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NIELSEN ROOT THEORY AND HOPF DEGREE THEORY 75
degree (see Theorems 5.4 and 5.5 below). Here is the original definition of
Hopf’s degree.
Definition 5.1 ([H2, Definition VIIc, p. 582]). Let f : (M, ∂M) → (N,
∂N) be a proper map between two n-manifolds and let c ∈ int N. Then the
Absolutgrad or absolute degree A(f) of f is the sum of the multiplicities, in
the sense of Definition 2.7, of its root classes.
The reason for Hopf’s introduction of the absolute degree is that it provides
an algebraic homotopy invariant that is closely linked to the very concrete
concept of the geometric degree. This geometric interpretation of the
absolute degree is based on the equality of the “algebraic” and “geometric”
degrees which, as Hopf explained in the introduction to [H2] (see page 563),
was the goal of that paper. (For maps of euclidean spaces, the equality had
already been established in [H1, Satz IX, p. 590].) The following definition
of the geometric degree is taken from [E, p. 372] and [Sk, p. 416]. It is a
restatement of what Hopf understood by a geometric degree.
Definition 5.2. Let f : (M, ∂M) → (N, ∂N) be a proper map between two
n-manifolds. Then the geometric degree G(f) of f is the least non-negative
integer for which there exists a closed n-ball B n ⊂ int N and a proper map
g : (M, ∂M) → (N, ∂N), which is homotopic to f under a proper homotopy,
such that g −1 (B n ) has G(f) components, and each component is mapped
by g homeomorphically onto B n .
Note that for proper maps such an integer always exists, as G(f) is
bounded above by the number of roots of any transverse map homotopic
to it.
The absolute and the geometric degree of a map are in essence concepts
from Nielsen root theory. This fact follows immediately from the definition of
these degrees in 5.1 and 5.2 and from the definition of the minimum number
of transverse roots and the transverse Nielsen root number in (3.2) and
(3.3), but we state it explicitly in the next theorem in order to emphasize
and clarify the connection between Nielsen root theory and Hopf degree
theory.
Theorem 5.3. Let f : (M, ∂M) → (N, ∂N) be a proper map between two
n-manifold and let c be any point in int N. Then A(f) = N ∩| (f; c) and
G(f) = MR ∩| (f; c), that is, the absolute degree equals the transverse root
Nielsen number and the geometric degree equals the least number of roots
which are transverse at c for all maps in the proper homotopy class of f.
One can see from the introduction of his paper [H2] that Hopf was very
well aware of the fact brought out by Theorem 5.3, that the problem of
finding the geometric degree is of a similar nature to the problem of finding