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Atoroidal

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296: 91:(i.e., the image of the map on fundamental group induced by an inclusion of a boundary component). The terminology is not standardized, and different authors require atoroidal 3-manifolds to satisfy certain additional restrictions. For instance: 99:) gives a definition of atoroidality that combines both geometric and algebraic aspects, in terms of maps from a torus to the manifold and the induced maps on the fundamental group. He then notes that for 134:. He makes the same restriction on geometrically atoroidal manifolds (which he calls topologically atoroidal) and in addition requires them to avoid incompressible boundary-parallel embedded 77: 337: 130:) requires the algebraic variant of atoroidal manifolds (which he calls simply atoroidal) to avoid being one of three kinds of 84: 274: 244: 213: 182: 52: 100: 103: 236: 205: 31:. There are two major variations in this terminology: an essential torus may be defined geometrically, as an 330: 361: 323: 40: 230: 168: 264: 199: 356: 8: 138:. With these definitions, the two kinds of atoroidality are equivalent except on certain 88: 270: 240: 209: 178: 174: 80: 36: 311: 260: 139: 123: 307: 350: 135: 131: 17: 24: 120:) uses the geometric definition, restricted to irreducible manifolds. 32: 303: 47: 113:) uses the algebraic definition without additional restrictions. 295: 269:, Progress in Mathematics, vol. 183, Springer, p. 6, 43: 28: 173:, De Gruyter Expositions in Mathematics, vol. 32, 55: 201:
The hyperbolization theorem for fibered 3-manifolds
170:
Conformal Geometry of Discrete Groups and Manifolds
71: 348: 106:3-manifolds this gives the algebraic definition. 72:{\displaystyle \mathbb {Z} \times \mathbb {Z} } 331: 146:A 3-manifold that is not atoroidal is called 46:, or it may be defined algebraically, as a 338: 324: 27:is one that does not contain an essential 204:, Contemporary Mathematics, vol. 7, 65: 57: 266:Hyperbolic Manifolds and Discrete Groups 259: 166: 127: 96: 235:, Mathematical surveys and monographs, 349: 290: 228: 197: 117: 110: 13: 14: 373: 232:The Ricci Flow: Geometric aspects 294: 253: 222: 191: 160: 1: 237:American Mathematical Society 206:American Mathematical Society 153: 310:. You can help Knowledge by 7: 167:Apanasov, Boris N. (2000), 109:Jean-Pierre Otal ( 10: 378: 289: 198:Otal, Jean-Pierre (2001), 95:Boris Apanasov ( 116:Bennett Chow ( 104:boundary-incompressible 229:Chow, Bennett (2007), 73: 74: 124:Michael Kapovich 53: 89:peripheral subgroup 69: 319: 318: 261:Kapovich, Michael 175:Walter de Gruyter 140:Seifert manifolds 81:fundamental group 37:boundary parallel 369: 340: 333: 326: 304:geometry-related 298: 291: 281: 279: 257: 251: 249: 226: 220: 218: 195: 189: 187: 164: 78: 76: 75: 70: 68: 60: 377: 376: 372: 371: 370: 368: 367: 366: 347: 346: 345: 344: 287: 285: 284: 277: 258: 254: 247: 239:, p. 436, 227: 223: 216: 196: 192: 185: 177:, p. 294, 165: 161: 156: 64: 56: 54: 51: 50: 12: 11: 5: 375: 365: 364: 362:Geometry stubs 359: 343: 342: 335: 328: 320: 317: 316: 299: 283: 282: 275: 252: 245: 221: 214: 208:, p. ix, 190: 183: 158: 157: 155: 152: 144: 143: 121: 114: 107: 67: 63: 59: 41:incompressible 9: 6: 4: 3: 2: 374: 363: 360: 358: 355: 354: 352: 341: 336: 334: 329: 327: 322: 321: 315: 313: 309: 306:article is a 305: 300: 297: 293: 292: 288: 278: 276:9780817649135 272: 268: 267: 262: 256: 248: 246:9780821839461 242: 238: 234: 233: 225: 217: 215:9780821821534 211: 207: 203: 202: 194: 186: 184:9783110808056 180: 176: 172: 171: 163: 159: 151: 149: 141: 137: 136:Klein bottles 133: 129: 125: 122: 119: 115: 112: 108: 105: 102: 98: 94: 93: 92: 90: 86: 82: 61: 49: 45: 42: 38: 34: 30: 26: 23: 19: 312:expanding it 301: 286: 265: 255: 231: 224: 200: 193: 169: 162: 147: 145: 132:fiber bundle 83:that is not 21: 15: 357:3-manifolds 101:irreducible 18:mathematics 351:Categories 154:References 25:3-manifold 85:conjugate 62:× 22:atoroidal 263:(2009), 148:toroidal 48:subgroup 33:embedded 126: ( 79:of its 273:  243:  212:  181:  35:, non- 302:This 87:to a 44:torus 29:torus 20:, an 308:stub 271:ISBN 241:ISBN 210:ISBN 179:ISBN 128:2009 118:2007 111:2001 97:2000 39:, 16:In 353:: 150:. 339:e 332:t 325:v 314:. 280:. 250:. 219:. 188:. 142:. 66:Z 58:Z

Index

mathematics
3-manifold
torus
embedded
boundary parallel
incompressible
torus
subgroup
fundamental group
conjugate
peripheral subgroup
2000
irreducible
boundary-incompressible
2001
2007
Michael Kapovich
2009
fiber bundle
Klein bottles
Seifert manifolds
Conformal Geometry of Discrete Groups and Manifolds
Walter de Gruyter
ISBN
9783110808056
The hyperbolization theorem for fibered 3-manifolds
American Mathematical Society
ISBN
9780821821534
The Ricci Flow: Geometric aspects

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