1544:
1328:
144:
1888:
1693:
1039:
1220:
showed that every element of the stable homotopy groups of spheres can be expressed using composition products and higher Toda brackets in terms of certain well known elements, called Hopf elements.
2174:
689:
2104:
1766:
589:
1984:
2239:
1212:
of these elements. The Toda bracket is not quite an element of a stable homotopy group, because it is only defined up to addition of composition products of certain other elements.
408:
2030:
1937:
1210:
872:
518:
1105:
874:
is not uniquely defined up to homotopy, because there was some choice in choosing the maps from the cones. Changing these maps changes the Toda bracket by adding elements of
447:
311:
2207:
1579:
1438:
1399:
1367:
1169:
1137:
1792:
473:
344:
202:
176:
1238:
54:
2271:
2136:
2057:
1613:
1426:
750:
249:
1716:
948:
910:
834:
814:
794:
770:
633:
613:
538:
364:
273:
226:
727:
1807:
1621:
2521:
2435:
2138:
are permanent cycles. Moreover, these Massey products have a lift to a motivic Adams spectral sequence giving an element in the Toda bracket
1052:, where multiplication (called composition product) is given by composition of representing maps, and any element of non-zero degree is
953:
There are also higher Toda brackets of several elements, defined when suitable lower Toda brackets vanish. This parallels the theory of
978:
2141:
641:
2513:
21:
2066:
1721:
546:
2548:
1942:
2212:
2060:
372:
252:
1997:
1904:
1177:
839:
478:
2359:
1794:
1585:
1334:
1229:
1045:
1078:
970:
417:
281:
150:
1539:{\displaystyle W{\stackrel {f}{\ \to \ }}X{\stackrel {i}{\ \to \ }}C_{f}{\stackrel {q}{\ \to \ }}W}
2376:
2338:
2319:
1216:
used the composition product and Toda brackets to label many of the elements of homotopy groups.
1049:
2179:
1994:
There is a convergence theorem originally due to Moss which states that special Massey products
1552:
1372:
1340:
1142:
1110:
1323:{\displaystyle W{\stackrel {f}{\ \to \ }}X{\stackrel {g}{\ \to \ }}Y{\stackrel {h}{\ \to \ }}Z}
139:{\displaystyle W{\stackrel {f}{\ \to \ }}X{\stackrel {g}{\ \to \ }}Y{\stackrel {h}{\ \to \ }}Z}
2517:
2485:
2431:
2402:
2311:
1771:
452:
323:
181:
155:
2475:
2465:
2423:
2392:
2368:
2303:
2531:
2497:
2445:
2388:
2244:
2109:
2035:
1591:
1404:
2527:
2493:
2441:
2419:
2384:
732:
231:
1701:
954:
915:
877:
819:
799:
779:
755:
618:
598:
523:
349:
258:
211:
2397:
697:
2542:
2323:
205:
2505:
2453:
1213:
25:
2337:
Isaksen, Daniel C.; Wang, Guozhen; Xu, Zhouli (2020-06-17). "More stable stems".
2456:(1973), "The nilpotency of elements of the stable homotopy groups of spheres",
958:
2489:
2315:
2470:
1053:
2406:
28:, who defined them and used them to compute homotopy groups of spheres in (
1883:{\displaystyle C_{f}{\stackrel {q}{\ \to \ }}W{\stackrel {b}{\ \to \ }}Z}
1688:{\displaystyle X{\stackrel {i}{\ \to \ }}C_{f}{\stackrel {a}{\ \to \ }}Y}
317:
2480:
2427:
2380:
2307:
2291:
2372:
2343:
1232:
the Toda bracket can be defined as follows. Again, suppose that
1034:{\displaystyle \pi _{\ast }^{S}=\bigoplus _{k\geq 0}\pi _{k}^{S}}
2357:
Cohen, Joel M. (1968), "The decomposition of stable homotopy.",
2063:
contain a permanent cycle, meaning has an associated element in
2416:
Stable homotopy groups of spheres. A computer-assisted approach
20:
is an operation on homotopy classes of maps, in particular on
2418:, Lecture Notes in Mathematics, vol. 1423, Berlin:
2292:"Secondary compositions and the Adams spectral sequence"
2169:{\displaystyle \langle \alpha ,\beta ,\gamma \rangle }
965:
The Toda bracket for stable homotopy groups of spheres
2247:
2215:
2182:
2144:
2112:
2069:
2038:
2000:
1945:
1907:
1810:
1774:
1724:
1704:
1624:
1594:
1555:
1441:
1407:
1375:
1343:
1241:
1180:
1145:
1113:
1081:
981:
918:
880:
842:
822:
802:
782:
758:
735:
700:
644:
621:
601:
549:
526:
481:
455:
420:
375:
352:
326:
284:
261:
234:
214:
184:
158:
57:
1224:
The Toda bracket for general triangulated categories
149:
is a sequence of maps between spaces, such that the
684:{\displaystyle \langle f,g,h\rangle \colon SW\to Z}
2265:
2233:
2201:
2168:
2130:
2098:
2051:
2024:
1978:
1931:
1882:
1786:
1760:
1710:
1687:
1607:
1573:
1538:
1420:
1393:
1361:
1322:
1204:
1163:
1131:
1099:
1033:
942:
904:
866:
828:
808:
788:
764:
744:
721:
683:
627:
607:
583:
532:
512:
467:
441:
402:
358:
338:
305:
267:
243:
220:
196:
170:
138:
2510:Composition methods in homotopy groups of spheres
729:of homotopy classes of maps from the suspension
346:to a trivial map, which when post-composed with
2512:, Annals of Mathematics Studies, vol. 49,
1044:of the stable homotopy groups of spheres is a
2414:Kochman, Stanley O. (1990), "Toda brackets",
8:
2458:Journal of the Mathematical Society of Japan
2163:
2145:
2019:
2001:
1926:
1908:
1199:
1181:
861:
843:
663:
645:
475:to a trivial map, which when composed with
2099:{\displaystyle \pi _{*}^{s}(\mathbb {S} )}
1761:{\displaystyle h\circ a\circ i=h\circ g=0}
584:{\displaystyle G\circ C_{f}\colon CW\to Z}
2479:
2469:
2396:
2342:
2246:
2214:
2187:
2181:
2143:
2111:
2089:
2088:
2079:
2074:
2068:
2043:
2037:
1999:
1979:{\displaystyle \operatorname {hom} (W,Z)}
1944:
1906:
1869:
1858:
1856:
1855:
1835:
1824:
1822:
1821:
1815:
1809:
1773:
1723:
1703:
1674:
1663:
1661:
1660:
1654:
1642:
1631:
1629:
1628:
1623:
1599:
1593:
1554:
1516:
1505:
1503:
1502:
1496:
1484:
1473:
1471:
1470:
1459:
1448:
1446:
1445:
1440:
1412:
1406:
1374:
1342:
1309:
1298:
1296:
1295:
1284:
1273:
1271:
1270:
1259:
1248:
1246:
1245:
1240:
1179:
1144:
1112:
1091:
1086:
1080:
1025:
1020:
1004:
991:
986:
980:
917:
879:
841:
821:
801:
781:
757:
734:
699:
643:
620:
600:
560:
548:
525:
486:
480:
454:
419:
374:
351:
325:
283:
260:
233:
213:
183:
157:
125:
114:
112:
111:
100:
89:
87:
86:
75:
64:
62:
61:
56:
2282:
1057:
41:
2234:{\displaystyle \alpha ,\beta ,\gamma }
403:{\displaystyle h\circ F\colon CW\to Z}
2025:{\displaystyle \langle a,b,c\rangle }
1932:{\displaystyle \langle f,g,h\rangle }
1217:
1205:{\displaystyle \langle f,g,h\rangle }
867:{\displaystyle \langle f,g,h\rangle }
694:representing an element in the group
48:) for more information. Suppose that
7:
513:{\displaystyle C_{f}\colon CW\to CX}
45:
29:
275:. Then we get a (non-unique) map
2290:Moss, R. Michael F. (1970-08-01).
1901:is (a choice of) the Toda bracket
414:Similarly we get a non-unique map
14:
1100:{\displaystyle \pi _{\ast }^{S}}
1432:so we obtain an exact triangle
1333:is a sequence of morphism in a
442:{\displaystyle G\colon CX\to Z}
306:{\displaystyle F\colon CW\to Y}
2093:
2085:
1973:
1964:
1958:
1952:
1862:
1852:
1846:
1828:
1667:
1635:
1533:
1527:
1509:
1477:
1452:
1302:
1277:
1252:
934:
919:
899:
884:
716:
701:
675:
595:By joining these two cones on
575:
501:
433:
394:
297:
118:
93:
68:
1:
449:induced by a homotopy from
2565:
2514:Princeton University Press
2202:{\displaystyle \pi _{*,*}}
1574:{\displaystyle g\circ f=0}
1394:{\displaystyle h\circ g=0}
1362:{\displaystyle g\circ f=0}
1164:{\displaystyle g\cdot h=0}
1132:{\displaystyle f\cdot g=0}
615:and the maps from them to
22:homotopy groups of spheres
2296:Mathematische Zeitschrift
1228:In the case of a general
2106:, assuming the elements
1787:{\displaystyle h\circ a}
468:{\displaystyle h\circ g}
339:{\displaystyle g\circ f}
197:{\displaystyle h\circ g}
171:{\displaystyle g\circ f}
2061:Adams spectral sequence
2267:
2235:
2203:
2170:
2132:
2100:
2053:
2026:
1980:
1933:
1884:
1795:factors (non-uniquely)
1788:
1762:
1712:
1689:
1609:
1586:factors (non-uniquely)
1575:
1540:
1422:
1395:
1363:
1324:
1206:
1165:
1133:
1101:
1035:
944:
906:
868:
830:
810:
790:
766:
746:
723:
685:
629:
609:
585:
540:, gives another map,
534:
520:, the cone of the map
514:
469:
443:
404:
360:
340:
307:
269:
245:
222:
198:
172:
140:
2471:10.2969/jmsj/02540707
2360:Annals of Mathematics
2268:
2266:{\displaystyle a,b,c}
2236:
2204:
2171:
2133:
2131:{\displaystyle a,b,c}
2101:
2054:
2052:{\displaystyle E_{r}}
2027:
1981:
1934:
1885:
1789:
1763:
1718:. Then, the relation
1713:
1690:
1610:
1608:{\displaystyle C_{f}}
1576:
1541:
1423:
1421:{\displaystyle C_{f}}
1396:
1364:
1335:triangulated category
1325:
1230:triangulated category
1207:
1166:
1134:
1102:
1036:
945:
907:
869:
831:
811:
791:
767:
747:
724:
686:
630:
610:
586:
535:
515:
470:
444:
405:
361:
341:
308:
270:
246:
223:
199:
173:
141:
2245:
2213:
2180:
2142:
2110:
2067:
2036:
1998:
1943:
1905:
1808:
1772:
1722:
1702:
1622:
1592:
1553:
1439:
1405:
1373:
1341:
1239:
1178:
1143:
1111:
1079:
979:
916:
878:
840:
820:
800:
780:
756:
733:
698:
642:
619:
599:
547:
524:
479:
453:
418:
373:
350:
324:
282:
259:
232:
212:
182:
156:
55:
16:In mathematics, the
2084:
2032:of elements in the
1990:Convergence theorem
1428:denote the cone of
1096:
1030:
996:
2428:10.1007/BFb0083797
2422:, pp. 12–34,
2308:10.1007/BF01129978
2263:
2231:
2199:
2166:
2128:
2096:
2070:
2049:
2022:
1976:
1929:
1880:
1784:
1758:
1708:
1685:
1605:
1571:
1536:
1418:
1391:
1359:
1320:
1202:
1161:
1129:
1097:
1082:
1031:
1016:
1015:
982:
940:
902:
864:
826:
806:
786:
762:
745:{\displaystyle SW}
742:
719:
681:
625:
605:
581:
530:
510:
465:
439:
400:
356:
336:
303:
265:
244:{\displaystyle CA}
241:
218:
194:
168:
136:
2523:978-0-691-09586-8
2437:978-3-540-52468-7
2363:, Second Series,
1874:
1867:
1861:
1840:
1833:
1827:
1711:{\displaystyle a}
1679:
1672:
1666:
1647:
1640:
1634:
1521:
1514:
1508:
1489:
1482:
1476:
1464:
1457:
1451:
1314:
1307:
1301:
1289:
1282:
1276:
1264:
1257:
1251:
1000:
943:{\displaystyle f}
905:{\displaystyle h}
829:{\displaystyle h}
809:{\displaystyle g}
789:{\displaystyle f}
765:{\displaystyle Z}
628:{\displaystyle Z}
608:{\displaystyle W}
533:{\displaystyle f}
359:{\displaystyle h}
268:{\displaystyle A}
221:{\displaystyle A}
130:
123:
117:
105:
98:
92:
80:
73:
67:
2556:
2534:
2500:
2483:
2473:
2448:
2409:
2400:
2349:
2348:
2346:
2334:
2328:
2327:
2287:
2272:
2270:
2269:
2264:
2240:
2238:
2237:
2232:
2208:
2206:
2205:
2200:
2198:
2197:
2175:
2173:
2172:
2167:
2137:
2135:
2134:
2129:
2105:
2103:
2102:
2097:
2092:
2083:
2078:
2058:
2056:
2055:
2050:
2048:
2047:
2031:
2029:
2028:
2023:
1985:
1983:
1982:
1977:
1938:
1936:
1935:
1930:
1889:
1887:
1886:
1881:
1876:
1875:
1873:
1868:
1865:
1859:
1857:
1842:
1841:
1839:
1834:
1831:
1825:
1823:
1820:
1819:
1793:
1791:
1790:
1785:
1767:
1765:
1764:
1759:
1717:
1715:
1714:
1709:
1694:
1692:
1691:
1686:
1681:
1680:
1678:
1673:
1670:
1664:
1662:
1659:
1658:
1649:
1648:
1646:
1641:
1638:
1632:
1630:
1614:
1612:
1611:
1606:
1604:
1603:
1580:
1578:
1577:
1572:
1545:
1543:
1542:
1537:
1523:
1522:
1520:
1515:
1512:
1506:
1504:
1501:
1500:
1491:
1490:
1488:
1483:
1480:
1474:
1472:
1466:
1465:
1463:
1458:
1455:
1449:
1447:
1427:
1425:
1424:
1419:
1417:
1416:
1400:
1398:
1397:
1392:
1368:
1366:
1365:
1360:
1329:
1327:
1326:
1321:
1316:
1315:
1313:
1308:
1305:
1299:
1297:
1291:
1290:
1288:
1283:
1280:
1274:
1272:
1266:
1265:
1263:
1258:
1255:
1249:
1247:
1211:
1209:
1208:
1203:
1170:
1168:
1167:
1162:
1138:
1136:
1135:
1130:
1106:
1104:
1103:
1098:
1095:
1090:
1075:are elements of
1046:supercommutative
1040:
1038:
1037:
1032:
1029:
1024:
1014:
995:
990:
949:
947:
946:
941:
911:
909:
908:
903:
873:
871:
870:
865:
835:
833:
832:
827:
815:
813:
812:
807:
795:
793:
792:
787:
771:
769:
768:
763:
751:
749:
748:
743:
728:
726:
725:
722:{\displaystyle }
720:
690:
688:
687:
682:
635:, we get a map
634:
632:
631:
626:
614:
612:
611:
606:
590:
588:
587:
582:
565:
564:
539:
537:
536:
531:
519:
517:
516:
511:
491:
490:
474:
472:
471:
466:
448:
446:
445:
440:
409:
407:
406:
401:
365:
363:
362:
357:
345:
343:
342:
337:
312:
310:
309:
304:
274:
272:
271:
266:
250:
248:
247:
242:
227:
225:
224:
219:
208:. Given a space
203:
201:
200:
195:
177:
175:
174:
169:
145:
143:
142:
137:
132:
131:
129:
124:
121:
115:
113:
107:
106:
104:
99:
96:
90:
88:
82:
81:
79:
74:
71:
65:
63:
2564:
2563:
2559:
2558:
2557:
2555:
2554:
2553:
2549:Homotopy theory
2539:
2538:
2524:
2504:
2452:
2438:
2420:Springer-Verlag
2413:
2373:10.2307/1970586
2356:
2353:
2352:
2336:
2335:
2331:
2289:
2288:
2284:
2279:
2243:
2242:
2211:
2210:
2183:
2178:
2177:
2140:
2139:
2108:
2107:
2065:
2064:
2039:
2034:
2033:
1996:
1995:
1992:
1941:
1940:
1903:
1902:
1811:
1806:
1805:
1770:
1769:
1720:
1719:
1700:
1699:
1650:
1620:
1619:
1595:
1590:
1589:
1551:
1550:
1492:
1437:
1436:
1408:
1403:
1402:
1371:
1370:
1339:
1338:
1237:
1236:
1226:
1176:
1175:
1141:
1140:
1109:
1108:
1077:
1076:
977:
976:
967:
955:Massey products
914:
913:
876:
875:
838:
837:
818:
817:
798:
797:
778:
777:
754:
753:
731:
730:
696:
695:
640:
639:
617:
616:
597:
596:
556:
545:
544:
522:
521:
482:
477:
476:
451:
450:
416:
415:
371:
370:
348:
347:
322:
321:
280:
279:
257:
256:
230:
229:
210:
209:
180:
179:
154:
153:
53:
52:
38:
12:
11:
5:
2562:
2560:
2552:
2551:
2541:
2540:
2537:
2536:
2522:
2502:
2464:(4): 707–732,
2450:
2436:
2411:
2367:(2): 305–320,
2351:
2350:
2329:
2302:(4): 283–310.
2281:
2280:
2278:
2275:
2262:
2259:
2256:
2253:
2250:
2230:
2227:
2224:
2221:
2218:
2196:
2193:
2190:
2186:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2127:
2124:
2121:
2118:
2115:
2095:
2091:
2087:
2082:
2077:
2073:
2046:
2042:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
1991:
1988:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1891:
1890:
1879:
1872:
1864:
1854:
1851:
1848:
1845:
1838:
1830:
1818:
1814:
1783:
1780:
1777:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1707:
1696:
1695:
1684:
1677:
1669:
1657:
1653:
1645:
1637:
1627:
1602:
1598:
1570:
1567:
1564:
1561:
1558:
1547:
1546:
1535:
1532:
1529:
1526:
1519:
1511:
1499:
1495:
1487:
1479:
1469:
1462:
1454:
1444:
1415:
1411:
1390:
1387:
1384:
1381:
1378:
1358:
1355:
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1331:
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1319:
1312:
1304:
1294:
1287:
1279:
1269:
1262:
1254:
1244:
1225:
1222:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1160:
1157:
1154:
1151:
1148:
1128:
1125:
1122:
1119:
1116:
1094:
1089:
1085:
1042:
1041:
1028:
1023:
1019:
1013:
1010:
1007:
1003:
999:
994:
989:
985:
966:
963:
939:
936:
933:
930:
927:
924:
921:
901:
898:
895:
892:
889:
886:
883:
863:
860:
857:
854:
851:
848:
845:
825:
805:
785:
761:
741:
738:
718:
715:
712:
709:
706:
703:
692:
691:
680:
677:
674:
671:
668:
665:
662:
659:
656:
653:
650:
647:
624:
604:
593:
592:
580:
577:
574:
571:
568:
563:
559:
555:
552:
529:
509:
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503:
500:
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489:
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461:
458:
438:
435:
432:
429:
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423:
412:
411:
399:
396:
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387:
384:
381:
378:
355:
335:
332:
329:
314:
313:
302:
299:
296:
293:
290:
287:
264:
240:
237:
217:
193:
190:
187:
167:
164:
161:
147:
146:
135:
128:
120:
110:
103:
95:
85:
78:
70:
60:
37:
34:
24:, named after
13:
10:
9:
6:
4:
3:
2:
2561:
2550:
2547:
2546:
2544:
2533:
2529:
2525:
2519:
2515:
2511:
2507:
2506:Toda, Hiroshi
2503:
2499:
2495:
2491:
2487:
2482:
2477:
2472:
2467:
2463:
2459:
2455:
2454:Nishida, Goro
2451:
2447:
2443:
2439:
2433:
2429:
2425:
2421:
2417:
2412:
2408:
2404:
2399:
2394:
2390:
2386:
2382:
2378:
2374:
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2366:
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2355:
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2321:
2317:
2313:
2309:
2305:
2301:
2297:
2293:
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2283:
2276:
2274:
2260:
2257:
2254:
2251:
2248:
2228:
2225:
2222:
2219:
2216:
2209:for elements
2194:
2191:
2188:
2184:
2160:
2157:
2154:
2151:
2148:
2125:
2122:
2119:
2116:
2113:
2080:
2075:
2071:
2062:
2059:-page of the
2044:
2040:
2016:
2013:
2010:
2007:
2004:
1989:
1987:
1970:
1967:
1961:
1955:
1949:
1946:
1939:in the group
1923:
1920:
1917:
1914:
1911:
1900:
1896:
1877:
1870:
1849:
1843:
1836:
1816:
1812:
1804:
1803:
1802:
1800:
1796:
1781:
1778:
1775:
1768:implies that
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1705:
1682:
1675:
1655:
1651:
1643:
1625:
1618:
1617:
1616:
1600:
1596:
1587:
1584:
1581:implies that
1568:
1565:
1562:
1559:
1556:
1549:The relation
1530:
1524:
1517:
1497:
1493:
1485:
1467:
1460:
1442:
1435:
1434:
1433:
1431:
1413:
1409:
1388:
1385:
1382:
1379:
1376:
1356:
1353:
1350:
1347:
1344:
1336:
1317:
1310:
1292:
1285:
1267:
1260:
1242:
1235:
1234:
1233:
1231:
1223:
1221:
1219:
1215:
1196:
1193:
1190:
1187:
1184:
1174:
1171:, there is a
1158:
1155:
1152:
1149:
1146:
1126:
1123:
1120:
1117:
1114:
1092:
1087:
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1055:
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1047:
1026:
1021:
1017:
1011:
1008:
1005:
1001:
997:
992:
987:
983:
975:
974:
973:
972:
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960:
956:
951:
937:
931:
928:
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922:
896:
893:
890:
887:
881:
858:
855:
852:
849:
846:
823:
803:
783:
775:
772:, called the
759:
739:
736:
713:
710:
707:
704:
678:
672:
669:
666:
660:
657:
654:
651:
648:
638:
637:
636:
622:
602:
578:
572:
569:
566:
561:
557:
553:
550:
543:
542:
541:
527:
507:
504:
498:
495:
492:
487:
483:
462:
459:
456:
436:
430:
427:
424:
421:
397:
391:
388:
385:
382:
379:
376:
369:
368:
367:
366:gives a map
353:
333:
330:
327:
319:
316:induced by a
300:
294:
291:
288:
285:
278:
277:
276:
262:
254:
238:
235:
215:
207:
206:nullhomotopic
191:
188:
185:
165:
162:
159:
152:
133:
126:
108:
101:
83:
76:
58:
51:
50:
49:
47:
43:
35:
33:
31:
27:
23:
19:
2509:
2461:
2457:
2415:
2364:
2358:
2332:
2299:
2295:
2285:
1993:
1898:
1894:
1892:
1798:
1697:
1582:
1548:
1429:
1332:
1227:
1218:Cohen (1968)
1214:Hiroshi Toda
1173:Toda bracket
1172:
1072:
1068:
1064:
1062:
1058:Nishida 1973
1043:
968:
952:
774:Toda bracket
773:
693:
594:
413:
315:
151:compositions
148:
42:Kochman 1990
39:
26:Hiroshi Toda
18:Toda bracket
17:
15:
2481:2433/220059
251:denote the
2344:2001.04511
2277:References
1337:such that
971:direct sum
959:cohomology
836:. The map
36:Definition
2490:0025-5645
2324:122909581
2316:1432-1823
2229:γ
2223:β
2217:α
2195:∗
2189:∗
2185:π
2164:⟩
2161:γ
2155:β
2149:α
2146:⟨
2076:∗
2072:π
2020:⟩
2002:⟨
1950:
1927:⟩
1909:⟨
1893:for some
1863:→
1829:→
1779:∘
1747:∘
1735:∘
1729:∘
1698:for some
1668:→
1636:→
1560:∘
1510:→
1478:→
1453:→
1380:∘
1348:∘
1303:→
1278:→
1253:→
1200:⟩
1182:⟨
1150:⋅
1118:⋅
1088:∗
1084:π
1054:nilpotent
1018:π
1009:≥
1002:⨁
988:∗
984:π
862:⟩
844:⟨
676:→
667::
664:⟩
646:⟨
576:→
567::
554:∘
502:→
493::
460:∘
434:→
425::
395:→
386::
380:∘
331:∘
298:→
289::
204:are both
189:∘
163:∘
119:→
94:→
69:→
46:Toda 1962
30:Toda 1962
2543:Category
2508:(1962),
2407:16591550
2241:lifting
1797:through
1588:through
318:homotopy
2532:0143217
2498:0341485
2446:1052407
2389:0231377
2381:1970586
1897:. This
1048:graded
2530:
2520:
2496:
2488:
2444:
2434:
2405:
2398:224450
2395:
2387:
2379:
2322:
2314:
1866:
1860:
1832:
1826:
1671:
1665:
1639:
1633:
1513:
1507:
1481:
1475:
1456:
1450:
1401:. Let
1306:
1300:
1281:
1275:
1256:
1250:
816:, and
228:, let
122:
116:
97:
91:
72:
66:
44:) or (
2377:JSTOR
2339:arXiv
2320:S2CID
1107:with
320:from
40:See (
2518:ISBN
2486:ISSN
2432:ISBN
2403:PMID
2312:ISSN
1801:as
1615:as
1369:and
1139:and
1071:and
1067:and
1050:ring
969:The
912:and
253:cone
178:and
2476:hdl
2466:doi
2424:doi
2393:PMC
2369:doi
2304:doi
2300:115
2176:in
1947:hom
1063:If
1060:).
957:in
776:of
752:to
255:of
32:).
2545::
2528:MR
2526:,
2516:,
2494:MR
2492:,
2484:,
2474:,
2462:25
2460:,
2442:MR
2440:,
2430:,
2401:,
2391:,
2385:MR
2383:,
2375:,
2365:87
2318:.
2310:.
2298:.
2294:.
2273:.
1986:.
961:.
950:.
796:,
2535:.
2501:.
2478::
2468::
2449:.
2426::
2410:.
2371::
2347:.
2341::
2326:.
2306::
2261:c
2258:,
2255:b
2252:,
2249:a
2226:,
2220:,
2192:,
2158:,
2152:,
2126:c
2123:,
2120:b
2117:,
2114:a
2094:)
2090:S
2086:(
2081:s
2045:r
2041:E
2017:c
2014:,
2011:b
2008:,
2005:a
1974:)
1971:Z
1968:,
1965:]
1962:1
1959:[
1956:W
1953:(
1924:h
1921:,
1918:g
1915:,
1912:f
1899:b
1895:b
1878:Z
1871:b
1853:]
1850:1
1847:[
1844:W
1837:q
1817:f
1813:C
1799:W
1782:a
1776:h
1756:0
1753:=
1750:g
1744:h
1741:=
1738:i
1732:a
1726:h
1706:a
1683:Y
1676:a
1656:f
1652:C
1644:i
1626:X
1601:f
1597:C
1583:g
1569:0
1566:=
1563:f
1557:g
1534:]
1531:1
1528:[
1525:W
1518:q
1498:f
1494:C
1486:i
1468:X
1461:f
1443:W
1430:f
1414:f
1410:C
1389:0
1386:=
1383:g
1377:h
1357:0
1354:=
1351:f
1345:g
1318:Z
1311:h
1293:Y
1286:g
1268:X
1261:f
1243:W
1197:h
1194:,
1191:g
1188:,
1185:f
1159:0
1156:=
1153:h
1147:g
1127:0
1124:=
1121:g
1115:f
1093:S
1073:h
1069:g
1065:f
1056:(
1027:S
1022:k
1012:0
1006:k
998:=
993:S
938:f
935:]
932:Z
929:,
926:X
923:S
920:[
900:]
897:Y
894:,
891:W
888:S
885:[
882:h
859:h
856:,
853:g
850:,
847:f
824:h
804:g
784:f
760:Z
740:W
737:S
717:]
714:Z
711:,
708:W
705:S
702:[
679:Z
673:W
670:S
661:h
658:,
655:g
652:,
649:f
623:Z
603:W
591:.
579:Z
573:W
570:C
562:f
558:C
551:G
528:f
508:X
505:C
499:W
496:C
488:f
484:C
463:g
457:h
437:Z
431:X
428:C
422:G
410:.
398:Z
392:W
389:C
383:F
377:h
354:h
334:f
328:g
301:Y
295:W
292:C
286:F
263:A
239:A
236:C
216:A
192:g
186:h
166:f
160:g
134:Z
127:h
109:Y
102:g
84:X
77:f
59:W
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