under fairly mild conditions, that our lower bound is sharp. Some corollaries concerning minimum fixed point sets for ordinary Nielsen numbers of a fibre map are given.

Groupoids and the Mayer-Vietoris sequence

141. The object of this paper, which goes back to an earlier version by the first author (1972), is to generalize the sequence (1.1) al&d the operations to a situation of a Mayer-Vietoris type. The prototype of such a sequence in a non-abelian situation was given somewhat obscurely in

Nielsen type numbers for periodic points on nonconnected spaces

Abstract Two homotopy invariant Nielsen type numbers exist for periodic points of a self-map ƒ: X → X of a compact connected ANR X, namely NP n (ƒ) , the Nielsen type number of period n, and NΦ n (ƒ) , the Nielsen type number of the nth iterate. The first is a lower bound for the number of periodic points of least period n, and the second a lower bound for the number of periodic points of all periods m ⩽ n. Both these Nielsen type numbers are extended here to the case where the ANR X is no longer connected. Calculations and many examples are given.

Nielsen type numbers for periodic points on pairs of spaces

Abstract Two homotopy invariant Nielsen type numbers exist for periodic points of a self-map ƒ: X → X of a compact connected ANR X , namely NP n (ƒ) , the Nielsen type number of period n , and NΦ n (ƒ) , the Nielsen type number of the n th iterate. The first is a lower bound for the number of periodic points of least period n , and the second a lower bound for the number of periodic points of all periods m ⩽ n . Both these Nielsen type numbers are extended here to the case of maps ƒ: (X, A) → (X, A) of pairs of compact ANRs. Calculations and many examples are given.

Fibre Techniques in Nielsen Periodic Point Theory On Solvmanifolds III: Calculations

. This third paper of the series gives the necessarily lengthy illustrations of the main results of the first two, delayed until now for reasons of space. The illustrations in question concern calculations of the Nielsen type periodic point numbers N P n (f) and N φn (f) for self maps f of solvmanifolds. We indicate that for low dimensional solvmanifolds, we can often give formulae (as opposed to algorithms) for these numbers, which of course include formulae for the ordinary Nielsen numbers N (f n). We give a complete analysis of all maps on two very different example generalizations of the Klein bottle K 2. Both examples admit non-weakly Jiang maps, which is where the more complex calculations occur. Our methods employ matrix theory and modular arguments with periodic matrices. Among other things, our results include the promised completion of the more difficult calculations on K 2 itself, as well as a generalization of a surprising result first observed in Nielsen periodic point theory on K 2. More precisely we show for each m > 1 that there is a solvmanifold S of dimension m, and a self map f on S for which, N P n (f) = N (f n) for an infinite number of n. We also discuss the generality of our considerations, indicating that the type of map and solvmanifold given here and the methods used, provide the data needed to make calculations for all maps on all solvmanifolds. Finally we indicate that data that allow the exhibited modular patterns, though widely available, do not hold universally.

Periodic points on the complement

Abstract Let f :(X,A)→(X,A) be a self-map of a pair of compact ANRs, with X connected. In 1989 the second author studied fixed point theory on the complement. He defined a number N(f;X−A) which is a lower bound for the number of fixed points on X−A of maps g that are homotopic to f as a map of pairs. In this paper we generalize these ideas from fixed point theory to periodic point theory, and define two Nielsen type numbers for periodic points on the complement. We give a number of examples and follow a particular one through to show, that for this type of example one of the Nielsen type periodic numbers is given by a formula, and the other is algorithmic. We also highlight a computational simplification of the modified fundamental group approach (which we use here) over the covering space approach.