36:
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1648:, then on the boundary of the ball, draw geodesic lines connecting the points to the north and south pole. Now identify spherical triangles by identifying the north pole to the south pole and the points
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The lens space L(2;5) consists of the "lens" between the red and yellow walls using a double rotation that aligns the slits. Five "lens" regions are shown in the picture in total.
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were not homeomorphic even though they have isomorphic fundamental groups and the same homology, though they do not have the same homotopy type. Other lens spaces (such as
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as in case 2., they are "obviously" homeomorphic, as one can easily produce a homeomorphism. It is harder to show that these are the only homeomorphic lens spaces.
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which is symmetric. (Locally symmetric spaces are symmetric spaces that are quotiented by an isometry that has no fixed points; lens spaces meet this definition.)
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The double-rotation that identifies the walls of the lens space. In this stereographic view, the double-rotation rotates both around the z-axis and along it.
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be a closed curve in the lens space which lifts to a knot in the universal cover of the lens space. If the lifted knot has a trivial
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alone, and the simplest examples of closed manifolds whose homeomorphism type is not determined by their homotopy type.
2453:, Pure and Applied Mathematics 89, Translated from the German edition of 1934, Academic Press Inc. New York (1980)
2334:
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1850:. Fill in the bipyramid to make it solid and give the triangles on the boundary the same identification as above.
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Classifications up to homeomorphism and homotopy equivalence are known, as follows. The three-dimensional spaces
1838:
directly above and below the center of the polygon. Construct the bipyramid by joining each point of the regular
1294:{\displaystyle (z_{1},\ldots ,z_{n})\mapsto (e^{2\pi iq_{1}/p}\cdot z_{1},\ldots ,e^{2\pi iq_{n}/p}\cdot z_{n}).}
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Salvatore, Paolo; Longoni, Riccardo (2005), "Configuration spaces are not homotopy invariant",
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in 1908. They were the first known examples of 3-manifolds which were not determined by their
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The invariant that gives the homotopy classification of 3-dimensional lens spaces is the
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2210:) to be a homeomorphism classification. In modern terms, lens spaces are determined by
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788:{\displaystyle (z_{1},z_{2})\mapsto (e^{2\pi i/p}\cdot z_{1},e^{2\pi iq/p}\cdot z_{2})}
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3-manifold that is a quotient of S³ by ℤ/p actions: (z,w) ↦ (exp(2πi/p)z, exp(2πiq/p)w)
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In the 3-manifold case, a lens space can be visualized as the result of gluing two
2461:
2358:; Yasukhara, Akira (2003), "Symmetry of Links and Classification of Lens Spaces",
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is often defined to be a solid ball with the following identification: first mark
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Another related definition is to view the solid ball as the following solid
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Brody, E. J. (1960), "The topological classification of the lens spaces",
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Ueber die topologischen
Invarianten mehrdimensionaler Mannigfaltigkeiten
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There is a complete classification of three-dimensional lens spaces, by
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equally spaced points on the equator of the solid ball, denote them
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The homeomorphism classification is more subtle, and is given by
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This can be generalized to higher dimensions as follows: Let
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2177:{\displaystyle q_{1}\equiv \pm q_{2}^{\pm 1}{\pmod {p}}}
2106:{\displaystyle q_{1}\equiv \pm q_{2}^{\pm 1}{\pmod {p}}}
1780:. The resulting space is homeomorphic to the lens space
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Alternative definitions of three-dimensional lens spaces
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2008:{\displaystyle q_{1}q_{2}\equiv \pm n^{2}{\pmod {p}}}
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2466:Monatsh. fuer Math. und Phys. 19, 1–118 (1908) (
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128:. The term often refers to a specific class of
57:but its sources remain unclear because it lacks
2217:type, and there are no normal invariants (like
2247:Another invariant is the homotopy type of the
1376:The fundamental group of all the lens spaces
8:
2507:(undergraduate dissertation), archived from
2199:
1854:Classification of 3-dimensional lens spaces
2391:(1935), "Homotopieringe und Linsenräume",
1465:{\displaystyle \mathbb {Z} /p\mathbb {Z} }
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88:Learn how and when to remove this message
1430:{\displaystyle L(p;q_{1},\ldots ,q_{n})}
1058:{\displaystyle L(p;q_{1},\ldots q_{n})}
2207:
888:{\displaystyle p,q_{1},\ldots ,q_{n}}
7:
241:in 1919 showed that the lens spaces
2166:
2095:
1997:
1946:homotopy equivalent if and only if
2473:
2343:Notes on basic 3-manifold topology
408:The three-dimensional lens spaces
190:The three-dimensional lens spaces
25:
2036:{\displaystyle n\in \mathbb {N} }
1549:The three dimensional lens space
180:{\displaystyle S^{2}\times S^{1}}
1362:{\displaystyle L(p;q)=L(p;1,q).}
1000:{\displaystyle \mathbb {C} ^{n}}
593:{\displaystyle \mathbb {C} ^{2}}
143:of their boundaries. Often the
34:
2159:
2088:
1990:
657:generated by the homeomorphism
500:-actions. More precisely, let
2234:Przytycki & Yasukhara 2003
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1124:{\displaystyle \mathbb {Z} /p}
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623:{\displaystyle \mathbb {Z} /p}
493:{\displaystyle \mathbb {Z} /p}
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385:of manifolds as distinct from
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1:
2501:A Short Survey of Lens Spaces
2393:Abh. Math. Sem. Univ. Hamburg
2289:Graduate Texts in Mathematics
1822:: construct a planar regular
2347:(Explains classification of
2253:Salvatore & Longoni 2005
2232:classification is given in (
2202:) as a classification up to
2046:homeomorphic if and only if
1304:In three dimensions we have
2584:
2335:Cambridge University Press
1935:{\displaystyle L(p;q_{2})}
1893:{\displaystyle L(p;q_{1})}
895:be integers such that the
2498:Watkins, Matthew (1990),
2435:10.1016/j.top.2004.11.002
1773:{\displaystyle a_{i+q+1}}
1504:locally symmetric spaces
1094:{\displaystyle S^{2n-1}}
971:{\displaystyle S^{2n-1}}
798:is free. The resulting
43:This article includes a
2206:, but it was shown in (
1734:{\displaystyle a_{i+1}}
1701:{\displaystyle a_{i+q}}
1641:{\displaystyle a_{p-1}}
72:more precise citations.
2532:Lens spaces: a history
2480:
2450:A textbook of topology
2219:characteristic classes
2178:
2107:
2037:
2009:
1936:
1894:
1809:
1808:{\displaystyle L(p;q)}
1774:
1735:
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1578:
1577:{\displaystyle L(p;q)}
1535:
1534:{\displaystyle L(2;1)}
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978:as the unit sphere in
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833:{\displaystyle L(p;q)}
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571:as the unit sphere in
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544:integers and consider
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494:
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437:
436:{\displaystyle L(p;q)}
375:
374:{\displaystyle L(7;2)}
340:
339:{\displaystyle L(7;1)}
305:
304:{\displaystyle L(5;2)}
270:
269:{\displaystyle L(5;1)}
219:
218:{\displaystyle L(p;q)}
181:
113:
105:
2540:at the Manifold Atlas
2534:at the Manifold Atlas
2528:at the Manifold Atlas
2481:
2351:up to homeomorphism.)
2298:Annals of Mathematics
2285:Topology and Geometry
2198:. This was given in (
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2010:
1937:
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1668:{\displaystyle a_{i}}
1643:
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1608:{\displaystyle a_{0}}
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1492:{\displaystyle q_{i}}
1467:
1432:
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1296:
1131:-action generated by
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917:
915:{\displaystyle q_{i}}
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650:{\displaystyle S^{3}}
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564:{\displaystyle S^{3}}
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463:{\displaystyle S^{3}}
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2269:Spherical 3-manifold
2249:configuration spaces
2242:Alexander polynomial
2196:Reidemeister torsion
2189:torsion linking form
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398:Reidemeister torsion
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2488:English translation
2479:{\displaystyle \S }
2361:Geometriae Dedicata
2356:Przytycki, Józef H.
2223:surgery obstruction
2157:
2086:
1472:independent of the
1065:is the quotient of
225:were introduced by
120:is an example of a
2476:
2405:10.1007/BF02940717
2389:Reidemeister, Kurt
2330:Algebraic Topology
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1007:. The lens space
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387:algebraic topology
383:geometric topology
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45:list of references
2445:William Threlfall
2200:Reidemeister 1935
1842:sided polygon to
1830:. Put two points
935:{\displaystyle p}
533:{\displaystyle q}
513:{\displaystyle p}
443:are quotients of
394:fundamental group
235:fundamental group
122:topological space
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16:(Redirected from
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2538:Fake lens spaces
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2204:PL homeomorphism
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1502:Lens spaces are
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2374:10.1023/A:10240
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2340:Allen Hatcher,
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49:related reading
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2520:External links
2518:
2517:
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2495:
2492:John Stillwell
2475:
2454:
2438:
2419:(2): 375–380,
2408:
2399:(1): 102–109,
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2352:
2338:
2322:
2305:(1): 163–184,
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2230:knot-theoretic
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2115:
2114:
2101:
2098:
2094:
2091:
2084:
2081:
2076:
2072:
2068:
2065:
2060:
2056:
2044:
2031:
2027:
2024:
2003:
2000:
1996:
1993:
1986:
1982:
1978:
1975:
1970:
1966:
1960:
1956:
1931:
1926:
1922:
1918:
1915:
1912:
1909:
1889:
1884:
1880:
1876:
1873:
1870:
1867:
1855:
1852:
1804:
1801:
1798:
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1789:
1767:
1764:
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1755:
1751:
1728:
1725:
1722:
1718:
1695:
1692:
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1685:
1662:
1658:
1635:
1632:
1629:
1625:
1602:
1598:
1573:
1570:
1567:
1564:
1561:
1558:
1546:
1543:
1530:
1527:
1524:
1521:
1518:
1515:
1486:
1482:
1460:
1456:
1452:
1447:
1426:
1421:
1417:
1413:
1410:
1407:
1402:
1398:
1394:
1391:
1388:
1385:
1373:
1370:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
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1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1302:
1301:
1290:
1287:
1282:
1278:
1274:
1269:
1265:
1259:
1255:
1251:
1248:
1245:
1241:
1237:
1234:
1231:
1226:
1222:
1218:
1213:
1209:
1203:
1199:
1195:
1192:
1189:
1185:
1181:
1178:
1175:
1170:
1166:
1162:
1159:
1156:
1151:
1147:
1143:
1120:
1116:
1111:
1088:
1085:
1082:
1079:
1075:
1054:
1049:
1045:
1041:
1038:
1033:
1029:
1025:
1022:
1019:
1016:
994:
989:
965:
962:
959:
956:
952:
931:
909:
905:
882:
878:
874:
871:
868:
863:
859:
855:
852:
829:
826:
823:
820:
817:
814:
802:is called the
800:quotient space
796:
795:
784:
779:
775:
771:
766:
762:
758:
755:
752:
749:
745:
741:
736:
732:
728:
723:
719:
715:
712:
709:
705:
701:
698:
695:
690:
686:
682:
677:
673:
669:
644:
640:
619:
615:
610:
587:
582:
558:
554:
529:
509:
489:
485:
480:
457:
453:
432:
429:
426:
423:
420:
417:
405:
402:
370:
367:
364:
361:
358:
355:
335:
332:
329:
326:
323:
320:
300:
297:
294:
291:
288:
285:
265:
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259:
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250:
214:
211:
208:
205:
202:
199:
174:
170:
166:
161:
157:
139:together by a
96:
95:
53:external links
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2580:
2569:
2566:
2564:
2561:
2560:
2558:
2549:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2523:
2519:
2514:on 2006-09-25
2510:
2503:
2502:
2496:
2493:
2489:
2465:
2463:
2458:
2455:
2452:
2451:
2446:
2442:
2439:
2436:
2432:
2427:
2422:
2418:
2414:
2409:
2406:
2402:
2398:
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2390:
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2375:
2371:
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2363:
2362:
2357:
2353:
2350:
2346:
2344:
2339:
2336:
2333:
2331:
2326:
2325:Allen Hatcher
2323:
2320:
2316:
2312:
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2304:
2300:
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2293:
2290:
2286:
2282:
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2250:
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2220:
2216:
2214:
2209:
2205:
2201:
2197:
2192:
2190:
2185:
2167:
2163:
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2150:
2145:
2141:
2137:
2134:
2129:
2125:
2096:
2092:
2082:
2079:
2074:
2070:
2066:
2063:
2058:
2054:
2045:
2025:
2022:
1998:
1994:
1984:
1980:
1976:
1973:
1968:
1964:
1958:
1954:
1945:
1944:
1943:
1924:
1920:
1916:
1913:
1907:
1882:
1878:
1874:
1871:
1865:
1853:
1851:
1849:
1845:
1841:
1837:
1833:
1829:
1825:
1821:
1816:
1799:
1796:
1793:
1787:
1765:
1762:
1759:
1756:
1753:
1749:
1726:
1723:
1720:
1716:
1693:
1690:
1687:
1683:
1660:
1656:
1633:
1630:
1627:
1623:
1600:
1596:
1587:
1568:
1565:
1562:
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1544:
1542:
1525:
1522:
1519:
1513:
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1500:
1484:
1480:
1454:
1450:
1419:
1415:
1411:
1408:
1405:
1400:
1396:
1392:
1389:
1383:
1371:
1369:
1356:
1350:
1347:
1344:
1341:
1338:
1332:
1329:
1323:
1320:
1317:
1311:
1288:
1280:
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1272:
1267:
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1235:
1232:
1229:
1224:
1220:
1216:
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1190:
1187:
1183:
1168:
1164:
1160:
1157:
1154:
1149:
1145:
1134:
1133:
1132:
1118:
1114:
1086:
1083:
1080:
1077:
1073:
1047:
1043:
1039:
1036:
1031:
1027:
1023:
1020:
1014:
992:
963:
960:
957:
954:
950:
942:and consider
929:
907:
903:
880:
876:
872:
869:
866:
861:
857:
853:
850:
841:
824:
821:
818:
812:
805:
801:
777:
773:
769:
764:
760:
756:
753:
750:
747:
743:
739:
734:
730:
726:
721:
717:
713:
710:
707:
703:
688:
684:
680:
675:
671:
660:
659:
658:
642:
638:
617:
613:
585:
556:
552:
543:
527:
507:
487:
483:
455:
451:
427:
424:
421:
415:
403:
401:
399:
395:
390:
388:
384:
365:
362:
359:
353:
330:
327:
324:
318:
295:
292:
289:
283:
260:
257:
254:
248:
240:
236:
232:
228:
209:
206:
203:
197:
188:
172:
168:
164:
159:
155:
146:
142:
141:homeomorphism
138:
133:
131:
127:
123:
119:
110:
102:
92:
89:
81:
78:February 2012
71:
67:
61:
60:
54:
50:
46:
41:
32:
31:
19:
2509:the original
2500:
2460:
2448:
2426:math/0401075
2416:
2412:
2396:
2392:
2368:(1): 57–61,
2365:
2359:
2348:
2341:
2328:
2302:
2296:
2284:
2246:
2237:
2227:
2212:
2193:
2186:
2116:
1857:
1847:
1843:
1839:
1835:
1831:
1823:
1817:
1585:
1548:
1501:
1375:
1303:
1101:by the free
842:
803:
797:
600:. Then the
407:
391:
189:
134:
117:
115:
84:
75:
64:Please help
56:
2563:3-manifolds
2526:Lens spaces
2287:, Springer
2281:Glen Bredon
630:-action on
130:3-manifolds
126:mathematics
70:introducing
18:Lens spaces
2557:Categories
2544:lens space
2490:(2008) by
2291:139, 1993.
2275:References
2208:Brody 1960
1372:Properties
804:lens space
404:Definition
137:solid tori
118:lens space
2568:Manifolds
2474:§
2151:±
2138:±
2135:≡
2080:±
2067:±
2064:≡
2026:∈
2015:for some
1977:±
1974:≡
1820:bipyramid
1631:−
1409:…
1273:⋅
1247:π
1233:…
1217:⋅
1191:π
1177:↦
1158:…
1084:−
1040:…
961:−
870:…
770:⋅
751:π
727:⋅
711:π
697:↦
165:×
2413:Topology
2263:See also
2215:homotopy
231:homology
145:3-sphere
2382:1988423
2337:, 2002.
2319:1969884
2236:): let
1828:polygon
542:coprime
66:improve
2380:
2349:L(p,q)
2317:
2213:simple
1826:sided
2512:(PDF)
2505:(PDF)
2421:arXiv
2315:JSTOR
2301:, 2,
2221:) or
1942:are:
1741:with
1675:with
51:, or
2548:nLab
2486:20)
2443:and
1900:and
1846:and
1834:and
1708:and
520:and
396:and
346:and
276:and
233:and
147:and
2546:on
2431:doi
2401:doi
2370:doi
2307:doi
2251:– (
2164:mod
2117:If
2093:mod
1995:mod
1615:to
1437:is
540:be
470:by
2559::
2459:,
2447:,
2429:,
2417:44
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2397:11
2395:,
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2259:.
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400:.
389:.
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2464:,
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2345:.
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2309::
2238:C
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1959:1
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