For printing - MSP

More documents

Recommendations

Info

78 ROBERT F. BROWN AND HELGA SCHIRMER G(f) = d but A(f) = 0 if d is even and A(f) = 1 if d is odd. The computation of the geometric degree for all surface maps is an open problem. The sharpness of the Nielsen root number N(f; c) is a more delicate property as the conclusions of Theorems 6.1 and 6.2 do not hold with N(f; c) in place of N ∩| (f; c). This was already known to Hopf who constructed examples, for every j > 4 as well as for j = 1, of a map f from the double torus to the torus such that N(f; c) = j but f cannot be homotoped to a map with only N(f; c) roots (see [H2, Satz XVa, p. 610 and Satz XVb, p. 623]). Using results of Hopf, Lin [L, §4] constructed a map from the double torus to the torus with N(f; c) = 3 and MR[f; c] = 4. For maps between orientable closed surfaces, necessary and sufficient conditions for the sharpness of the root Nielsen number been recently been found by D.L. Gonçalves and H. Zieschang [GZ1] and [GZ2]. They proved: Theorem 6.3 (Gonçalves and Zieschang). Let M and N be closed orientable surfaces of genus h and k respectively. Let f : M → N be a map such that j is finite, then N(f; c) is sharp if and only if | deg(f)| ≤ 2h − 2 + j 2k − 1 . Some new results concerning the sharpness of N(f; c) for maps of surfaces can easily be obtained from our calculations of the two Nielsen root numbers in §3. We can use the fact that if N(f; c) = N ∩| (f; c) and N ∩| (f; c) is sharp then N(f; c) must also be sharp. So an inspection of Theorems 3.11, 3.12 and 6.1 immediately yields: Theorem 6.4. If f is a map between two closed surfaces which is not orientation-true, then N(f; c) is sharp. The map f : T 2 #P 2 → T 2 of Example 3.19 has exactly |d| roots at c if c ∈ D. Theorem 6.4 implies that if c ∈ D, then there is a map homotopic to f that has |d| roots at c (compare Example 4.6). Theorems 6.3 and 6.4 give necessary and sufficient conditions for the sharpness of N(f; c) for all maps between closed surfaces, with the exception of the case of orientation-true maps between non-orientable surfaces. The Nielsen root number may be sharp in this case also. For instance, for any c ∈ P 2 , the map f : K → P 2 of Example 3.16 is homotopic to a map g with one root at c because f is one-to-one at two points and P 2 is homogeneous. However, it is likely that, for orientation-true maps between non-orientable manifolds, N(f; c) will only be sharp for some maps, as is true in the orientable case. We contribute a partial solution to this problem in the following theorem, which again follows immediately from Theorems 3.11, 3.12 and 6.1.
NIELSEN ROOT THEORY AND HOPF DEGREE THEORY 79 Theorem 6.5. Let f be an orientation-true map between two non-orientable closed surfaces. Then N(f; c) is sharp if either j is infinite or if | deg( f)| ≤ 1. Finally, Theorems 3.11, 3.12 and 6.2 yield some results concerning the sharpness of N(f; c) in the case where the boundaries of the surfaces M, N are non-empty. We omit the details. References [Bd] G. Bredon, An Introduction to Compact Transformation Groups, Academic Press, New York, 1972. [Bk1] R. Brooks, Coincidences, roots and fixed points, Doctoral Dissertation, Univ. of California, Los Angeles, 1967. [Bk2] , On the sharpness of the ∆2 and ∆1 Nielsen numbers, J. Reine Angew. Math., 259 (1973), 101-108. [BB] R. Brooks and R. Brown, A lower bound for the ∆-Nielsen number, Trans. Amer. Math. Soc., 143 (1969), 555-564. [BO] R. Brooks and C. Odenthal, Nielsen numbers for roots of maps of aspherical manifolds, Pacific J. Math., 170 (1995), 405-420. [Bw] L.E.J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann., 71 (1911), 97-115. [Bn1] R. Brown, The Lefschetz Fixed Point Theorem, Scott-Foresman, 1971. [Bn2] , A middle-distance look at root theory, in ‘Nielsen Theory and Reidemeister Torsion’, Banach Center Publications, 49 (1999), 29-41. [BS1] R. Brown and H. Schirmer, Correction to “Nielsen coincidence theory and coincidence-producing maps for manifolds with boundary”, Top. Appl., 67 (1995), 233-234. [BS2] , Nielsen theory of roots of maps of pairs, Top. Appl., 92 (1999), 247-274. [DJ] R. Dobrenko and J. Jezierski, The coincidence Nielsen theory on non-orientable manifolds, Rocky Mt. J. Math., 23 (1993), 67-85. [Do] A. Dold, Lectures on Algebraic Topology, 2nd edition, Springer-Verlag, Berlin, 1980. [E] D.B.A. Epstein, The degree of a map, Proc. London Math. Soc., 16 (1966), 369-383. [GJ] D. Goncalves and J. Jezierski, Lefschetz coincidence formula on non-orientable manifolds, Fund. Math., 153 (1997), 1-23. [GZ1] D. Goncalves and H. Zieschang, Equations in free groups and coincidences of mappings on surfaces, preprint. [GZ2] , Equations in free groups and coincidences of mappings on surfaces, II, preprint. [H1] H. Hopf, Zur Topologie der Abbildungen von Mannigfaltigkeiten, Erster Teil, Math. Ann., 100 (1928), 579-608. [H2] , Zur Topologie der Abbildungen von Mannigfaltigkeiten, Zweiter Teil, Math. Ann., 102 (1930), 562-623.
Page 1 and 2:
Pacific Journal of Mathematics Volu
Page 3 and 4:
PACIFIC JOURNAL OF MATHEMATICS Vol.
Page 5 and 6:
EIGENVALUES ASYMPTOTICS 3 (i) If pm
Page 7 and 8:
EIGENVALUES ASYMPTOTICS 5 Corollary
Page 9 and 10:
Thus, by the Itô formula, we have
Page 11 and 12:
Therefore, K1(t) ≡ t 2−d/2 ≤
Page 13 and 14:
EIGENVALUES ASYMPTOTICS 11 The chan
Page 15 and 16:
EIGENVALUES ASYMPTOTICS 13 Here Res
Page 17 and 18:
Page 19 and 20:
IMAGINARY QUADRATIC FIELDS 17 Table
Page 21 and 22:
IMAGINARY QUADRATIC FIELDS 19 and t
Page 23 and 24:
IMAGINARY QUADRATIC FIELDS 21 Propo
Page 25 and 26:
〈σ 〈σ〉 〈σ, τ〉 2 〈1
Page 27 and 28:
IMAGINARY QUADRATIC FIELDS 25 Then
Page 29 and 30: IMAGINARY QUADRATIC FIELDS 27 were
Page 31 and 32: IMAGINARY QUADRATIC FIELDS 29 Thus
Page 33: IMAGINARY QUADRATIC FIELDS 31 [3] ,
Page 36 and 37: 34 CARINA BOYALLIAN spherical serie
Page 38 and 39: 36 CARINA BOYALLIAN Here I G MAN (
Page 40 and 41: 38 CARINA BOYALLIAN where k ≤ [ n
Page 42 and 43: 40 CARINA BOYALLIAN The case G = Sp
Page 44 and 45: 42 CARINA BOYALLIAN we can, since i
Page 46 and 47: 44 CARINA BOYALLIAN and this formul
Page 48 and 49: 46 CARINA BOYALLIAN Here, J(ν) int
Page 50 and 51: 48 CARINA BOYALLIAN [V] D. Vogan, R
Page 52 and 53: 50 ROBERT F. BROWN AND HELGA SCHIRM
Page 84 and 85: 82 GILLES CARRON à l’infini s’
Page 86 and 87: 84 GILLES CARRON isométriques au d
Page 88 and 89: 86 GILLES CARRON Comme D est ellipt
Page 90 and 91: 88 GILLES CARRON Réciproquement si
Page 92 and 93: 90 GILLES CARRON où H est la proje
Page 94 and 95: 92 GILLES CARRON 2.a. Exemples repo
Page 96 and 97: 94 GILLES CARRON alors si σ ∈ C
Page 98 and 99: 96 GILLES CARRON Ainsi s’il exist
Page 100 and 101: 98 GILLES CARRON Proposition 2.7. N
Page 102 and 103: 100 GILLES CARRON l’espace des 1
Page 104 and 105: 102 GILLES CARRON compact. De même
Page 106 and 107: 104 GILLES CARRON 4. Application du
Page 108 and 109: 106 GILLES CARRON Dans le cas où l
Page 111 and 112: PACIFIC JOURNAL OF MATHEMATICS Vol.
Page 113 and 114: BRAIDED OPERATOR COMMUTATORS 111 Re
Page 115 and 116: BRAIDED OPERATOR COMMUTATORS 113 Re
Page 117 and 118: BRAIDED OPERATOR COMMUTATORS 115 3.
Page 119 and 120: Then, for 1 < k ≤ m + 1, we have
Page 121 and 122: BRAIDED OPERATOR COMMUTATORS 119 Mo
Page 123 and 124: BRAIDED OPERATOR COMMUTATORS 121 It
Page 125: E-mail address: prosk@imath.kiev.ua
Page 128 and 129: 124 DANIEL J. MADDEN times. Further
Page 130 and 131:
126 DANIEL J. MADDEN Further, s t
Page 132 and 133:
128 DANIEL J. MADDEN Now the whole
Page 134 and 135:
130 DANIEL J. MADDEN and β = 1 −
Page 136 and 137:
132 DANIEL J. MADDEN We can simplif
Page 138 and 139:
134 DANIEL J. MADDEN surd, but we n
Page 140 and 141:
136 DANIEL J. MADDEN and d = d(u, v
Page 142 and 143:
138 DANIEL J. MADDEN (6l + 7) k +
Page 144 and 145:
140 DANIEL J. MADDEN Then the conti
Page 146 and 147:
142 DANIEL J. MADDEN Thus n + 1 1 =
Page 148 and 149:
144 DANIEL J. MADDEN If we take b =
Page 150 and 151:
146 DANIEL J. MADDEN b, 2b(2bn + 1)
Page 153 and 154:
Page 155 and 156:
TANGLES, 2-HANDLE ADDITIONS AND DEH
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
m n n n n TANGLES, 2-HANDLE ADDITIO
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
SOME CHARACTERIZATION, UNIQUENESS A
Page 183 and 184:
and SOME CHARACTERIZATION, UNIQUENE
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
Page 201 and 202:
Page 203 and 204:
SYMPLECTIC SUBMANIFOLDS FROM SURFAC
Page 205 and 206:
Page 207 and 208:
Page 209:
Page 212 and 213:
208 PAULO TIRAO Theorem. The affine
Page 214 and 215:
210 PAULO TIRAO It follows from the
Page 216 and 217:
212 PAULO TIRAO K ρ1 (0,−1,1) =
Page 218 and 219:
214 PAULO TIRAO Theorem 2.7. Let ρ
Page 220 and 221:
216 PAULO TIRAO Notation: By A = [
Page 222 and 223:
218 PAULO TIRAO Now is not difficul
Page 224 and 225:
220 PAULO TIRAO Regard H n (Z2 ⊕
Page 226 and 227:
222 PAULO TIRAO is an isomorphism.
Page 228 and 229:
224 PAULO TIRAO Lemma 4.2. Let Λ1
Page 230 and 231:
226 PAULO TIRAO (c)1 B1 B2 B3 B1 0
Page 232 and 233:
228 PAULO TIRAO ⎛ ⎜ B2 = ⎜
Page 234 and 235:
230 PAULO TIRAO being those in whic
Page 236 and 237:
232 PAULO TIRAO Hence, if ρ = m1χ
Page 239 and 240:
Page 241 and 242:
(1.6) HÖLDER REGULARITY FOR ∂ 23
Page 243 and 244:
HÖLDER REGULARITY FOR ∂ 239 can
Page 245 and 246:
HÖLDER REGULARITY FOR ∂ 241 wher
Page 247 and 248:
HÖLDER REGULARITY FOR ∂ 243 Theo
Page 249 and 250:
Note (4.7) bD∩U ′ i dzA j J H
Page 251 and 252:
HÖLDER REGULARITY FOR ∂ 247 p. 1
Page 253 and 254:
similarly, we can prove HÖLDER REG
Page 255 and 256:
HÖLDER REGULARITY FOR ∂ 251 For
Page 257 and 258:
Now if n1 ≥ 2, l ∈ S1, then (4.
Page 259 and 260:
HÖLDER REGULARITY FOR ∂ 255 Refe
Page 261 and 262:
Guidelines for Authors Authors may
show all

For printing - MSP

Create successful ePaper yourself

Delete template?

Save as template?