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NIELSEN ROOT THEORY AND HOPF DEGREE THEORY 55
as a map from P 2 − int D to D ⊂ T 2 . Since the generator of π1(T 2 #P 2 , x0)
represented by an orientation-reversing loop is mapped into the contractible
set D, we see that f is of Type III.
The remainder of this section will be devoted to defining, for f : (M, ∂M)
→ (N, ∂N) a proper map of any type, the multiplicity of a root class of f
at c ∈ int N.
Points x1, x2 ∈ f −1 (c) are in the same root class of f at c if there is a path
w : I → M from x1 to x2 such that f ◦ w is a contractible loop at c (see [Kg,
Chapter V.B]). (This definition goes back to Hopf [H2, Definition V, p. 575],
where a root class is called a “Schicht”.) Since f : (M, ∂M) → (N, ∂N) is
proper, the root classes are compact subsets of M and there are only finitely
many of them. Let V ⊂ int N be a contractible neighborhood of c. Since f
is boundary-preserving, f −1 (V ) is contained in int M. Let R be a root class
of f at c and let U be an open subset of f −1 (V ) such that U ∩ f −1 (c) = R.
Since U is an open subset of M, it is a manifold, that is a space locally
homeomorphic to R n , but it is not necessarily connected.
We shall first assume that U is an oriented manifold. If M is itself an oriented
manifold, then the orientation of U is the restriction of the orientation
of M. The neighborhood V is contractible, so it is an orientable manifold
and we choose an orientation for it, selecting the restriction of the orientation
of N if that manifold is oriented. The integer-valued local degree of
f|U : U → V over c is defined; it is denoted by deg c(f|U) [Do, Definition 4.2,
p. 267]. If U0 ⊂ U, an open subset containing R, is oriented by restricting
the orientation of U, then deg c(f|U0) = deg c(f|U). Consequently, if U1 and
U2 are open subsets of f −1 (V ) containing R that are oriented so that their
orientations agree on U1 ∩ U2, then deg c(f|U1) = deg c(f|U2). Moreover, if
V0 ⊂ V is also a contractible neighborhood of c, and U ⊂ f −1 (V0) then, if
V0 is oriented by restricting the orientation of V , it follows that deg c(f|U)
has the same value if we view f|U as a map into V0 as it does if we view
f|U as a map into V .
The following remark describes the relationship between the cohomological
degree and the local degree.
Remark 2.5. The definition above of the cohomological degree deg(f) of
a proper map f : (M, ∂M) → (N, ∂N) of oriented manifolds made use of
fundamental classes [int N] ∈ ˇ H n (int N) and [W ] ∈ ˇ H n (W ), where W =
M − f −1 (∂N). Duality [Do, Prop. 7.14, p. 297] gives us corresponding
elements of singular homology {int N} ∈ H0(int N) and {W } ∈ H0(W ) so
that, for the homology transfer homomorphism f! [Do, Equation 10.7, p.
310], we have f!{int N} = deg(f){W }. Consequently, for f∗ : H0(W ) →
H0(int N), we see that f∗f!{int N} = deg(f){int N}. On the other hand,
[Do, Prop. 10.10, p. 312] implies that f∗f!{int N} = deg c(f|W ) {int N}.