A Lower Bound for the Δ-Nielsen Number
Robin B. S. Brooks, Robert F. Brown · Transactions of the American Mathematical Society · 1969
Introduction. This paper is concerned with the of solutions of three kinds of equations. Let f, g: X-Y and h: X-X be maps, and let yo E Y. The equations we will study are (1)f(x) =g(x), (2)f(x) =yo, and (3) h(x) =x. In [2], the first author defined a lower bound N for the of solutions of these equations which remains such a lower bound whenf, g, and h are moved through homotopies. The N is called the A-Nielsen of the equation. It is not, in general, possible to compute the A-Nielsen for particular spaces and maps directly from its definition. An integer R which can be computed algebraically just from a knowledge of the maps induced on the fundamental groups was defined in [2], and it was proved that N? R. In some cases it could be shown that N= R. We will define a positive integer J, which is also easier to compute than N, and which has the property that J? N. Bounding N between integers we know something about improves our chances of determining N. Furthermore, J is of interest in itself because it is also a lower bound (though a poorer one) for the of solutions. We will further prove that, under mild additional hypotheses, J divides N, R, and an appropriately defined number for the equation we are considering. In the case of the study of fixed points, that is, a solution to h(x) = x, there has been a lower bound of this kind, N(h), the Nielsen of h, for many years [6], [8]. Unfortunately, N(h) is not, in general, the same as the A-Nielsen N, in fact N(h) < N. However, we are still able to prove that J< N(h) ? R and that, under an additional hypothesis, J divides both N(h) and the classical Lefschetz L(h). In ?11, we will introduce our basic assumptions and will summarize those definitions and results that we will be using from [2]. The definition of J and the proofs of the results indicated above occupy ?111. ?IV is devoted to applications of these general theorems to the three kinds of equations.