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NIELSEN ROOT THEORY AND HOPF DEGREE THEORY 63
is a root class R = f −1 (c), which is open in f −1 (c). Since the cardinality
of q −1 (c) is j and f proper implies that f −1 (c) is compact, there cannot be
infinitely many root classes, so j is finite.
As a consequence of Lemmas 3.6 and 3.7 we can now restrict our attention
to the remaining cases, in which f is an orientable map and j is finite.
For the proof of the next lemma, we will use the following set of covering
spaces of M and N, and of basepoint-preserving lifts of f to these covering
spaces; compare [E, p. 370].
M ′ , x ′ 0
p ′
⏐
M, x0
⏐
ep
M, x0
f ′
−−−→ N ′ , c ′
ef
⏐
⏐
q ′
−−−→ N, c
⏐
⏐
eq
bf
−−−→ N, c
The spaces and maps in this diagram are obtained in the following manner.
(1) p: M → M is the orientable covering of M. Hence M = M if M is
orientable, but M is a 2-sheeted covering of M if M is not orientable.
This covering space corresponds to the subgroup of π1(M, x0) which
is generated by the orientation-preserving loop classes of M.
(2) q : N → N is the minimal covering space of N with the property that
f ◦ p: M → N has a lift f : M → N. Hence N corresponds to the
subgroup of π1( N, c) which is generated by the images under f of
all orientation-preserving loop classes of M, and the homomorphism
fπ : π1( M, x0) → π1( N, c) is onto.
(3) q ′ : N ′ → N is the orientable covering of N. Hence N ′ corresponds to
the subgroup of π1( N, c) which is generated by the images under f of
all orientation-preserving loop classes of M which have an orientationpreserving
image in N.
(4) p ′ : M ′ → M is the minimal cover of M with the property that f ◦
p ′ : M ′ → N has a lift to f ′ : M ′ → N ′ . Hence M ′ corresponds to the
subgroup of π1( M, x0) which is generated by all orientation-preserving
loop classes of M which have an orientation-preserving image in N
under f, and the homomorphism f ′ π : π1(M ′ , x ′ 0 ) → π1(N ′ , c ′ ) is onto.
Equivalently, p ′ : M ′ → M is the pullback of q ′ : N ′ → N over M by
means of f.