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48 CARINA BOYALLIAN [V] D. Vogan, Representations of Real Reductive Lie Groups, Progress in Mathematics, Birkhauser, Vol. 15, 1981. [V2] , Unitarizability of certain series of representations, Annals of Math., 119 (1984), 141-187. Received May 5, 1999 and revised December 9, 1999. This research was partially supported by Fundación Antorchas and CONICET. Famaf - Universidad Nacional de Córdoba Ciudad Universitaria, Córdoba (5000) Argentina E-mail address: boyallia@mate.uncor.edu
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
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Page 5 and 6: EIGENVALUES ASYMPTOTICS 3 (i) If pm
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Page 19 and 20: IMAGINARY QUADRATIC FIELDS 17 Table
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Page 25 and 26: 〈σ 〈σ〉 〈σ, τ〉 2 〈1
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Page 36 and 37: 34 CARINA BOYALLIAN spherical serie
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Page 52 and 53: 50 ROBERT F. BROWN AND HELGA SCHIRM
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98 GILLES CARRON Proposition 2.7. N
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100 GILLES CARRON l’espace des 1
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102 GILLES CARRON compact. De même
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104 GILLES CARRON 4. Application du
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106 GILLES CARRON Dans le cas où l
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PACIFIC JOURNAL OF MATHEMATICS Vol.
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BRAIDED OPERATOR COMMUTATORS 111 Re
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BRAIDED OPERATOR COMMUTATORS 113 Re
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BRAIDED OPERATOR COMMUTATORS 115 3.
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Then, for 1 < k ≤ m + 1, we have
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BRAIDED OPERATOR COMMUTATORS 119 Mo
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BRAIDED OPERATOR COMMUTATORS 121 It
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E-mail address: prosk@imath.kiev.ua
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124 DANIEL J. MADDEN times. Further
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126 DANIEL J. MADDEN Further, s t
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128 DANIEL J. MADDEN Now the whole
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130 DANIEL J. MADDEN and β = 1 −
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132 DANIEL J. MADDEN We can simplif
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134 DANIEL J. MADDEN surd, but we n
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136 DANIEL J. MADDEN and d = d(u, v
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138 DANIEL J. MADDEN (6l + 7) k +
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140 DANIEL J. MADDEN Then the conti
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142 DANIEL J. MADDEN Thus n + 1 1 =
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144 DANIEL J. MADDEN If we take b =
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146 DANIEL J. MADDEN b, 2b(2bn + 1)
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TANGLES, 2-HANDLE ADDITIONS AND DEH
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m n n n n TANGLES, 2-HANDLE ADDITIO
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SOME CHARACTERIZATION, UNIQUENESS A
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and SOME CHARACTERIZATION, UNIQUENE
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SYMPLECTIC SUBMANIFOLDS FROM SURFAC
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208 PAULO TIRAO Theorem. The affine
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210 PAULO TIRAO It follows from the
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212 PAULO TIRAO K ρ1 (0,−1,1) =
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214 PAULO TIRAO Theorem 2.7. Let ρ
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216 PAULO TIRAO Notation: By A = [
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218 PAULO TIRAO Now is not difficul
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220 PAULO TIRAO Regard H n (Z2 ⊕
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222 PAULO TIRAO is an isomorphism.
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224 PAULO TIRAO Lemma 4.2. Let Λ1
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226 PAULO TIRAO (c)1 B1 B2 B3 B1 0
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228 PAULO TIRAO ⎛ ⎜ B2 = ⎜
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230 PAULO TIRAO being those in whic
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232 PAULO TIRAO Hence, if ρ = m1χ
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(1.6) HÖLDER REGULARITY FOR ∂ 23
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HÖLDER REGULARITY FOR ∂ 239 can
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HÖLDER REGULARITY FOR ∂ 241 wher
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HÖLDER REGULARITY FOR ∂ 243 Theo
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Note (4.7) bD∩U ′ i dzA j J H
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HÖLDER REGULARITY FOR ∂ 247 p. 1
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similarly, we can prove HÖLDER REG
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HÖLDER REGULARITY FOR ∂ 251 For
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Now if n1 ≥ 2, l ∈ S1, then (4.
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HÖLDER REGULARITY FOR ∂ 255 Refe
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