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NIELSEN ROOT THEORY AND HOPF DEGREE THEORY 67
classes have multiplicity |m(R)| = 0, and so deg c(f|U) = 0 also. Therefore
deg(f) = (deg c(f|UR): R is a root class of f) = 0 as claimed.
Maps that are homotopic by a proper homotopy induce the same homomorphism
of Čech cohomology with compact supports [Do, p. 290], so the
cohomological degree is a proper homotopy invariant. Therefore, Theorems
3.11 and 3.12 imply
Corollary 3.14. Let f : (M, ∂M) → (N, ∂N) be a proper map, then N(f; c)
and N ∩| (f; c) are proper homotopy invariants. Moreover, the values of both
N(f; c) and N ∩| (f; c) are independent of the choice of c ∈ int N.
We conclude this section with some examples of maps with non-zero Nielsen
root numbers.
Example 3.15. Let f : S n → S n be a map of degree d = 0 between two
n-spheres, where n ≥ 2. As j = 1, Theorem 3.13 shows that N(f; c) = 1
and N ∩| (f; c) = |d|, and so N ∩| (f; c) N(f; c) if |d| > 1. More generally,
it follows from Theorem 3.13 that N ∩| (f; c) = |d| is strictly greater than
N(f; c) = 1 for any map f : (M, ∂M) → (N, ∂N) between two orientable
n-manifolds if fπ is an epimorphism and | deg(f)| = |d| > 1. Examples
with N ∩| (f; c) = N(f; c) for maps of non-orientable manifolds can readily
be constructed by using cartesian products. To be specific, let f : P 2 ×S 2 →
P 2 ×S 2 be the product map f = f1 ×f2 with f1 the identity and f2 of degree
d with |d| > 1. Then f is of Type I, j = 1, f = f, M = N = S 2 × S 2 and
f = f1 × f2 where f1 is the identity and f2 = f2. Now | deg( f)| = |d| = 0 so
N(f; c) = 1 by Theorem 3.11 whereas Theorem 3.12 implies that N ∩| (f; c) =
|d| > 1.
Example 3.16. A different type of example of a map f of non-orientable
manifolds for which N ∩| (f; c) = N(f; c) is illustrated by the following. Rep-
resent the Klein bottle by K = P 2 #S 2 #P 2 , then a rotation of S 2 interchanging
the copies of P 2 defines an action of Z/2 on K with two fixed
points. Therefore, the homomorphism of fundamental groups induced by
the quotient map f : K → P 2 is onto [Bd, Cor. 6.3, p. 91]. Thus P 2 = P 2
and f is the lift f : T 2 → S 2 of f to the oriented covers. By inspection,
N ∩| ( f; c) = 2, and so we obtain from Theorem 3.13 that f is a map of degree
±2. Since the map f is of Type I, Theorems 3.11 and 3.12 imply that
N(f; c) = 1 and N ∩| (f; c) = 2.
Example 3.17. As in Example 2.2(b), let f : M → N be the covering map
of the orientable cover of a non-orientable manifold N. Since j = 2 and
deg( f) = 1, Theorem 3.11 implies that N(f; c) = 2 and Theorem 3.12
implies that N ∩| (f; c) = 2 also.