For printing - MSP

PACIFIC JOURNAL OF MATHEMATICS
Vol. 198, No. 1, 2001
NIELSEN ROOT THEORY AND HOPF DEGREE THEORY
Robert F. Brown and Helga Schirmer
The Nielsen root number N(f; c) of a map f : M → N at
a point c ∈ N is a homotopy invariant lower bound for the
number of roots at c, that is, for the cardinality of f −1 (c).
There is a formula for calculating N(f; c) if M and N are
closed oriented manifolds of the same dimension. We extend
the calculation of N(f; c) to manifolds that are not orientable,
and also to manifolds that have non-empty boundaries and are
not compact, provided that the map f is boundary-preserving
and proper. Because of its connection with degree theory, we
introduce the transverse Nielsen root number for maps transverse
to c, obtain computational results for it in the same setting,
and prove that the two Nielsen root numbers are sharp
lower bounds in dimensions other than 2. We apply these
extended root theory results to the degree theory for maps
of not necessarily orientable manifolds introduced by Hopf in
1930. Thus we re-establish, in a new and modern treatment,
the relationship of Hopf’s Absolutgrad and the geometric degree
with homotopy invariants of Nielsen root theory, a relationship
that is present in Hopf’s work but not in subsequent
re-examinations of Hopf’s degree theory.
1. Introduction.
The goal of this paper is two-fold. We will extend results from Nielsen root
theory for maps between orientable n-manifolds so as to remove the orientability
hypothesis. Then we will use the extended theory to re-establish
the connection between Nielsen root theory and two variants of the degree
of a map, namely, Hopf’s Absolutgrad and the geometric degree. By using
methods from present-day Nielsen theory, we will provide new ways of
understanding some of the basic concepts of Hopf’s theory as well as more
direct proofs for some of the results. We next describe these goals in more
detail.
If f : M → N is a map between two manifolds and c ∈ N, then a root
of f at c is a point in f −1 (c). The Nielsen root number N(f; c) is a lower
bound for the cardinality of f −1 (c), and it is homotopy invariant. While it
is possible to define N(f; c) even if M and N are not manifolds, it is usually
not possible to compute it in such general settings. If, however, M and N
49