401:, a theorem which had also been conjectured first by Hopf. One of the lines of attacks is by looking for manifolds with more symmetry. It is particular for example that all known manifolds of positive sectional curvature allow for an isometric circle action. The corresponding vector field is called a
1826:
There is a conjecture which relates to the Hopf sign conjecture but which does not refer to
Riemannian geometry at all. Aspherical manifolds are connected manifolds for which all higher homotopy groups disappear. The Euler characteristic then should satisfy the same condition as a negatively curved
1817:
Finally, one can ask why one would be interested in such a special case like the Hopf product conjecture. Hopf himself was motivated by problems from physics. When Hopf started to work in the mid 1920s, the theory of relativity was only 10 years old and it sparked a great deal of interest in
569:
discusses the conjecture in his book, and points to the work of Hopf from the 1920s which was influenced by such type of questions. The conjectures are listed as problem 8 (positive curvature case) and 10 (negative curvature case) in ``Yau's problems" of 1982.
1414:
are the only simply-connected 4-manifolds which are known to admit a metric of positive curvature. Wolfgang Ziller once conjectured this might be the full list and that in dimension 5, the only simply-connected 5-manifold of positive curvature is the 5-sphere
2499:
1721:
with positive sectional curvature. So, when looking at the evidence and the work done so far, it appears that the Hopf question most likely will be answered as the statement "There is no metric of positive curvature on
2001:
544:
714:
1812:
1764:
1715:
1599:
1547:
1332:
1659:
is not possible as then the manifold has to be a sphere.) A general reference for manifolds with non-negative sectional curvature giving many examples is as well as. A related conjecture is that
840:
223:
1898:
2310:
Jean-Claude
Hausmann, Geometric Hopfian and non-Hopfian situations. Geometry and topology (Athens, Ga., 1985), 157–166, Lecture Notes in Pure and Appl. Math., 105, Dekker, New York, 1987
1976:(a PhD student of Morse) from a year earlier. The problem to generalize this to higher dimensions was for some time known as the Hopf conjecture too. In any case, this is now a theorem:
1904:
There can not be a direct relation to the
Riemannian case as there are aspherical manifolds that are not homeomorphic to a smooth Riemannian manifold with negative sectional curvature.
1927:
There had been a bit of confusion about the word "Hopf conjecture" as an unrelated mathematician
Eberhard Hopf and contemporary of Heinz Hopf worked on topics like geodesic flows. (
888:
768:
395:
357:
2438:
2090:
1503:
1412:
578:
There are analogue conjectures if the curvature is allowed to become zero too. The statement should still be attributed to Hopf (for example in a talk given in 1953 in Italy).
1966:
1657:
1628:
1471:
1442:
1380:
2353:
Wolfgang Ziller, Riemannian
Manifolds with Positive Sectional Curvature, Lecture given in Guanajuato of 2010 in: Geometry of Manifolds with Non-negative Sectional Curvature,
1164:
1059:
1024:
956:
277:
2519:
2278:
1121:
989:
921:
2139:
2039:
1272:
1191:
641:
111:
1348:
One does not know any example of a compact, simply-connected manifold of nonnegative sectional curvature which does not admit a metric of strictly positive curvature.
319:
1245:
1086:
461:
432:
2298:
1211:
1339:
The conjecture was popularized in the book of
Gromoll, Klingenberg and Meyer from 1968, and was prominently displayed as Problem 1 in Yau's list of problems.
2480:
Wolfgang Ziller, Examples of
Riemannian manifolds with non-negative sectional curvature, Surv. Differ. Geom., 11, pages 63-102, International Press, 2007
1766:" because so far, the theorems of Bourguignon (perturbation result near product metric), Hopf (codimension 1), Weinstein (codimension 2) as well as the
2588:
1968:
has no conjugate points, then it must be flat (the Gauss curvature is zero everywhere). The theorem of
Eberhard Hopf generalized a theorem of
493:
280:
1818:
differential geometry, especially in global structure of 4-manifolds, as such manifolds appear in cosmology as models of the universe.
2536:
2489:
C. Escher and W. Ziller, Topology of non-negatively curved manifolds", Annals of Global
Analysis and Geometry, 46, pages 23-55, 2014
2243:
2208:
119:
654:
2354:
2332:
2200:
2444:
2268:, Sulla geometria riemanniana globale della superficie, Rendiconti del Seminario matematico e fisico di Milano, 1953, pages 48-63
2096:
1131:
Hopf asked whether every continuous self-map of an oriented closed manifold of degree 1 is necessarily a homotopy equivalence.
397:. For sufficiently pinched positive curvature manifolds, the Hopf conjecture (in the positive curvature case) follows from the
2573:
1773:
1770:
excluding pinched positive curvature metrics, point towards this outcome. The construction of a positive curvature metric on
1725:
1676:
1560:
1508:
1293:
773:
2375:
1553:
that in the neighborhood of the product metric, there is no metric of positive curvature. It is also known from work of
1505:
are the only simply connected positive curvature 4-manifolds would settle the Hopf product conjecture. Back to the case
2133:
2173:
550:
1814:
would certainly be a surprise in global differential geometry, but it is not excluded yet that such a metric exists.
549:
On the history of the problem: the first written explicit appearance of the conjecture is in the proceedings of the
2235:
1916:
149:
1839:
1444:. Of course, solving the Hopf product conjecture would settle the Yau question. Also the Ziller conjecture that
123:
2367:
1550:
139:
845:
2155:
1919:
conjectured that the same inequality holds for a non-positively curved piecewise
Euclidean (PE) manifold.
719:
362:
324:
2408:
2060:
21:
1476:
1385:
1942:
1633:
1604:
1447:
1418:
1356:
2583:
604:
592:
402:
74:
58:
1137:
2324:
1833:
1029:
994:
926:
600:
596:
588:
584:
405:. The conjecture (for the positive curvature case) has also been proved for manifolds of dimension
236:
135:
70:
66:
54:
50:
40:
The Hopf conjecture is an open problem in global Riemannian geometry. It goes back to questions of
1091:
961:
893:
226:
1343:
formulated there an interesting new observation (which could be reformulated as a conjecture).
2578:
2515:
2290:
2239:
2204:
1996:
1973:
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131:
80:
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2010:
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R.L. Bishop and S.I. Goldberg, Some implications on the generalized Gauss-Bonnet theorem,
2249:
2115:
2018:
1668:
of rank greater than one cannot carry a Riemannian metric of positive sectional curvature.
1665:
437:
408:
648:
This version was stated as such as Question 1 in the paper or then in a paper of Chern.
291:
in 1955.). For manifolds of dimension 6 or higher the conjecture is open. An example of
2396:
2320:
2227:
2052:
1936:
1767:
1554:
1340:
1213:
factors through a non-trivial covering space, contradicting the degree-1 assumption.
1196:
398:
2567:
2511:
2192:
2035:
1969:
1928:
1601:
exists with positive curvature, then this Riemannian manifold can not be embedded in
1217:
566:
292:
2370:(1975), "Some constructions related to H. Hopf's conjecture on product manifolds",
1912:
464:
230:
2038:, Positive sectional curvatures does not imply positive Gauss-Bonnet integrand,
1978:
A Riemannian metric without conjugate points on the n-dimensional torus is flat.
558:
284:
2500:
Proceedings of the National Academy of Sciences of the United States of America
2374:, Proceedings of Symposia in Pure Mathematics, vol. 27, Providence, R.I.:
2265:
2168:
2135:
The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank
1932:
562:
143:
127:
41:
29:
2458:
2400:
2110:
2056:
1822:
Thurston conjecture on aspherical manifolds (extension of Hopf's conjecture)
553:, which is a paper based on talks, Heinz Hopf gave in the spring of 1931 in
476:
321:. The positive curvature case is known to hold however for hypersurfaces in
1935:
are unrelated and might never have met even so they were both students of
1836:
of even dimension. Then its Euler characteristic satisfies the inequality
1999:(1966). "On curvature and characteristic classes of a Riemann manifold".
554:
25:
539:{\displaystyle k-(\operatorname {rank} G-\operatorname {rank} H)\leq 5.}
295:
had shown that the Chern–Gauss–Bonnet integrand can become negative for
2547:
2014:
229:, assuring that the orientation cover is simply connected, so that the
651:
An example for which the conjecture is confirmed is for the product
2002:
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
1939:). There is a theorem of Eberhard Hopf stating that if the 2-torus
2234:, Annals of Mathematics Studies, vol. 102, Princeton, N.J.:
1630:. (It follows already from a result of Hopf that an embedding in
1285:
Another famous question of Hopf is the Hopf product conjecture:
2331:. Lecture Notes in Mathematics. Vol. 55. Berlin-New York:
709:{\displaystyle M=M_{1}\times M_{2}\times \cdots \times M_{d}}
574:
Non-negatively or non-positively curved Riemannian manifolds
546:
Some references about manifolds with some symmetry are and
1827:
manifold is conjectured to satisfy in Riemannian geometry:
2171:(1932), "Differentialgeometry und topologische Gestalt",
1923:(Unrelated:) Riemannian metrics with no conjugate points
20:
may refer to one of several conjectural statements from
1807:{\displaystyle \mathbb {S} ^{2}\times \mathbb {S} ^{2}}
1759:{\displaystyle \mathbb {S} ^{2}\times \mathbb {S} ^{2}}
1710:{\displaystyle \mathbb {S} ^{2}\times \mathbb {S} ^{2}}
1594:{\displaystyle \mathbb {S} ^{2}\times \mathbb {S} ^{2}}
1542:{\displaystyle \mathbb {S} ^{2}\times \mathbb {S} ^{2}}
1327:{\displaystyle \mathbb {S} ^{2}\times \mathbb {S} ^{2}}
279:. For 4-manifolds, the statement also follows from the
1907:
This topological version of Hopf conjecture is due to
2411:
2063:
1945:
1842:
1776:
1728:
1679:
1636:
1607:
1563:
1511:
1479:
1450:
1421:
1388:
1359:
1296:
1253:
1226:
1199:
1172:
1140:
1094:
1067:
1032:
997:
964:
929:
896:
848:
835:{\displaystyle \chi (M)=\prod _{k=1}^{d}\chi (M_{k})}
776:
722:
657:
613:
496:
440:
411:
365:
327:
301:
239:
152:
83:
2174:
Jahresbericht der Deutschen Mathematiker-Vereinigung
890:, the sign conjecture is confirmed in that case (if
36:
Positively or negatively curved Riemannian manifolds
2498:E. Hopf, Closed Surfaces without conjugate points,
2154:L. Kennard, "On the Hopf conjecture with symmetry,
2432:
2084:
1960:
1892:
1806:
1758:
1709:
1651:
1622:
1593:
1541:
1497:
1465:
1436:
1406:
1374:
1326:
1266:
1239:
1205:
1185:
1158:
1115:
1080:
1053:
1018:
983:
950:
915:
882:
834:
762:
708:
635:
538:
455:
426:
389:
351:
313:
271:
217:
105:
2533:Riemannian tori without conjugate points are flat
2520:Transactions of the American Mathematical Society
2279:Transactions of the American Mathematical Society
1281:Product conjecture for the product of two spheres
2140:Proceedings of the American Mathematical Society
2040:Proceedings of the American Mathematical Society
716:of 2-dimensional manifolds with curvature sign
359:(Hopf) or codimension two surfaces embedded in
2299:Bulletin of the American Mathematical Society
1277:There are, however, some non-Hopfian groups.
218:{\displaystyle b_{0}-b_{1}+b_{2}-b_{3}+b_{4}}
8:
1893:{\displaystyle (-1)^{k}\chi (M^{2k})\geq 0.}
757:
736:
1216:This implies that the conjecture holds for
475:admitting an isometric action of a compact
130:, this follows from the finiteness of the
2457:
2418:
2414:
2413:
2410:
2109:
2070:
2066:
2065:
2062:
1952:
1948:
1947:
1944:
1872:
1856:
1841:
1798:
1794:
1793:
1783:
1779:
1778:
1775:
1750:
1746:
1745:
1735:
1731:
1730:
1727:
1701:
1697:
1696:
1686:
1682:
1681:
1678:
1643:
1639:
1638:
1635:
1614:
1610:
1609:
1606:
1585:
1581:
1580:
1570:
1566:
1565:
1562:
1533:
1529:
1528:
1518:
1514:
1513:
1510:
1489:
1485:
1482:
1481:
1478:
1457:
1453:
1452:
1449:
1428:
1424:
1423:
1420:
1398:
1394:
1391:
1390:
1387:
1366:
1362:
1361:
1358:
1318:
1314:
1313:
1303:
1299:
1298:
1295:
1258:
1252:
1231:
1225:
1198:
1177:
1171:
1139:
1093:
1072:
1066:
1031:
996:
969:
963:
928:
901:
895:
874:
864:
853:
847:
823:
807:
796:
775:
770:. As the Euler characteristic satisfies
727:
721:
700:
681:
668:
656:
627:
612:
495:
439:
410:
372:
368:
367:
364:
334:
330:
329:
326:
300:
257:
244:
238:
209:
196:
183:
170:
157:
151:
97:
82:
2197:A panoramic view of Riemannian geometry
1988:
1334:carry a metric with positive curvature?
2518:, Manifolds without conjugate points,
2132:Thomas PĂĽttmann and Catherine Searle,
2222:
2220:
883:{\displaystyle \prod _{k=1}^{d}e_{k}}
471:-dimensional torus and for manifolds
7:
1166:of degree 1 induces a surjection on
44:from 1931. A modern formulation is:
763:{\displaystyle e_{k}\in \{-1,0,1\}}
390:{\displaystyle \mathbb {R} ^{2d+2}}
352:{\displaystyle \mathbb {R} ^{2d+1}}
122:, these statements follow from the
2433:{\displaystyle \mathbb {R} ^{n+2}}
2085:{\displaystyle \mathbb {R} ^{n+2}}
2057:"Positively curved n-manifolds in
1274:and thus a homotopy equivalence.
14:
2537:Geometric and Functional Analysis
2531:Dmitri Burago and Sergei Ivanov,
1498:{\displaystyle \mathbb {CP} ^{2}}
1407:{\displaystyle \mathbb {CP} ^{2}}
1382:and the complex projective plane
1220:, as for them one then gets that
482:with principal isotropy subgroup
2445:Journal of Differential Geometry
2329:Riemannsche Geometrie im Grossen
2232:Seminar on Differential Geometry
2097:Journal of Differential Geometry
1961:{\displaystyle \mathbb {T} ^{2}}
1652:{\displaystyle \mathbb {R} ^{5}}
1623:{\displaystyle \mathbb {R} ^{6}}
1466:{\displaystyle \mathbb {S} ^{4}}
1437:{\displaystyle \mathbb {S} ^{5}}
1375:{\displaystyle \mathbb {S} ^{4}}
1134:It is easy to see that any map
1881:
1865:
1853:
1843:
1159:{\displaystyle f\colon M\to M}
1150:
1104:
1098:
1042:
1036:
1007:
1001:
939:
933:
829:
816:
786:
780:
624:
614:
595:. A compact, (2d)-dimensional
527:
503:
146:the Euler characteristic with
94:
84:
1:
2589:Unsolved problems in geometry
2376:American Mathematical Society
1557:that if a metric is given on
1061:for odd d, and if one of the
1054:{\displaystyle \chi (M)<0}
1019:{\displaystyle \chi (M)>0}
951:{\displaystyle \chi (M)>0}
272:{\displaystyle b_{1}=b_{3}=0}
583:A compact, even-dimensional
49:A compact, even-dimensional
2230:(1982), "Problem section",
1821:
1673:This would also imply that
1549:: it is known from work of
551:German Mathematical Society
2605:
2327:; Meyer, Wolfgang (1968).
2301:, 72, pages 167-2019, 1966
2281:, 112, pages 508-545, 1964
2236:Princeton University Press
1116:{\displaystyle \chi (M)=0}
984:{\displaystyle e_{k}<0}
916:{\displaystyle e_{k}>0}
281:Chern–Gauss–Bonnet theorem
128:four-dimensional manifolds
2522:, 51, pages 362-386, 1942
2158:, 17, 2013, pages 563-593
1353:At present, the 4-sphere
287:in 1955 (written down by
2542:(1994), no. 3, 259-269,
2368:Bourguignon, Jean-Pierre
1267:{\displaystyle \pi _{1}}
1186:{\displaystyle \pi _{1}}
636:{\displaystyle (-1)^{d}}
106:{\displaystyle (-1)^{d}}
2502:, 34, page 47-51 (1948)
2156:Geometry & Topology
2145:(2001), no. 1, 163-166.
1832:Suppose M is a closed,
1551:Jean-Pierre Bourguignon
463:admitting an isometric
2459:10.4310/jdg/1214429270
2434:
2111:10.4310/jdg/1214429270
2086:
1962:
1894:
1808:
1760:
1711:
1653:
1624:
1595:
1543:
1499:
1467:
1438:
1408:
1376:
1328:
1268:
1241:
1207:
1187:
1160:
1117:
1082:
1055:
1020:
985:
952:
917:
884:
869:
836:
812:
764:
710:
637:
540:
457:
428:
391:
353:
315:
314:{\displaystyle d>2}
273:
219:
140:Euler–Poincaré formula
107:
2574:Differential geometry
2435:
2372:Differential Geometry
2087:
1963:
1895:
1809:
1761:
1712:
1654:
1625:
1596:
1544:
1500:
1468:
1439:
1409:
1377:
1329:
1269:
1247:is an isomorphism on
1242:
1240:{\displaystyle f_{*}}
1208:
1188:
1161:
1127:Self-maps of degree 1
1118:
1083:
1081:{\displaystyle e_{k}}
1056:
1021:
986:
953:
918:
885:
849:
837:
792:
765:
711:
638:
565:in the fall of 1931.
541:
458:
429:
392:
354:
316:
274:
220:
108:
22:differential geometry
2409:
2325:Klingenberg, Wilhelm
2238:, pp. 669–706,
2061:
1943:
1840:
1774:
1726:
1677:
1634:
1605:
1561:
1509:
1477:
1448:
1419:
1386:
1357:
1294:
1251:
1224:
1197:
1170:
1138:
1092:
1065:
1030:
995:
962:
927:
894:
846:
842:which has the sign
774:
720:
655:
611:
605:Euler characteristic
593:Euler characteristic
494:
456:{\displaystyle 4k+4}
438:
427:{\displaystyle 4k+2}
409:
403:killing vector field
363:
325:
299:
237:
150:
124:Gauss–Bonnet theorem
81:
75:Euler characteristic
59:Euler characteristic
2401:"Positively curved
1834:aspherical manifold
1290:Can the 4-manifold
601:sectional curvature
597:Riemannian manifold
589:sectional curvature
585:Riemannian manifold
71:sectional curvature
67:Riemannian manifold
55:sectional curvature
51:Riemannian manifold
2548:10.1007/BF01896241
2430:
2378:, pp. 33–37,
2293:, The geometry of
2082:
2015:10.1007/BF02960745
1997:Chern, Shiing-Shen
1958:
1890:
1804:
1756:
1707:
1649:
1620:
1591:
1539:
1495:
1463:
1434:
1404:
1372:
1324:
1264:
1237:
1203:
1183:
1156:
1113:
1078:
1051:
1016:
981:
948:
913:
880:
832:
760:
706:
633:
599:with non-positive
587:with non-negative
536:
486:and cohomogeneity
453:
424:
387:
349:
311:
269:
215:
103:
2516:Gustav A. Hedlund
2291:Shiing-Shen Chern
1719:Riemannian metric
1206:{\displaystyle f}
591:has non-negative
289:Shiing-Shen Chern
132:fundamental group
2596:
2558:
2529:
2523:
2509:
2503:
2496:
2490:
2487:
2481:
2478:
2472:
2471:
2461:
2439:
2437:
2436:
2431:
2429:
2428:
2417:
2393:
2387:
2386:
2364:
2358:
2351:
2345:
2344:
2317:
2311:
2308:
2302:
2288:
2282:
2275:
2269:
2263:
2257:
2256:
2224:
2215:
2214:
2189:
2183:
2182:
2165:
2159:
2152:
2146:
2130:
2124:
2123:
2113:
2091:
2089:
2088:
2083:
2081:
2080:
2069:
2049:
2043:
2033:
2027:
2026:
1993:
1967:
1965:
1964:
1959:
1957:
1956:
1951:
1909:William Thurston
1899:
1897:
1896:
1891:
1880:
1879:
1861:
1860:
1813:
1811:
1810:
1805:
1803:
1802:
1797:
1788:
1787:
1782:
1765:
1763:
1762:
1757:
1755:
1754:
1749:
1740:
1739:
1734:
1716:
1714:
1713:
1708:
1706:
1705:
1700:
1691:
1690:
1685:
1658:
1656:
1655:
1650:
1648:
1647:
1642:
1629:
1627:
1626:
1621:
1619:
1618:
1613:
1600:
1598:
1597:
1592:
1590:
1589:
1584:
1575:
1574:
1569:
1548:
1546:
1545:
1540:
1538:
1537:
1532:
1523:
1522:
1517:
1504:
1502:
1501:
1496:
1494:
1493:
1488:
1472:
1470:
1469:
1464:
1462:
1461:
1456:
1443:
1441:
1440:
1435:
1433:
1432:
1427:
1413:
1411:
1410:
1405:
1403:
1402:
1397:
1381:
1379:
1378:
1373:
1371:
1370:
1365:
1333:
1331:
1330:
1325:
1323:
1322:
1317:
1308:
1307:
1302:
1273:
1271:
1270:
1265:
1263:
1262:
1246:
1244:
1243:
1238:
1236:
1235:
1212:
1210:
1209:
1204:
1192:
1190:
1189:
1184:
1182:
1181:
1165:
1163:
1162:
1157:
1122:
1120:
1119:
1114:
1087:
1085:
1084:
1079:
1077:
1076:
1060:
1058:
1057:
1052:
1025:
1023:
1022:
1017:
991:for all k, then
990:
988:
987:
982:
974:
973:
957:
955:
954:
949:
923:for all k, then
922:
920:
919:
914:
906:
905:
889:
887:
886:
881:
879:
878:
868:
863:
841:
839:
838:
833:
828:
827:
811:
806:
769:
767:
766:
761:
732:
731:
715:
713:
712:
707:
705:
704:
686:
685:
673:
672:
642:
640:
639:
634:
632:
631:
545:
543:
542:
537:
462:
460:
459:
454:
433:
431:
430:
425:
396:
394:
393:
388:
386:
385:
371:
358:
356:
355:
350:
348:
347:
333:
320:
318:
317:
312:
278:
276:
275:
270:
262:
261:
249:
248:
224:
222:
221:
216:
214:
213:
201:
200:
188:
187:
175:
174:
162:
161:
136:Poincaré duality
112:
110:
109:
104:
102:
101:
16:In mathematics,
2604:
2603:
2599:
2598:
2597:
2595:
2594:
2593:
2564:
2563:
2562:
2561:
2530:
2526:
2510:
2506:
2497:
2493:
2488:
2484:
2479:
2475:
2412:
2407:
2406:
2397:Weinstein, Alan
2395:
2394:
2390:
2366:
2365:
2361:
2352:
2348:
2333:Springer Verlag
2321:Gromoll, Detlef
2319:
2318:
2314:
2309:
2305:
2289:
2285:
2276:
2272:
2264:
2260:
2246:
2228:Yau, Shing-Tung
2226:
2225:
2218:
2211:
2191:
2190:
2186:
2167:
2166:
2162:
2153:
2149:
2131:
2127:
2064:
2059:
2058:
2053:Weinstein, Alan
2051:
2050:
2046:
2034:
2030:
1995:
1994:
1990:
1985:
1946:
1941:
1940:
1925:
1868:
1852:
1838:
1837:
1824:
1792:
1777:
1772:
1771:
1744:
1729:
1724:
1723:
1695:
1680:
1675:
1674:
1666:symmetric space
1637:
1632:
1631:
1608:
1603:
1602:
1579:
1564:
1559:
1558:
1527:
1512:
1507:
1506:
1480:
1475:
1474:
1451:
1446:
1445:
1422:
1417:
1416:
1389:
1384:
1383:
1360:
1355:
1354:
1312:
1297:
1292:
1291:
1283:
1254:
1249:
1248:
1227:
1222:
1221:
1195:
1194:
1193:; if not, then
1173:
1168:
1167:
1136:
1135:
1129:
1090:
1089:
1068:
1063:
1062:
1028:
1027:
1026:for even d and
993:
992:
965:
960:
959:
925:
924:
897:
892:
891:
870:
844:
843:
819:
772:
771:
723:
718:
717:
696:
677:
664:
653:
652:
623:
609:
608:
576:
492:
491:
436:
435:
407:
406:
366:
361:
360:
328:
323:
322:
297:
296:
253:
240:
235:
234:
227:Synge's theorem
205:
192:
179:
166:
153:
148:
147:
93:
79:
78:
61:. A compact, (2
38:
18:Hopf conjecture
12:
11:
5:
2602:
2600:
2592:
2591:
2586:
2581:
2576:
2566:
2565:
2560:
2559:
2524:
2504:
2491:
2482:
2473:
2427:
2424:
2421:
2416:
2405:-manifolds in
2388:
2359:
2346:
2312:
2303:
2283:
2270:
2258:
2244:
2216:
2209:
2193:Berger, Marcel
2184:
2160:
2147:
2125:
2079:
2076:
2073:
2068:
2044:
2028:
1987:
1986:
1984:
1981:
1974:Gustav Hedlund
1955:
1950:
1937:Erhard Schmidt
1924:
1921:
1902:
1901:
1889:
1886:
1883:
1878:
1875:
1871:
1867:
1864:
1859:
1855:
1851:
1848:
1845:
1823:
1820:
1801:
1796:
1791:
1786:
1781:
1768:sphere theorem
1753:
1748:
1743:
1738:
1733:
1704:
1699:
1694:
1689:
1684:
1671:
1670:
1646:
1641:
1617:
1612:
1588:
1583:
1578:
1573:
1568:
1555:Alan Weinstein
1536:
1531:
1526:
1521:
1516:
1492:
1487:
1484:
1460:
1455:
1431:
1426:
1401:
1396:
1393:
1369:
1364:
1351:
1350:
1341:Shing-Tung Yau
1337:
1336:
1321:
1316:
1311:
1306:
1301:
1282:
1279:
1261:
1257:
1234:
1230:
1218:Hopfian groups
1202:
1180:
1176:
1155:
1152:
1149:
1146:
1143:
1128:
1125:
1112:
1109:
1106:
1103:
1100:
1097:
1088:is zero, then
1075:
1071:
1050:
1047:
1044:
1041:
1038:
1035:
1015:
1012:
1009:
1006:
1003:
1000:
980:
977:
972:
968:
947:
944:
941:
938:
935:
932:
912:
909:
904:
900:
877:
873:
867:
862:
859:
856:
852:
831:
826:
822:
818:
815:
810:
805:
802:
799:
795:
791:
788:
785:
782:
779:
759:
756:
753:
750:
747:
744:
741:
738:
735:
730:
726:
703:
699:
695:
692:
689:
684:
680:
676:
671:
667:
663:
660:
646:
645:
630:
626:
622:
619:
616:
575:
572:
535:
532:
529:
526:
523:
520:
517:
514:
511:
508:
505:
502:
499:
452:
449:
446:
443:
423:
420:
417:
414:
399:sphere theorem
384:
381:
378:
375:
370:
346:
343:
340:
337:
332:
310:
307:
304:
283:as noticed by
268:
265:
260:
256:
252:
247:
243:
212:
208:
204:
199:
195:
191:
186:
182:
178:
173:
169:
165:
160:
156:
116:
115:
100:
96:
92:
89:
86:
69:with negative
65:)-dimensional
53:with positive
37:
34:
28:attributed to
13:
10:
9:
6:
4:
3:
2:
2601:
2590:
2587:
2585:
2582:
2580:
2577:
2575:
2572:
2571:
2569:
2556:
2553:
2549:
2545:
2541:
2538:
2534:
2528:
2525:
2521:
2517:
2513:
2512:Marston Morse
2508:
2505:
2501:
2495:
2492:
2486:
2483:
2477:
2474:
2469:
2465:
2460:
2455:
2451:
2447:
2446:
2441:
2425:
2422:
2419:
2404:
2398:
2392:
2389:
2385:
2381:
2377:
2373:
2369:
2363:
2360:
2356:
2350:
2347:
2342:
2338:
2334:
2330:
2326:
2322:
2316:
2313:
2307:
2304:
2300:
2297:-structures,
2296:
2292:
2287:
2284:
2280:
2274:
2271:
2267:
2262:
2259:
2255:
2251:
2247:
2245:0-691-08268-5
2241:
2237:
2233:
2229:
2223:
2221:
2217:
2212:
2210:3-540-65317-1
2206:
2202:
2198:
2194:
2188:
2185:
2180:
2176:
2175:
2170:
2164:
2161:
2157:
2151:
2148:
2144:
2141:
2137:
2136:
2129:
2126:
2121:
2117:
2112:
2107:
2103:
2099:
2098:
2093:
2077:
2074:
2071:
2054:
2048:
2045:
2041:
2037:
2036:Robert Geroch
2032:
2029:
2024:
2020:
2016:
2012:
2008:
2004:
2003:
1998:
1992:
1989:
1982:
1980:
1979:
1975:
1971:
1970:Marston Morse
1953:
1938:
1934:
1930:
1929:Eberhard Hopf
1922:
1920:
1918:
1917:Michael Davis
1914:
1910:
1905:
1900:
1887:
1884:
1876:
1873:
1869:
1862:
1857:
1849:
1846:
1835:
1830:
1829:
1828:
1819:
1815:
1799:
1789:
1784:
1769:
1751:
1741:
1736:
1720:
1702:
1692:
1687:
1669:
1667:
1662:
1661:
1660:
1644:
1615:
1586:
1576:
1571:
1556:
1552:
1534:
1524:
1519:
1490:
1458:
1429:
1399:
1367:
1349:
1346:
1345:
1344:
1342:
1335:
1319:
1309:
1304:
1288:
1287:
1286:
1280:
1278:
1275:
1259:
1255:
1232:
1228:
1219:
1214:
1200:
1178:
1174:
1153:
1147:
1144:
1141:
1132:
1126:
1124:
1110:
1107:
1101:
1095:
1073:
1069:
1048:
1045:
1039:
1033:
1013:
1010:
1004:
998:
978:
975:
970:
966:
945:
942:
936:
930:
910:
907:
902:
898:
875:
871:
865:
860:
857:
854:
850:
824:
820:
813:
808:
803:
800:
797:
793:
789:
783:
777:
754:
751:
748:
745:
742:
739:
733:
728:
724:
701:
697:
693:
690:
687:
682:
678:
674:
669:
665:
661:
658:
649:
644:
628:
620:
617:
606:
602:
598:
594:
590:
586:
581:
580:
579:
573:
571:
568:
567:Marcel Berger
564:
560:
556:
552:
547:
533:
530:
524:
521:
518:
515:
512:
509:
506:
500:
497:
489:
485:
481:
478:
474:
470:
466:
450:
447:
444:
441:
421:
418:
415:
412:
404:
400:
382:
379:
376:
373:
344:
341:
338:
335:
308:
305:
302:
294:
293:Robert Geroch
290:
286:
282:
266:
263:
258:
254:
250:
245:
241:
232:
231:Betti numbers
228:
210:
206:
202:
197:
193:
189:
184:
180:
176:
171:
167:
163:
158:
154:
145:
142:equating for
141:
137:
133:
129:
125:
121:
114:
98:
90:
87:
76:
72:
68:
62:
60:
57:has positive
56:
52:
47:
46:
45:
43:
35:
33:
31:
27:
23:
19:
2539:
2532:
2527:
2507:
2494:
2485:
2476:
2449:
2443:
2402:
2391:
2371:
2362:
2349:
2328:
2315:
2306:
2294:
2286:
2273:
2261:
2231:
2196:
2187:
2178:
2172:
2163:
2150:
2142:
2134:
2128:
2101:
2095:
2047:
2031:
2006:
2000:
1991:
1977:
1926:
1913:Ruth Charney
1906:
1903:
1831:
1825:
1816:
1672:
1663:
1352:
1347:
1338:
1289:
1284:
1276:
1215:
1133:
1130:
650:
647:
582:
577:
548:
487:
483:
479:
472:
468:
465:torus action
117:
64:
48:
39:
17:
15:
2584:Conjectures
2169:Hopf, Heinz
2009:: 117–126.
559:Switzerland
285:John Milnor
144:4-manifolds
2568:Categories
2452:(1): 1–4.
2266:Heinz Hopf
2104:(1): 1–4.
2042:, 54, 1976
1983:References
1933:Heinz Hopf
1717:admits no
1664:A compact
563:Bad Elster
490:such that
42:Heinz Hopf
30:Heinz Hopf
2181:: 209–228
1885:≥
1863:χ
1847:−
1790:×
1742:×
1693:×
1577:×
1525:×
1310:×
1256:π
1233:∗
1175:π
1151:→
1145::
1096:χ
1034:χ
999:χ
931:χ
851:∏
814:χ
794:∏
778:χ
740:−
734:∈
694:×
691:⋯
688:×
675:×
618:−
531:≤
522:
516:−
510:
501:−
477:Lie group
190:−
164:−
88:−
2579:Topology
2399:(1970).
2355:Springer
2201:Springer
2195:(2003).
2055:(1970).
643:or zero.
607:of sign
555:Fribourg
120:surfaces
77:of sign
26:topology
2555:1274115
2468:0264562
2384:0380906
2341:0229177
2254:0645728
2120:0264562
2023:0075647
958:and if
561:and at
233:vanish
2466:
2382:
2357:, 2014
2339:
2252:
2242:
2207:
2118:
2021:
126:. For
467:of a
134:and
2514:and
2240:ISBN
2205:ISBN
1972:and
1931:and
1915:and
1473:and
1046:<
1011:>
976:<
943:>
908:>
603:has
519:rank
507:rank
306:>
225:and
138:and
118:For
73:has
24:and
2544:doi
2454:doi
2143:130
2106:doi
2011:doi
1123:).
434:or
2570::
2552:MR
2550:,
2535:,
2464:MR
2462:.
2448:.
2442:.
2380:MR
2337:MR
2335:.
2323:;
2250:MR
2248:,
2219:^
2203:.
2199:.
2179:41
2177:,
2138:,
2116:MR
2114:.
2102:14
2100:.
2094:.
2019:MR
2017:.
2007:20
2005:.
1911:.
1888:0.
557:,
534:5.
32:.
2557:.
2546::
2540:4
2470:.
2456::
2450:4
2440:"
2426:2
2423:+
2420:n
2415:R
2403:n
2343:.
2295:G
2213:.
2122:.
2108::
2092:"
2078:2
2075:+
2072:n
2067:R
2025:.
2013::
1954:2
1949:T
1882:)
1877:k
1874:2
1870:M
1866:(
1858:k
1854:)
1850:1
1844:(
1800:2
1795:S
1785:2
1780:S
1752:2
1747:S
1737:2
1732:S
1703:2
1698:S
1688:2
1683:S
1645:5
1640:R
1616:6
1611:R
1587:2
1582:S
1572:2
1567:S
1535:2
1530:S
1520:2
1515:S
1491:2
1486:P
1483:C
1459:4
1454:S
1430:5
1425:S
1400:2
1395:P
1392:C
1368:4
1363:S
1320:2
1315:S
1305:2
1300:S
1260:1
1229:f
1201:f
1179:1
1154:M
1148:M
1142:f
1111:0
1108:=
1105:)
1102:M
1099:(
1074:k
1070:e
1049:0
1043:)
1040:M
1037:(
1014:0
1008:)
1005:M
1002:(
979:0
971:k
967:e
946:0
940:)
937:M
934:(
911:0
903:k
899:e
876:k
872:e
866:d
861:1
858:=
855:k
830:)
825:k
821:M
817:(
809:d
804:1
801:=
798:k
790:=
787:)
784:M
781:(
758:}
755:1
752:,
749:0
746:,
743:1
737:{
729:k
725:e
702:d
698:M
683:2
679:M
670:1
666:M
662:=
659:M
629:d
625:)
621:1
615:(
528:)
525:H
513:G
504:(
498:k
488:k
484:H
480:G
473:M
469:k
451:4
448:+
445:k
442:4
422:2
419:+
416:k
413:4
383:2
380:+
377:d
374:2
369:R
345:1
342:+
339:d
336:2
331:R
309:2
303:d
267:0
264:=
259:3
255:b
251:=
246:1
242:b
211:4
207:b
203:+
198:3
194:b
185:2
181:b
177:+
172:1
168:b
159:0
155:b
113:.
99:d
95:)
91:1
85:(
63:d
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