2213:
2635:
372:
900:
1572:
1101:
1326:
195:
1866:
1789:
1445:
766:
2056:
1453:
1963:
1675:
This composition passes to the quotient to give a well-defined map in terms of cohomology, this is the cup product. This approach explains the existence of a cup product for cohomology but not for homology:
2617:
905:
The cup product of two cocycles is again a cocycle, and the product of a coboundary with a cocycle (in either order) is a coboundary. The cup product operation induces a bilinear operation on cohomology,
996:
689:
466:
1227:
612:
2122:
1712:
1670:
2502:
2247:
1150:
552:
731:
152:
758:
183:
1625:
2394:
2161:
502:
2420:
2324:
2422:. By taking the images of the fundamental homology classes of these manifolds under inclusion, one can obtain a bilinear product on homology. This product is
2364:
2344:
2298:
2274:
1599:
2626:
can be defined in terms of intersections, shifting dimensions by 1, or alternatively in terms of a non-vanishing cup product on the complement of a link.
1012:
367:{\displaystyle (\alpha ^{p}\smile \beta ^{q})(\sigma )=\alpha ^{p}(\sigma \circ \iota _{0,1,...p})\cdot \beta ^{q}(\sigma \circ \iota _{p,p+1,...,p+q})}
1242:
1794:
1717:
895:{\displaystyle \delta (\alpha ^{p}\smile \beta ^{q})=\delta {\alpha ^{p}}\smile \beta ^{q}+(-1)^{p}(\alpha ^{p}\smile \delta {\beta ^{q}}).}
1567:{\displaystyle \displaystyle C^{\bullet }(X)\times C^{\bullet }(X)\to C^{\bullet }(X\times X){\overset {\Delta ^{*}}{\to }}C^{\bullet }(X)}
1358:
51:. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space
2201:
1968:
1878:
2509:
2799:
2785:
2771:
2763:
2220:
can be defined in terms of a non-vanishing cup product on the complement of a link. The complement of these two linked circles in
2167:
multiplication in the first cohomology group can be used to decompose the torus as a 2-cell diagram, thus having product equal to
912:
625:
396:
2367:
124:
2819:
2655:
The cup product is a binary (2-ary) operation; one can define a ternary (3-ary) and higher order operation called the
1161:
2065:
Cup products may be used to distinguish manifolds from wedges of spaces with identical cohomology groups. The space
560:
2824:
68:
2068:
1679:
1637:
2814:
2249:
deformation retracts to a wedge sum of a torus and 2-sphere, which has a non-vanishing cup product in degree 1.
2429:
2253:
For oriented manifolds, there is a geometric heuristic that "the cup product is dual to intersections."
1868:, which goes the wrong way round to allow us to define a product. This is however of use in defining the
2204:
differential forms belongs to the de Rham class of the cup product of the two original de Rham classes.
2758:
James R. Munkres, "Elements of
Algebraic Topology", Perseus Publishing, Cambridge Massachusetts (1984)
2223:
2660:
1123:
2423:
507:
92:
2740:
2193:
2189:
709:
130:
24:
1628:
736:
161:
1604:
2795:
2781:
2767:
2759:
2716:
2672:
2642:
1344:
378:
114:
2373:
1632:
1107:
80:
2139:
475:
2677:
2277:
103:
76:
64:
2399:
2303:
2777:
2687:
2656:
2650:
2638:
2623:
2349:
2329:
2283:
2259:
2217:
1584:
72:
2791:
2808:
2692:
2197:
1578:
1096:{\displaystyle \alpha ^{p}\smile \beta ^{q}=(-1)^{pq}(\beta ^{q}\smile \alpha ^{p})}
2641:
generalize cup product, allowing one to define "higher order linking numbers", the
1233:
2682:
1869:
100:
20:
2212:
1321:{\displaystyle f^{*}(\alpha \smile \beta )=f^{*}(\alpha )\smile f^{*}(\beta ),}
2734:
703:
2634:
1861:{\displaystyle \Delta _{*}\colon H_{\bullet }(X)\to H_{\bullet }(X\times X)}
1784:{\displaystyle \Delta ^{*}\colon H^{\bullet }(X\times X)\to H^{\bullet }(X)}
469:
2133:
1114:
390:
32:
2426:
to the cup product, in the sense that taking the
Poincaré pairings
1440:{\displaystyle \smile \colon H^{p}(X)\times H^{q}(X)\to H^{p+q}(X)}
2633:
2211:
2051:{\displaystyle u\smile (v_{1}+v_{2})=u\smile v_{1}+u\smile v_{2}.}
1958:{\displaystyle (u_{1}+u_{2})\smile v=u_{1}\smile v+u_{2}\smile v}
2663:, which is only partly defined (only defined for some triples).
1875:
Bilinearity follows from this presentation of cup product, i.e.
2612:{\displaystyle ^{*}\smile ^{*}=^{*}\in H^{i+j}(X,\mathbb {Z} )}
1006:
The cup product operation in cohomology satisfies the identity
2780:, "Topology and Geometry", Springer-Verlag, New York (1993)
2659:, which generalizes the cup product. This is a higher order
991:{\displaystyle H^{p}(X)\times H^{q}(X)\to H^{p+q}(X).}
2512:
2432:
2402:
2376:
2352:
2332:
2306:
2286:
2262:
2226:
2142:
2071:
1971:
1881:
1797:
1720:
1682:
1640:
1607:
1587:
1457:
1456:
1361:
1245:
1164:
1126:
1015:
915:
769:
739:
712:
628:
563:
510:
478:
399:
198:
164:
133:
2128:, but with a different cup product. In the case of
684:{\displaystyle \sigma \circ \iota _{p,p+1,...,p+q}}
461:{\displaystyle \iota _{S},S\subset \{0,1,...,p+q\}}
2611:
2496:
2414:
2388:
2358:
2338:
2318:
2292:
2268:
2241:
2155:
2116:
2050:
1957:
1860:
1783:
1706:
1664:
1619:
1593:
1566:
1439:
1320:
1221:
1144:
1095:
990:
894:
752:
725:
683:
606:
546:
496:
460:
366:
177:
146:
16:Turns the cohomology of a space into a graded ring
1222:{\displaystyle f^{*}\colon H^{*}(Y)\to H^{*}(X)}
2736:Informal talk in Derived Geometry (Jacob Lurie)
607:{\displaystyle \sigma \circ \iota _{0,1,...,p}}
2504:then there is the following equality :
67:. The cup product was introduced in work of
8:
2124:has the same cohomology groups as the torus
2117:{\displaystyle X:=S^{2}\vee S^{1}\vee S^{1}}
1707:{\displaystyle \Delta \colon X\to X\times X}
1665:{\displaystyle \Delta \colon X\to X\times X}
1106:so that the corresponding multiplication is
541:
511:
455:
419:
79:from 1935–1938, and, in full generality, by
2200:. In other words, the wedge product of two
1447:as induced from the following composition:
123:The construction starts with a product of
99:is a construction giving a product on the
2602:
2601:
2580:
2567:
2542:
2523:
2511:
2488:
2475:
2462:
2443:
2431:
2401:
2375:
2351:
2331:
2305:
2285:
2261:
2233:
2229:
2228:
2225:
2147:
2141:
2108:
2095:
2082:
2070:
2039:
2020:
1998:
1985:
1970:
1943:
1924:
1902:
1889:
1880:
1837:
1815:
1802:
1796:
1766:
1738:
1725:
1719:
1681:
1639:
1631:and the second is the map induced by the
1606:
1586:
1548:
1536:
1527:
1506:
1484:
1462:
1455:
1416:
1394:
1372:
1360:
1300:
1278:
1250:
1244:
1204:
1182:
1169:
1163:
1125:
1084:
1071:
1055:
1033:
1020:
1014:
964:
942:
920:
914:
879:
874:
862:
849:
827:
813:
808:
793:
780:
768:
744:
738:
717:
711:
639:
627:
574:
562:
509:
477:
404:
398:
319:
300:
263:
244:
219:
206:
197:
169:
163:
138:
132:
2497:{\displaystyle ^{*},^{*}\in H^{i},H^{j}}
2794:", Cambridge Publishing Company (2002)
2704:
2208:Cup product and geometric intersections
1355:It is possible to view the cup product
504:-simplex whose vertices are indexed by
2396:is again a submanifold of codimension
43:to form a composite cocycle of degree
472:of the simplex spanned by S into the
7:
2710:
2708:
2184:Cup product and differential forms
1799:
1722:
1683:
1641:
1533:
14:
2242:{\displaystyle \mathbb {R} ^{3}}
2175:where this is the base module).
2743:from the original on 2021-12-21
2717:"Cup Product and Intersections"
1155:is a continuous function, and
1117:, in the following sense: if
706:of the cup product of cochains
2606:
2592:
2564:
2551:
2539:
2532:
2520:
2513:
2459:
2452:
2440:
2433:
2004:
1978:
1908:
1882:
1855:
1843:
1830:
1827:
1821:
1778:
1772:
1759:
1756:
1744:
1692:
1650:
1560:
1554:
1529:
1524:
1512:
1499:
1496:
1490:
1474:
1468:
1434:
1428:
1409:
1406:
1400:
1384:
1378:
1312:
1306:
1290:
1284:
1268:
1256:
1216:
1210:
1197:
1194:
1188:
1145:{\displaystyle f\colon X\to Y}
1136:
1090:
1064:
1052:
1042:
982:
976:
957:
954:
948:
932:
926:
886:
855:
846:
836:
799:
773:
491:
479:
361:
306:
290:
250:
234:
228:
225:
199:
1:
1627:, where the first map is the
547:{\displaystyle \{0,...,p+q\}}
31:is a method of adjoining two
2136:associated to the copies of
1791:but would also induce a map
726:{\displaystyle \alpha ^{p}}
147:{\displaystyle \alpha ^{p}}
2841:
2733:Ciencias TV (2016-12-10),
2648:
2370:, then their intersection
2163:is degenerate, whereas in
2132:the multiplication of the
753:{\displaystyle \beta ^{q}}
178:{\displaystyle \beta ^{q}}
1620:{\displaystyle X\times X}
1331:for all classes α, β in
2389:{\displaystyle A\cap B}
2646:
2613:
2498:
2416:
2390:
2360:
2340:
2320:
2300:. If two submanifolds
2294:
2270:
2250:
2243:
2157:
2118:
2052:
1959:
1862:
1785:
1708:
1666:
1621:
1595:
1575:
1568:
1441:
1322:
1223:
1146:
1097:
992:
896:
754:
727:
685:
608:
548:
498:
462:
368:
179:
148:
2637:
2614:
2499:
2417:
2391:
2361:
2341:
2321:
2295:
2271:
2244:
2215:
2192:, the cup product of
2158:
2156:{\displaystyle S^{1}}
2119:
2053:
1960:
1863:
1786:
1709:
1667:
1622:
1596:
1569:
1449:
1442:
1323:
1236:in cohomology, then
1224:
1147:
1098:
993:
897:
755:
728:
686:
609:
549:
499:
497:{\displaystyle (p+q)}
463:
369:
180:
149:
2715:Hutchings, Michael.
2661:cohomology operation
2510:
2430:
2400:
2374:
2350:
2330:
2304:
2284:
2260:
2224:
2140:
2069:
1969:
1879:
1795:
1718:
1680:
1638:
1605:
1585:
1454:
1359:
1243:
1162:
1124:
1013:
913:
767:
737:
710:
699:of σ, respectively.
626:
561:
508:
476:
397:
196:
162:
131:
55:into a graded ring,
2415:{\displaystyle i+j}
2319:{\displaystyle A,B}
1339:). In other words,
1113:The cup product is
93:singular cohomology
2820:Algebraic topology
2792:Algebraic Topology
2647:
2609:
2494:
2412:
2386:
2356:
2336:
2316:
2290:
2266:
2251:
2239:
2196:is induced by the
2194:differential forms
2190:de Rham cohomology
2153:
2114:
2048:
1955:
1858:
1781:
1704:
1662:
1617:
1591:
1564:
1563:
1437:
1318:
1219:
1142:
1108:graded-commutative
1093:
988:
892:
750:
723:
681:
604:
544:
494:
458:
364:
175:
144:
25:algebraic topology
23:, specifically in
2825:Binary operations
2673:Singular homology
2643:Milnor invariants
2359:{\displaystyle j}
2339:{\displaystyle i}
2293:{\displaystyle n}
2269:{\displaystyle M}
2179:Other definitions
1594:{\displaystyle X}
1542:
1345:ring homomorphism
468:is the canonical
115:topological space
2832:
2790:Allen Hatcher, "
2751:
2750:
2749:
2748:
2730:
2724:
2723:
2721:
2712:
2618:
2616:
2615:
2610:
2605:
2591:
2590:
2572:
2571:
2547:
2546:
2528:
2527:
2503:
2501:
2500:
2495:
2493:
2492:
2480:
2479:
2467:
2466:
2448:
2447:
2421:
2419:
2418:
2413:
2395:
2393:
2392:
2387:
2365:
2363:
2362:
2357:
2345:
2343:
2342:
2337:
2325:
2323:
2322:
2317:
2299:
2297:
2296:
2291:
2276:be an oriented
2275:
2273:
2272:
2267:
2248:
2246:
2245:
2240:
2238:
2237:
2232:
2171:(more generally
2162:
2160:
2159:
2154:
2152:
2151:
2123:
2121:
2120:
2115:
2113:
2112:
2100:
2099:
2087:
2086:
2057:
2055:
2054:
2049:
2044:
2043:
2025:
2024:
2003:
2002:
1990:
1989:
1964:
1962:
1961:
1956:
1948:
1947:
1929:
1928:
1907:
1906:
1894:
1893:
1867:
1865:
1864:
1859:
1842:
1841:
1820:
1819:
1807:
1806:
1790:
1788:
1787:
1782:
1771:
1770:
1743:
1742:
1730:
1729:
1713:
1711:
1710:
1705:
1671:
1669:
1668:
1663:
1626:
1624:
1623:
1618:
1600:
1598:
1597:
1592:
1577:in terms of the
1573:
1571:
1570:
1565:
1553:
1552:
1543:
1541:
1540:
1528:
1511:
1510:
1489:
1488:
1467:
1466:
1446:
1444:
1443:
1438:
1427:
1426:
1399:
1398:
1377:
1376:
1327:
1325:
1324:
1319:
1305:
1304:
1283:
1282:
1255:
1254:
1228:
1226:
1225:
1220:
1209:
1208:
1187:
1186:
1174:
1173:
1151:
1149:
1148:
1143:
1102:
1100:
1099:
1094:
1089:
1088:
1076:
1075:
1063:
1062:
1038:
1037:
1025:
1024:
997:
995:
994:
989:
975:
974:
947:
946:
925:
924:
901:
899:
898:
893:
885:
884:
883:
867:
866:
854:
853:
832:
831:
819:
818:
817:
798:
797:
785:
784:
759:
757:
756:
751:
749:
748:
732:
730:
729:
724:
722:
721:
690:
688:
687:
682:
680:
679:
613:
611:
610:
605:
603:
602:
553:
551:
550:
545:
503:
501:
500:
495:
467:
465:
464:
459:
409:
408:
373:
371:
370:
365:
360:
359:
305:
304:
289:
288:
249:
248:
224:
223:
211:
210:
189:-cochain, then
184:
182:
181:
176:
174:
173:
153:
151:
150:
145:
143:
142:
81:Samuel Eilenberg
2840:
2839:
2835:
2834:
2833:
2831:
2830:
2829:
2815:Homology theory
2805:
2804:
2755:
2754:
2746:
2744:
2732:
2731:
2727:
2719:
2714:
2713:
2706:
2701:
2678:Homology theory
2669:
2653:
2639:Massey products
2632:
2630:Massey products
2622:Similarly, the
2576:
2563:
2538:
2519:
2508:
2507:
2484:
2471:
2458:
2439:
2428:
2427:
2398:
2397:
2372:
2371:
2348:
2347:
2328:
2327:
2326:of codimension
2302:
2301:
2282:
2281:
2278:smooth manifold
2258:
2257:
2227:
2222:
2221:
2210:
2186:
2181:
2143:
2138:
2137:
2104:
2091:
2078:
2067:
2066:
2063:
2035:
2016:
1994:
1981:
1967:
1966:
1939:
1920:
1898:
1885:
1877:
1876:
1833:
1811:
1798:
1793:
1792:
1762:
1734:
1721:
1716:
1715:
1678:
1677:
1636:
1635:
1603:
1602:
1583:
1582:
1579:chain complexes
1544:
1532:
1502:
1480:
1458:
1452:
1451:
1412:
1390:
1368:
1357:
1356:
1353:
1296:
1274:
1246:
1241:
1240:
1232:is the induced
1200:
1178:
1165:
1160:
1159:
1122:
1121:
1080:
1067:
1051:
1029:
1016:
1011:
1010:
1004:
960:
938:
916:
911:
910:
875:
858:
845:
823:
809:
789:
776:
765:
764:
740:
735:
734:
713:
708:
707:
635:
624:
623:
570:
559:
558:
506:
505:
474:
473:
400:
395:
394:
315:
296:
259:
240:
215:
202:
194:
193:
165:
160:
159:
134:
129:
128:
104:cohomology ring
89:
77:Hassler Whitney
69:J. W. Alexander
65:cohomology ring
17:
12:
11:
5:
2838:
2836:
2828:
2827:
2822:
2817:
2807:
2806:
2803:
2802:
2788:
2778:Glen E. Bredon
2775:
2753:
2752:
2725:
2703:
2702:
2700:
2697:
2696:
2695:
2690:
2688:Massey product
2685:
2680:
2675:
2668:
2665:
2657:Massey product
2651:Massey product
2649:Main article:
2631:
2628:
2624:linking number
2608:
2604:
2600:
2597:
2594:
2589:
2586:
2583:
2579:
2575:
2570:
2566:
2562:
2559:
2556:
2553:
2550:
2545:
2541:
2537:
2534:
2531:
2526:
2522:
2518:
2515:
2491:
2487:
2483:
2478:
2474:
2470:
2465:
2461:
2457:
2454:
2451:
2446:
2442:
2438:
2435:
2411:
2408:
2405:
2385:
2382:
2379:
2355:
2335:
2315:
2312:
2309:
2289:
2265:
2236:
2231:
2218:linking number
2209:
2206:
2185:
2182:
2180:
2177:
2150:
2146:
2111:
2107:
2103:
2098:
2094:
2090:
2085:
2081:
2077:
2074:
2062:
2059:
2047:
2042:
2038:
2034:
2031:
2028:
2023:
2019:
2015:
2012:
2009:
2006:
2001:
1997:
1993:
1988:
1984:
1980:
1977:
1974:
1954:
1951:
1946:
1942:
1938:
1935:
1932:
1927:
1923:
1919:
1916:
1913:
1910:
1905:
1901:
1897:
1892:
1888:
1884:
1857:
1854:
1851:
1848:
1845:
1840:
1836:
1832:
1829:
1826:
1823:
1818:
1814:
1810:
1805:
1801:
1780:
1777:
1774:
1769:
1765:
1761:
1758:
1755:
1752:
1749:
1746:
1741:
1737:
1733:
1728:
1724:
1714:induces a map
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1616:
1613:
1610:
1590:
1562:
1559:
1556:
1551:
1547:
1539:
1535:
1531:
1526:
1523:
1520:
1517:
1514:
1509:
1505:
1501:
1498:
1495:
1492:
1487:
1483:
1479:
1476:
1473:
1470:
1465:
1461:
1436:
1433:
1430:
1425:
1422:
1419:
1415:
1411:
1408:
1405:
1402:
1397:
1393:
1389:
1386:
1383:
1380:
1375:
1371:
1367:
1364:
1352:
1351:Interpretation
1349:
1343:is a (graded)
1329:
1328:
1317:
1314:
1311:
1308:
1303:
1299:
1295:
1292:
1289:
1286:
1281:
1277:
1273:
1270:
1267:
1264:
1261:
1258:
1253:
1249:
1230:
1229:
1218:
1215:
1212:
1207:
1203:
1199:
1196:
1193:
1190:
1185:
1181:
1177:
1172:
1168:
1153:
1152:
1141:
1138:
1135:
1132:
1129:
1104:
1103:
1092:
1087:
1083:
1079:
1074:
1070:
1066:
1061:
1058:
1054:
1050:
1047:
1044:
1041:
1036:
1032:
1028:
1023:
1019:
1003:
1000:
999:
998:
987:
984:
981:
978:
973:
970:
967:
963:
959:
956:
953:
950:
945:
941:
937:
934:
931:
928:
923:
919:
903:
902:
891:
888:
882:
878:
873:
870:
865:
861:
857:
852:
848:
844:
841:
838:
835:
830:
826:
822:
816:
812:
807:
804:
801:
796:
792:
788:
783:
779:
775:
772:
747:
743:
720:
716:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
645:
642:
638:
634:
631:
601:
598:
595:
592:
589:
586:
583:
580:
577:
573:
569:
566:
543:
540:
537:
534:
531:
528:
525:
522:
519:
516:
513:
493:
490:
487:
484:
481:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
418:
415:
412:
407:
403:
375:
374:
363:
358:
355:
352:
349:
346:
343:
340:
337:
334:
331:
328:
325:
322:
318:
314:
311:
308:
303:
299:
295:
292:
287:
284:
281:
278:
275:
272:
269:
266:
262:
258:
255:
252:
247:
243:
239:
236:
233:
230:
227:
222:
218:
214:
209:
205:
201:
172:
168:
158:-cochain and
141:
137:
88:
85:
63:), called the
15:
13:
10:
9:
6:
4:
3:
2:
2837:
2826:
2823:
2821:
2818:
2816:
2813:
2812:
2810:
2801:
2800:0-521-79540-0
2797:
2793:
2789:
2787:
2786:0-387-97926-3
2783:
2779:
2776:
2773:
2772:0-201-62728-0
2769:
2765:
2764:0-201-04586-9
2761:
2757:
2756:
2742:
2738:
2737:
2729:
2726:
2718:
2711:
2709:
2705:
2698:
2694:
2693:Torelli group
2691:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2670:
2666:
2664:
2662:
2658:
2652:
2644:
2640:
2636:
2629:
2627:
2625:
2620:
2598:
2595:
2587:
2584:
2581:
2577:
2573:
2568:
2560:
2557:
2554:
2548:
2543:
2535:
2529:
2524:
2516:
2505:
2489:
2485:
2481:
2476:
2472:
2468:
2463:
2455:
2449:
2444:
2436:
2425:
2424:Poincaré dual
2409:
2406:
2403:
2383:
2380:
2377:
2369:
2353:
2333:
2313:
2310:
2307:
2287:
2280:of dimension
2279:
2263:
2256:Indeed, let
2254:
2234:
2219:
2214:
2207:
2205:
2203:
2199:
2198:wedge product
2195:
2191:
2183:
2178:
2176:
2174:
2170:
2166:
2148:
2144:
2135:
2131:
2127:
2109:
2105:
2101:
2096:
2092:
2088:
2083:
2079:
2075:
2072:
2060:
2058:
2045:
2040:
2036:
2032:
2029:
2026:
2021:
2017:
2013:
2010:
2007:
1999:
1995:
1991:
1986:
1982:
1975:
1972:
1952:
1949:
1944:
1940:
1936:
1933:
1930:
1925:
1921:
1917:
1914:
1911:
1903:
1899:
1895:
1890:
1886:
1873:
1871:
1852:
1849:
1846:
1838:
1834:
1824:
1816:
1812:
1808:
1803:
1775:
1767:
1763:
1753:
1750:
1747:
1739:
1735:
1731:
1726:
1701:
1698:
1695:
1689:
1686:
1673:
1659:
1656:
1653:
1647:
1644:
1634:
1630:
1614:
1611:
1608:
1588:
1580:
1574:
1557:
1549:
1545:
1537:
1521:
1518:
1515:
1507:
1503:
1493:
1485:
1481:
1477:
1471:
1463:
1459:
1448:
1431:
1423:
1420:
1417:
1413:
1403:
1395:
1391:
1387:
1381:
1373:
1369:
1365:
1362:
1350:
1348:
1346:
1342:
1338:
1334:
1315:
1309:
1301:
1297:
1293:
1287:
1279:
1275:
1271:
1265:
1262:
1259:
1251:
1247:
1239:
1238:
1237:
1235:
1213:
1205:
1201:
1191:
1183:
1179:
1175:
1170:
1166:
1158:
1157:
1156:
1139:
1133:
1130:
1127:
1120:
1119:
1118:
1116:
1111:
1109:
1085:
1081:
1077:
1072:
1068:
1059:
1056:
1048:
1045:
1039:
1034:
1030:
1026:
1021:
1017:
1009:
1008:
1007:
1001:
985:
979:
971:
968:
965:
961:
951:
943:
939:
935:
929:
921:
917:
909:
908:
907:
889:
880:
876:
871:
868:
863:
859:
850:
842:
839:
833:
828:
824:
820:
814:
810:
805:
802:
794:
790:
786:
781:
777:
770:
763:
762:
761:
745:
741:
718:
714:
705:
700:
698:
694:
676:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
636:
632:
629:
621:
617:
599:
596:
593:
590:
587:
584:
581:
578:
575:
571:
567:
564:
555:
538:
535:
532:
529:
526:
523:
520:
517:
514:
488:
485:
482:
471:
452:
449:
446:
443:
440:
437:
434:
431:
428:
425:
422:
416:
413:
410:
405:
401:
392:
388:
384:
380:
377:where σ is a
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
326:
323:
320:
316:
312:
309:
301:
297:
293:
285:
282:
279:
276:
273:
270:
267:
264:
260:
256:
253:
245:
241:
237:
231:
220:
216:
212:
207:
203:
192:
191:
190:
188:
170:
166:
157:
139:
135:
126:
121:
119:
116:
112:
108:
105:
102:
98:
94:
86:
84:
82:
78:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
2766:(hardcover)
2745:, retrieved
2735:
2728:
2654:
2621:
2506:
2368:transversely
2255:
2252:
2187:
2172:
2168:
2164:
2129:
2125:
2064:
1874:
1674:
1576:
1450:
1354:
1340:
1336:
1332:
1330:
1234:homomorphism
1231:
1154:
1112:
1105:
1005:
904:
760:is given by
701:
696:
692:
619:
615:
557:Informally,
556:
386:
382:
376:
186:
155:
122:
117:
110:
106:
96:
90:
60:
56:
52:
48:
44:
40:
36:
28:
18:
2774:(paperback)
2683:Cap product
1870:cap product
1629:Künneth map
97:cup product
73:Eduard Čech
29:cup product
21:mathematics
2809:Categories
2747:2018-04-26
2699:References
2366:intersect
1115:functorial
1002:Properties
704:coboundary
620:front face
87:Definition
35:of degree
2574:∈
2569:∗
2558:∩
2544:∗
2530:⌣
2525:∗
2469:∈
2464:∗
2445:∗
2381:∩
2102:∨
2089:∨
2033:⌣
2014:⌣
1976:⌣
1950:⌣
1931:⌣
1912:⌣
1850:×
1839:∙
1831:→
1817:∙
1809::
1804:∗
1800:Δ
1768:∙
1760:→
1751:×
1740:∙
1732::
1727:∗
1723:Δ
1699:×
1693:→
1687::
1684:Δ
1657:×
1651:→
1645::
1642:Δ
1612:×
1550:∙
1538:∗
1534:Δ
1530:→
1519:×
1508:∙
1500:→
1486:∙
1478:×
1464:∙
1410:→
1388:×
1366::
1363:⌣
1310:β
1302:∗
1294:⌣
1288:α
1280:∗
1266:β
1263:⌣
1260:α
1252:∗
1206:∗
1198:→
1184:∗
1176::
1171:∗
1137:→
1131::
1082:α
1078:⌣
1069:β
1046:−
1031:β
1027:⌣
1018:α
958:→
936:×
877:β
872:δ
869:⌣
860:α
840:−
825:β
821:⌣
811:α
806:δ
791:β
787:⌣
778:α
771:δ
742:β
715:α
697:back face
637:ι
633:∘
630:σ
572:ι
568:∘
565:σ
470:embedding
417:⊂
402:ι
317:ι
313:∘
310:σ
298:β
294:⋅
261:ι
257:∘
254:σ
242:α
232:σ
217:β
213:⌣
204:α
167:β
136:α
83:in 1944.
2741:archived
2667:See also
2134:cochains
2061:Examples
1633:diagonal
379:singular
125:cochains
33:cocycles
691:is the
614:is the
391:simplex
113:) of a
2798:
2784:
2770:
2762:
2202:closed
101:graded
95:, the
27:, the
2720:(PDF)
185:is a
154:is a
127:: if
2796:ISBN
2782:ISBN
2768:ISBN
2760:ISBN
2346:and
2216:The
1965:and
1601:and
733:and
702:The
695:-th
622:and
618:-th
393:and
75:and
39:and
2188:In
1581:of
389:) -
91:In
19:In
2811::
2739:,
2707:^
2619:.
2076::=
1872:.
1672:.
1347:.
1110:.
554:.
385:+
120:.
71:,
47:+
2722:.
2645:.
2607:)
2603:Z
2599:,
2596:X
2593:(
2588:j
2585:+
2582:i
2578:H
2565:]
2561:B
2555:A
2552:[
2549:=
2540:]
2536:B
2533:[
2521:]
2517:A
2514:[
2490:j
2486:H
2482:,
2477:i
2473:H
2460:]
2456:B
2453:[
2450:,
2441:]
2437:A
2434:[
2410:j
2407:+
2404:i
2384:B
2378:A
2354:j
2334:i
2314:B
2311:,
2308:A
2288:n
2264:M
2235:3
2230:R
2173:M
2169:Z
2165:T
2149:1
2145:S
2130:X
2126:T
2110:1
2106:S
2097:1
2093:S
2084:2
2080:S
2073:X
2046:.
2041:2
2037:v
2030:u
2027:+
2022:1
2018:v
2011:u
2008:=
2005:)
2000:2
1996:v
1992:+
1987:1
1983:v
1979:(
1973:u
1953:v
1945:2
1941:u
1937:+
1934:v
1926:1
1922:u
1918:=
1915:v
1909:)
1904:2
1900:u
1896:+
1891:1
1887:u
1883:(
1856:)
1853:X
1847:X
1844:(
1835:H
1828:)
1825:X
1822:(
1813:H
1779:)
1776:X
1773:(
1764:H
1757:)
1754:X
1748:X
1745:(
1736:H
1702:X
1696:X
1690:X
1660:X
1654:X
1648:X
1615:X
1609:X
1589:X
1561:)
1558:X
1555:(
1546:C
1525:)
1522:X
1516:X
1513:(
1504:C
1497:)
1494:X
1491:(
1482:C
1475:)
1472:X
1469:(
1460:C
1435:)
1432:X
1429:(
1424:q
1421:+
1418:p
1414:H
1407:)
1404:X
1401:(
1396:q
1392:H
1385:)
1382:X
1379:(
1374:p
1370:H
1341:f
1337:Y
1335:(
1333:H
1316:,
1313:)
1307:(
1298:f
1291:)
1285:(
1276:f
1272:=
1269:)
1257:(
1248:f
1217:)
1214:X
1211:(
1202:H
1195:)
1192:Y
1189:(
1180:H
1167:f
1140:Y
1134:X
1128:f
1091:)
1086:p
1073:q
1065:(
1060:q
1057:p
1053:)
1049:1
1043:(
1040:=
1035:q
1022:p
986:.
983:)
980:X
977:(
972:q
969:+
966:p
962:H
955:)
952:X
949:(
944:q
940:H
933:)
930:X
927:(
922:p
918:H
890:.
887:)
881:q
864:p
856:(
851:p
847:)
843:1
837:(
834:+
829:q
815:p
803:=
800:)
795:q
782:p
774:(
746:q
719:p
693:q
677:q
674:+
671:p
668:,
665:.
662:.
659:.
656:,
653:1
650:+
647:p
644:,
641:p
616:p
600:p
597:,
594:.
591:.
588:.
585:,
582:1
579:,
576:0
542:}
539:q
536:+
533:p
530:,
527:.
524:.
521:.
518:,
515:0
512:{
492:)
489:q
486:+
483:p
480:(
456:}
453:q
450:+
447:p
444:,
441:.
438:.
435:.
432:,
429:1
426:,
423:0
420:{
414:S
411:,
406:S
387:q
383:p
381:(
362:)
357:q
354:+
351:p
348:,
345:.
342:.
339:.
336:,
333:1
330:+
327:p
324:,
321:p
307:(
302:q
291:)
286:p
283:.
280:.
277:.
274:,
271:1
268:,
265:0
251:(
246:p
238:=
235:)
229:(
226:)
221:q
208:p
200:(
187:q
171:q
156:p
140:p
118:X
111:X
109:(
107:H
61:X
59:(
57:H
53:X
49:q
45:p
41:q
37:p
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