36:
3060:
2705:
1880:
832:
must be all be zero. Then one notes that, in general, the
Lefschetz number can also be computed using the alternating sum of the matrix traces of the aforementioned linear maps (this is true for almost exactly the same reason that the
415:
1526:
966:
1142:
can be thought of as an identity matrix, and so each trace term is simply the dimension of the appropriate homology group. Thus the
Lefschetz number of the identity map is equal to the alternating sum of the
2889:
1233:
2457:
223:
1599:
1425:
841:
for the relation to the Euler characteristic). In the particular case of a fixed-point-free simplicial map, all of the diagonal values are zero, and thus the traces are all zero.
490:
558:
2561:
2401:
1069:
1006:
1753:
1553:
710:
283:
3165:
2737:
2320:
2245:
2854:
1140:
1177:
656:
2881:
2777:
2757:
2484:
2355:
2015:
1988:
1937:
1910:
1379:
1352:
1322:
1295:
2553:
2128:
2820:
2797:
2524:
2504:
2285:
2265:
2216:
2196:
2176:
2148:
2102:
2082:
2059:
2039:
1957:
1745:
1725:
1701:
1681:
1658:
1638:
1265:
1093:
1026:
871:
830:
802:
782:
762:
730:
680:
621:
601:
581:
514:
446:
311:
303:
253:
160:
132:
1430:
3091:
879:
3290:
3285:
3127:
808:(i.e., it sends each simplex to a different simplex). This means that the diagonal values of the matrices of the linear maps induced on the
79:
57:
658:. In fact, since the Lefschetz number has been defined at the homology level, the conclusion can be extended to say that any map
741:
3055:{\displaystyle \#X(k)=q^{\dim X}\sum _{i}(-1)^{i}\mathrm {tr} ((\Phi _{q}^{-1})^{*}|H^{i}({\bar {X}},\mathbb {Q} _{\ell })).}
105:
3280:
3263:
1185:
1096:
3258:
1244:
170:
2406:
101:
50:
44:
195:
3086:
2700:{\displaystyle \#X(k)=\sum _{i}(-1)^{i}\mathrm {tr} (F_{q}^{*}|H_{c}^{i}({\bar {X}},\mathbb {Q} _{\ell })).}
1558:
1384:
451:
2289:
529:
135:
61:
2360:
1031:
3209:
3253:
1875:{\displaystyle \Lambda _{f,g}=\sum (-1)^{k}\mathrm {tr} (D_{X}\circ g^{*}\circ D_{Y}^{-1}\circ f_{*}),}
974:
1148:
834:
2710:
This formula involves the trace of the
Frobenius on the Ă©tale cohomology, with compact supports, of
3081:
2018:
809:
1531:
688:
261:
3226:
3184:
2823:
233:
2713:
2296:
2221:
2832:
1118:
3204:
3156:
3123:
1614:
1153:
1108:
1072:
493:
425:
174:
163:
112:
17:
3218:
3174:
626:
3238:
3196:
3137:
2859:
2762:
2742:
2462:
2333:
1993:
1966:
1915:
1888:
1357:
1330:
1300:
1273:
181:
fixed point must have rather special topological properties (like a rotation of a circle).
3234:
3192:
3133:
3119:
3069:
2529:
2150:
be the identity map gives a simpler result, which we now know as the fixed-point theorem.
517:
410:{\displaystyle \Lambda _{f}:=\sum _{k\geq 0}(-1)^{k}\mathrm {tr} (H_{k}(f,\mathbb {Q} )),}
27:
Counts the fixed points of a continuous mapping from a compact topological space to itself
2107:
1613:). Lefschetz's focus was not on fixed points of maps, but rather on what are now called
2805:
2782:
2509:
2489:
2270:
2250:
2201:
2181:
2161:
2133:
2087:
2067:
2044:
2024:
1942:
1730:
1710:
1686:
1666:
1643:
1623:
1250:
1078:
1011:
856:
815:
805:
787:
767:
747:
715:
665:
606:
586:
566:
499:
431:
288:
238:
229:
145:
139:
117:
3065:
This formula involves usual cohomology, rather than cohomology with compact supports.
3274:
3111:
109:
1144:
421:
1521:{\displaystyle f_{*}\colon H_{0}(D^{n},\mathbb {Q} )\to H_{0}(D^{n},\mathbb {Q} )}
732:
has fixed points, as is the case for the identity map on odd-dimensional spheres.
93:
961:{\displaystyle \sum _{x\in \mathrm {Fix} (f)}\mathrm {ind} (f,x)=\Lambda _{f},}
1960:
1112:
1268:
659:
1661:
177:. A weak version of the theorem is enough to show that a mapping without
3230:
3188:
3222:
3179:
3160:
2555:. The Lefschetz trace formula holds in this context, and reads:
2064:
Lefschetz proved that if the coincidence number is nonzero, then
835:
Euler characteristic has a definition in terms of homology groups
523:
A simple version of the
Lefschetz fixed-point theorem states: if
3161:"Intersections and transformations of complexes and manifolds"
583:
has at least one fixed point, i.e., there exists at least one
29:
2104:
have a coincidence point. He noted in his paper that letting
1354:
is compact and triangulable, all its homology groups except
838:
3068:
The
Lefschetz trace formula can also be generalized to
685:
Note however that the converse is not true in general:
764:
has no fixed points, then (possibly after subdividing
2892:
2862:
2835:
2808:
2785:
2765:
2745:
2716:
2564:
2532:
2512:
2492:
2465:
2409:
2363:
2336:
2299:
2273:
2253:
2224:
2204:
2184:
2164:
2136:
2110:
2090:
2070:
2047:
2027:
1996:
1969:
1945:
1918:
1891:
1756:
1733:
1713:
1689:
1669:
1646:
1626:
1561:
1534:
1528:, whose trace is one; all this together implies that
1433:
1387:
1360:
1333:
1303:
1276:
1253:
1188:
1156:
1121:
1081:
1034:
1014:
977:
882:
859:
818:
790:
770:
750:
718:
691:
668:
629:
609:
589:
569:
532:
502:
454:
434:
314:
291:
264:
241:
198:
148:
120:
1228:{\displaystyle \Lambda _{\mathrm {id} }=\chi (X).\ }
3118:. Vol. 200 (2nd ed.). Berlin, New York:
3054:
2875:
2848:
2814:
2791:
2771:
2751:
2731:
2699:
2547:
2518:
2498:
2478:
2451:
2395:
2349:
2314:
2279:
2259:
2239:
2210:
2190:
2170:
2142:
2122:
2096:
2076:
2053:
2033:
2009:
1982:
1951:
1931:
1904:
1874:
1739:
1719:
1695:
1675:
1652:
1632:
1593:
1547:
1520:
1419:
1373:
1346:
1316:
1289:
1259:
1247:, which states that every continuous map from the
1243:The Lefschetz fixed-point theorem generalizes the
1227:
1171:
1134:
1087:
1063:
1020:
1000:
960:
865:
849:A stronger form of the theorem, also known as the
824:
796:
776:
756:
724:
704:
674:
650:
615:
595:
575:
552:
508:
484:
440:
409:
297:
277:
247:
217:
154:
126:
3166:Transactions of the American Mathematical Society
2826:, this formula can be rewritten in terms of the
1609:Lefschetz presented his fixed-point theorem in (
1115:can be easily computed by realizing that each
8:
1147:of the space, which in turn is equal to the
2452:{\displaystyle x_{1}^{q},\ldots ,x_{n}^{q}}
2178:be a variety defined over the finite field
1239:Relation to the Brouwer fixed-point theorem
189:For a formal statement of the theorem, let
873:has only finitely many fixed points, then
3178:
3092:Holomorphic Lefschetz fixed-point formula
3037:
3033:
3032:
3017:
3016:
3007:
2998:
2992:
2979:
2974:
2956:
2950:
2931:
2915:
2891:
2867:
2861:
2840:
2834:
2807:
2784:
2764:
2744:
2718:
2717:
2715:
2682:
2678:
2677:
2662:
2661:
2652:
2647:
2638:
2632:
2627:
2612:
2606:
2587:
2563:
2531:
2511:
2491:
2470:
2464:
2443:
2438:
2419:
2414:
2408:
2387:
2368:
2362:
2341:
2335:
2301:
2300:
2298:
2272:
2252:
2226:
2225:
2223:
2203:
2183:
2163:
2135:
2109:
2089:
2069:
2046:
2026:
2001:
1995:
1974:
1968:
1944:
1923:
1917:
1896:
1890:
1860:
1844:
1839:
1826:
1813:
1798:
1792:
1761:
1755:
1732:
1712:
1688:
1668:
1645:
1625:
1610:
1585:
1572:
1560:
1539:
1533:
1511:
1510:
1501:
1488:
1474:
1473:
1464:
1451:
1438:
1432:
1411:
1398:
1386:
1365:
1359:
1338:
1332:
1308:
1302:
1281:
1275:
1252:
1194:
1193:
1187:
1155:
1126:
1120:
1080:
1035:
1033:
1013:
978:
976:
949:
916:
894:
887:
881:
858:
817:
789:
769:
749:
717:
696:
690:
667:
628:
608:
588:
568:
549:
537:
531:
501:
475:
474:
459:
453:
433:
394:
393:
378:
363:
357:
332:
319:
313:
290:
269:
263:
240:
214:
197:
147:
119:
80:Learn how and when to remove this message
218:{\displaystyle f\colon X\rightarrow X\,}
43:This article includes a list of general
3103:
2526:; the set of such points is denoted by
1963:groups with rational coefficients, and
3207:(1937). "On the fixed point formula".
1594:{\displaystyle f\colon D^{n}\to D^{n}}
1420:{\displaystyle f\colon D^{n}\to D^{n}}
169:The counting is subject to an imputed
485:{\displaystyle H_{k}(X,\mathbb {Q} )}
7:
1324:must have at least one fixed point.
1103:Relation to the Euler characteristic
1095:. From this theorem one deduces the
553:{\displaystyle \Lambda _{f}\neq 0\,}
420:the alternating (finite) sum of the
2396:{\displaystyle x_{1},\ldots ,x_{n}}
1555:is non-zero for any continuous map
1381:are zero, and every continuous map
1064:{\displaystyle \mathrm {ind} (f,x)}
804:is homotopic to a fixed-point-free
2971:
2960:
2957:
2893:
2837:
2616:
2613:
2565:
1802:
1799:
1758:
1536:
1198:
1195:
1190:
1042:
1039:
1036:
985:
982:
979:
946:
923:
920:
917:
901:
898:
895:
693:
534:
367:
364:
316:
266:
49:it lacks sufficient corresponding
25:
1001:{\displaystyle \mathrm {Fix} (f)}
2357:, maps a point with coordinates
742:simplicial approximation theorem
34:
2856:, which acts as the inverse of
1939:is the homomorphism induced by
166:, who first stated it in 1926.
138:of the induced mappings on the
3291:Theorems in algebraic topology
3286:Theory of continuous functions
3116:Lectures on algebraic topology
3046:
3043:
3022:
3013:
2999:
2989:
2967:
2964:
2947:
2937:
2905:
2899:
2723:
2691:
2688:
2667:
2658:
2639:
2620:
2603:
2593:
2577:
2571:
2542:
2536:
2403:to the point with coordinates
2306:
2231:
1866:
1806:
1789:
1779:
1578:
1515:
1494:
1481:
1478:
1457:
1404:
1216:
1210:
1166:
1160:
1058:
1046:
1008:is the set of fixed points of
995:
989:
939:
927:
911:
905:
639:
633:
479:
465:
401:
398:
384:
371:
354:
344:
208:
18:Grothendieck–Lefschetz formula
1:
1327:This can be seen as follows:
100:is a formula that counts the
98:Lefschetz fixed-point theorem
2739:with values in the field of
2267:to the algebraic closure of
1705:Lefschetz coincidence number
1548:{\displaystyle \Lambda _{f}}
1107:The Lefschetz number of the
705:{\displaystyle \Lambda _{f}}
278:{\displaystyle \Lambda _{f}}
173:at a fixed point called the
3259:Encyclopedia of Mathematics
2459:. Thus the fixed points of
1703:of the same dimension, the
1245:Brouwer fixed-point theorem
682:has a fixed point as well.
3307:
2732:{\displaystyle {\bar {X}}}
2486:are exactly the points of
2315:{\displaystyle {\bar {X}}}
2240:{\displaystyle {\bar {X}}}
1683:to an orientable manifold
2849:{\displaystyle \Phi _{q}}
1427:induces the identity map
1135:{\displaystyle f_{\ast }}
1172:{\displaystyle \chi (X)}
810:simplicial chain complex
3087:Lefschetz zeta function
740:First, by applying the
134:to itself by means of
64:more precise citations.
3142:, Proposition VII.6.6.
3056:
2877:
2850:
2816:
2793:
2779:is a prime coprime to
2773:
2753:
2733:
2701:
2549:
2520:
2500:
2480:
2453:
2397:
2351:
2316:
2290:Frobenius endomorphism
2281:
2261:
2247:be the base change of
2241:
2212:
2192:
2172:
2144:
2124:
2098:
2078:
2055:
2035:
2011:
1984:
1953:
1933:
1906:
1876:
1741:
1721:
1697:
1677:
1654:
1634:
1595:
1549:
1522:
1421:
1375:
1348:
1318:
1291:
1261:
1229:
1173:
1136:
1089:
1065:
1022:
1002:
962:
867:
851:Lefschetz–Hopf theorem
845:Lefschetz–Hopf theorem
826:
798:
778:
758:
726:
706:
676:
652:
651:{\displaystyle f(x)=x}
617:
597:
577:
554:
510:
486:
442:
411:
299:
279:
255:to itself. Define the
249:
219:
156:
128:
3210:Annals of Mathematics
3057:
2878:
2876:{\displaystyle F_{q}}
2851:
2817:
2794:
2774:
2772:{\displaystyle \ell }
2759:-adic numbers, where
2754:
2752:{\displaystyle \ell }
2734:
2702:
2550:
2521:
2501:
2481:
2479:{\displaystyle F_{q}}
2454:
2398:
2352:
2350:{\displaystyle F_{q}}
2317:
2282:
2262:
2242:
2213:
2193:
2173:
2145:
2125:
2099:
2079:
2056:
2036:
2012:
2010:{\displaystyle D_{Y}}
1985:
1983:{\displaystyle D_{X}}
1954:
1934:
1932:{\displaystyle g_{*}}
1907:
1905:{\displaystyle f_{*}}
1877:
1742:
1722:
1698:
1678:
1655:
1635:
1596:
1550:
1523:
1422:
1376:
1374:{\displaystyle H_{0}}
1349:
1347:{\displaystyle D^{n}}
1319:
1317:{\displaystyle D^{n}}
1292:
1290:{\displaystyle D^{n}}
1262:
1230:
1174:
1137:
1097:Poincaré–Hopf theorem
1090:
1066:
1023:
1003:
963:
868:
827:
799:
779:
759:
727:
707:
677:
653:
618:
598:
578:
555:
511:
487:
443:
412:
300:
280:
250:
220:
157:
129:
3281:Fixed-point theorems
3082:Fixed-point theorems
3072:over finite fields.
2890:
2860:
2833:
2828:arithmetic Frobenius
2806:
2783:
2763:
2743:
2714:
2562:
2548:{\displaystyle X(k)}
2530:
2510:
2506:with coordinates in
2490:
2463:
2407:
2361:
2334:
2297:
2271:
2251:
2222:
2202:
2182:
2162:
2134:
2108:
2088:
2068:
2045:
2025:
1994:
1967:
1943:
1916:
1889:
1754:
1731:
1711:
1687:
1667:
1644:
1624:
1559:
1532:
1431:
1385:
1358:
1331:
1301:
1274:
1251:
1186:
1154:
1149:Euler characteristic
1119:
1079:
1032:
1012:
975:
880:
857:
816:
788:
768:
748:
744:, one shows that if
716:
712:may be zero even if
689:
666:
627:
607:
587:
567:
530:
500:
452:
432:
312:
289:
262:
239:
196:
162:. It is named after
146:
118:
3254:"Lefschetz formula"
2987:
2657:
2637:
2448:
2424:
2324:geometric Frobenius
2123:{\displaystyle X=Y}
1852:
1660:from an orientable
1099:for vector fields.
1075:of the fixed point
424:of the linear maps
3205:Lefschetz, Solomon
3157:Lefschetz, Solomon
3052:
2970:
2936:
2873:
2846:
2812:
2789:
2769:
2749:
2729:
2697:
2643:
2623:
2592:
2545:
2516:
2496:
2476:
2449:
2434:
2410:
2393:
2347:
2312:
2277:
2257:
2237:
2208:
2188:
2168:
2140:
2120:
2094:
2074:
2051:
2031:
2007:
1980:
1949:
1929:
1902:
1872:
1835:
1737:
1717:
1693:
1673:
1650:
1630:
1615:coincidence points
1605:Historical context
1591:
1545:
1518:
1417:
1371:
1344:
1314:
1287:
1257:
1225:
1169:
1132:
1085:
1061:
1018:
998:
958:
915:
863:
853:, states that, if
822:
794:
774:
754:
722:
702:
672:
648:
613:
593:
573:
550:
506:
482:
438:
407:
343:
295:
275:
245:
234:triangulable space
215:
152:
124:
106:continuous mapping
3129:978-3-540-10369-1
3025:
2927:
2815:{\displaystyle X}
2792:{\displaystyle q}
2726:
2670:
2583:
2519:{\displaystyle k}
2499:{\displaystyle X}
2309:
2280:{\displaystyle k}
2260:{\displaystyle X}
2234:
2218:elements and let
2211:{\displaystyle q}
2191:{\displaystyle k}
2171:{\displaystyle X}
2143:{\displaystyle g}
2097:{\displaystyle g}
2077:{\displaystyle f}
2054:{\displaystyle Y}
2034:{\displaystyle X}
2021:isomorphisms for
1952:{\displaystyle g}
1740:{\displaystyle g}
1720:{\displaystyle f}
1696:{\displaystyle Y}
1676:{\displaystyle X}
1653:{\displaystyle g}
1633:{\displaystyle f}
1260:{\displaystyle n}
1224:
1088:{\displaystyle x}
1021:{\displaystyle f}
883:
866:{\displaystyle f}
825:{\displaystyle X}
797:{\displaystyle f}
777:{\displaystyle X}
757:{\displaystyle f}
736:Sketch of a proof
725:{\displaystyle f}
675:{\displaystyle f}
616:{\displaystyle X}
596:{\displaystyle x}
576:{\displaystyle f}
509:{\displaystyle X}
494:singular homology
441:{\displaystyle f}
328:
298:{\displaystyle f}
248:{\displaystyle X}
175:fixed-point index
164:Solomon Lefschetz
155:{\displaystyle X}
127:{\displaystyle X}
113:topological space
90:
89:
82:
16:(Redirected from
3298:
3267:
3242:
3200:
3182:
3143:
3141:
3108:
3070:algebraic stacks
3061:
3059:
3058:
3053:
3042:
3041:
3036:
3027:
3026:
3018:
3012:
3011:
3002:
2997:
2996:
2986:
2978:
2963:
2955:
2954:
2935:
2926:
2925:
2882:
2880:
2879:
2874:
2872:
2871:
2855:
2853:
2852:
2847:
2845:
2844:
2821:
2819:
2818:
2813:
2798:
2796:
2795:
2790:
2778:
2776:
2775:
2770:
2758:
2756:
2755:
2750:
2738:
2736:
2735:
2730:
2728:
2727:
2719:
2706:
2704:
2703:
2698:
2687:
2686:
2681:
2672:
2671:
2663:
2656:
2651:
2642:
2636:
2631:
2619:
2611:
2610:
2591:
2554:
2552:
2551:
2546:
2525:
2523:
2522:
2517:
2505:
2503:
2502:
2497:
2485:
2483:
2482:
2477:
2475:
2474:
2458:
2456:
2455:
2450:
2447:
2442:
2423:
2418:
2402:
2400:
2399:
2394:
2392:
2391:
2373:
2372:
2356:
2354:
2353:
2348:
2346:
2345:
2321:
2319:
2318:
2313:
2311:
2310:
2302:
2286:
2284:
2283:
2278:
2266:
2264:
2263:
2258:
2246:
2244:
2243:
2238:
2236:
2235:
2227:
2217:
2215:
2214:
2209:
2197:
2195:
2194:
2189:
2177:
2175:
2174:
2169:
2149:
2147:
2146:
2141:
2129:
2127:
2126:
2121:
2103:
2101:
2100:
2095:
2083:
2081:
2080:
2075:
2061:, respectively.
2060:
2058:
2057:
2052:
2040:
2038:
2037:
2032:
2019:Poincaré duality
2016:
2014:
2013:
2008:
2006:
2005:
1989:
1987:
1986:
1981:
1979:
1978:
1958:
1956:
1955:
1950:
1938:
1936:
1935:
1930:
1928:
1927:
1911:
1909:
1908:
1903:
1901:
1900:
1881:
1879:
1878:
1873:
1865:
1864:
1851:
1843:
1831:
1830:
1818:
1817:
1805:
1797:
1796:
1772:
1771:
1746:
1744:
1743:
1738:
1726:
1724:
1723:
1718:
1702:
1700:
1699:
1694:
1682:
1680:
1679:
1674:
1659:
1657:
1656:
1651:
1639:
1637:
1636:
1631:
1600:
1598:
1597:
1592:
1590:
1589:
1577:
1576:
1554:
1552:
1551:
1546:
1544:
1543:
1527:
1525:
1524:
1519:
1514:
1506:
1505:
1493:
1492:
1477:
1469:
1468:
1456:
1455:
1443:
1442:
1426:
1424:
1423:
1418:
1416:
1415:
1403:
1402:
1380:
1378:
1377:
1372:
1370:
1369:
1353:
1351:
1350:
1345:
1343:
1342:
1323:
1321:
1320:
1315:
1313:
1312:
1296:
1294:
1293:
1288:
1286:
1285:
1269:closed unit disk
1266:
1264:
1263:
1258:
1234:
1232:
1231:
1226:
1222:
1203:
1202:
1201:
1178:
1176:
1175:
1170:
1141:
1139:
1138:
1133:
1131:
1130:
1094:
1092:
1091:
1086:
1070:
1068:
1067:
1062:
1045:
1027:
1025:
1024:
1019:
1007:
1005:
1004:
999:
988:
967:
965:
964:
959:
954:
953:
926:
914:
904:
872:
870:
869:
864:
831:
829:
828:
823:
803:
801:
800:
795:
783:
781:
780:
775:
763:
761:
760:
755:
731:
729:
728:
723:
711:
709:
708:
703:
701:
700:
681:
679:
678:
673:
657:
655:
654:
649:
622:
620:
619:
614:
602:
600:
599:
594:
582:
580:
579:
574:
559:
557:
556:
551:
542:
541:
515:
513:
512:
507:
491:
489:
488:
483:
478:
464:
463:
447:
445:
444:
439:
416:
414:
413:
408:
397:
383:
382:
370:
362:
361:
342:
324:
323:
304:
302:
301:
296:
284:
282:
281:
276:
274:
273:
257:Lefschetz number
254:
252:
251:
246:
224:
222:
221:
216:
185:Formal statement
161:
159:
158:
153:
133:
131:
130:
125:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
3306:
3305:
3301:
3300:
3299:
3297:
3296:
3295:
3271:
3270:
3252:
3249:
3223:10.2307/1968838
3203:
3180:10.2307/1989171
3155:
3152:
3147:
3146:
3130:
3120:Springer-Verlag
3110:
3109:
3105:
3100:
3078:
3031:
3003:
2988:
2946:
2911:
2888:
2887:
2883:on cohomology:
2863:
2858:
2857:
2836:
2831:
2830:
2824:equidimensional
2804:
2803:
2781:
2780:
2761:
2760:
2741:
2740:
2712:
2711:
2676:
2602:
2560:
2559:
2528:
2527:
2508:
2507:
2488:
2487:
2466:
2461:
2460:
2405:
2404:
2383:
2364:
2359:
2358:
2337:
2332:
2331:
2330:), denoted by
2295:
2294:
2269:
2268:
2249:
2248:
2220:
2219:
2200:
2199:
2180:
2179:
2160:
2159:
2156:
2132:
2131:
2106:
2105:
2086:
2085:
2066:
2065:
2043:
2042:
2023:
2022:
1997:
1992:
1991:
1970:
1965:
1964:
1941:
1940:
1919:
1914:
1913:
1892:
1887:
1886:
1856:
1822:
1809:
1788:
1757:
1752:
1751:
1729:
1728:
1709:
1708:
1685:
1684:
1665:
1664:
1642:
1641:
1622:
1621:
1620:Given two maps
1607:
1581:
1568:
1557:
1556:
1535:
1530:
1529:
1497:
1484:
1460:
1447:
1434:
1429:
1428:
1407:
1394:
1383:
1382:
1361:
1356:
1355:
1334:
1329:
1328:
1304:
1299:
1298:
1277:
1272:
1271:
1249:
1248:
1241:
1189:
1184:
1183:
1179:. Thus we have
1152:
1151:
1122:
1117:
1116:
1105:
1077:
1076:
1030:
1029:
1010:
1009:
973:
972:
945:
878:
877:
855:
854:
847:
814:
813:
786:
785:
766:
765:
746:
745:
738:
714:
713:
692:
687:
686:
664:
663:
625:
624:
605:
604:
585:
584:
565:
564:
533:
528:
527:
498:
497:
455:
450:
449:
430:
429:
374:
353:
315:
310:
309:
287:
286:
265:
260:
259:
237:
236:
232:from a compact
194:
193:
187:
144:
143:
140:homology groups
116:
115:
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
3304:
3302:
3294:
3293:
3288:
3283:
3273:
3272:
3269:
3268:
3248:
3247:External links
3245:
3244:
3243:
3217:(4): 819–822.
3201:
3151:
3148:
3145:
3144:
3128:
3112:Dold, Albrecht
3102:
3101:
3099:
3096:
3095:
3094:
3089:
3084:
3077:
3074:
3063:
3062:
3051:
3048:
3045:
3040:
3035:
3030:
3024:
3021:
3015:
3010:
3006:
3001:
2995:
2991:
2985:
2982:
2977:
2973:
2969:
2966:
2962:
2959:
2953:
2949:
2945:
2942:
2939:
2934:
2930:
2924:
2921:
2918:
2914:
2910:
2907:
2904:
2901:
2898:
2895:
2870:
2866:
2843:
2839:
2822:is smooth and
2811:
2788:
2768:
2748:
2725:
2722:
2708:
2707:
2696:
2693:
2690:
2685:
2680:
2675:
2669:
2666:
2660:
2655:
2650:
2646:
2641:
2635:
2630:
2626:
2622:
2618:
2615:
2609:
2605:
2601:
2598:
2595:
2590:
2586:
2582:
2579:
2576:
2573:
2570:
2567:
2544:
2541:
2538:
2535:
2515:
2495:
2473:
2469:
2446:
2441:
2437:
2433:
2430:
2427:
2422:
2417:
2413:
2390:
2386:
2382:
2379:
2376:
2371:
2367:
2344:
2340:
2308:
2305:
2276:
2256:
2233:
2230:
2207:
2187:
2167:
2155:
2152:
2139:
2119:
2116:
2113:
2093:
2073:
2050:
2030:
2004:
2000:
1977:
1973:
1948:
1926:
1922:
1899:
1895:
1883:
1882:
1871:
1868:
1863:
1859:
1855:
1850:
1847:
1842:
1838:
1834:
1829:
1825:
1821:
1816:
1812:
1808:
1804:
1801:
1795:
1791:
1787:
1784:
1781:
1778:
1775:
1770:
1767:
1764:
1760:
1747:is defined as
1736:
1716:
1692:
1672:
1649:
1629:
1611:Lefschetz 1926
1606:
1603:
1588:
1584:
1580:
1575:
1571:
1567:
1564:
1542:
1538:
1517:
1513:
1509:
1504:
1500:
1496:
1491:
1487:
1483:
1480:
1476:
1472:
1467:
1463:
1459:
1454:
1450:
1446:
1441:
1437:
1414:
1410:
1406:
1401:
1397:
1393:
1390:
1368:
1364:
1341:
1337:
1311:
1307:
1284:
1280:
1256:
1240:
1237:
1236:
1235:
1221:
1218:
1215:
1212:
1209:
1206:
1200:
1197:
1192:
1168:
1165:
1162:
1159:
1129:
1125:
1104:
1101:
1084:
1060:
1057:
1054:
1051:
1048:
1044:
1041:
1038:
1017:
997:
994:
991:
987:
984:
981:
969:
968:
957:
952:
948:
944:
941:
938:
935:
932:
929:
925:
922:
919:
913:
910:
907:
903:
900:
897:
893:
890:
886:
862:
846:
843:
821:
806:simplicial map
793:
773:
753:
737:
734:
721:
699:
695:
671:
647:
644:
641:
638:
635:
632:
612:
592:
572:
561:
560:
548:
545:
540:
536:
520:coefficients.
505:
481:
477:
473:
470:
467:
462:
458:
437:
418:
417:
406:
403:
400:
396:
392:
389:
386:
381:
377:
373:
369:
366:
360:
356:
352:
349:
346:
341:
338:
335:
331:
327:
322:
318:
294:
272:
268:
244:
230:continuous map
226:
225:
213:
210:
207:
204:
201:
186:
183:
151:
123:
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3303:
3292:
3289:
3287:
3284:
3282:
3279:
3278:
3276:
3265:
3261:
3260:
3255:
3251:
3250:
3246:
3240:
3236:
3232:
3228:
3224:
3220:
3216:
3212:
3211:
3206:
3202:
3198:
3194:
3190:
3186:
3181:
3176:
3172:
3168:
3167:
3162:
3158:
3154:
3153:
3149:
3139:
3135:
3131:
3125:
3121:
3117:
3113:
3107:
3104:
3097:
3093:
3090:
3088:
3085:
3083:
3080:
3079:
3075:
3073:
3071:
3066:
3049:
3038:
3028:
3019:
3008:
3004:
2993:
2983:
2980:
2975:
2951:
2943:
2940:
2932:
2928:
2922:
2919:
2916:
2912:
2908:
2902:
2896:
2886:
2885:
2884:
2868:
2864:
2841:
2829:
2825:
2809:
2800:
2786:
2766:
2746:
2720:
2694:
2683:
2673:
2664:
2653:
2648:
2644:
2633:
2628:
2624:
2607:
2599:
2596:
2588:
2584:
2580:
2574:
2568:
2558:
2557:
2556:
2539:
2533:
2513:
2493:
2471:
2467:
2444:
2439:
2435:
2431:
2428:
2425:
2420:
2415:
2411:
2388:
2384:
2380:
2377:
2374:
2369:
2365:
2342:
2338:
2329:
2328:the Frobenius
2325:
2303:
2292:
2291:
2274:
2254:
2228:
2205:
2185:
2165:
2153:
2151:
2137:
2117:
2114:
2111:
2091:
2071:
2062:
2048:
2028:
2020:
2002:
1998:
1975:
1971:
1962:
1946:
1924:
1920:
1912:is as above,
1897:
1893:
1869:
1861:
1857:
1853:
1848:
1845:
1840:
1836:
1832:
1827:
1823:
1819:
1814:
1810:
1793:
1785:
1782:
1776:
1773:
1768:
1765:
1762:
1750:
1749:
1748:
1734:
1714:
1706:
1690:
1670:
1663:
1647:
1627:
1618:
1616:
1612:
1604:
1602:
1586:
1582:
1573:
1569:
1565:
1562:
1540:
1507:
1502:
1498:
1489:
1485:
1470:
1465:
1461:
1452:
1448:
1444:
1439:
1435:
1412:
1408:
1399:
1395:
1391:
1388:
1366:
1362:
1339:
1335:
1325:
1309:
1305:
1282:
1278:
1270:
1267:-dimensional
1254:
1246:
1238:
1219:
1213:
1207:
1204:
1182:
1181:
1180:
1163:
1157:
1150:
1146:
1145:Betti numbers
1127:
1123:
1114:
1110:
1102:
1100:
1098:
1082:
1074:
1055:
1052:
1049:
1015:
992:
955:
950:
942:
936:
933:
930:
908:
891:
888:
884:
876:
875:
874:
860:
852:
844:
842:
840:
836:
819:
811:
807:
791:
771:
751:
743:
735:
733:
719:
697:
683:
669:
661:
645:
642:
636:
630:
610:
590:
570:
546:
543:
538:
526:
525:
524:
521:
519:
503:
495:
471:
468:
460:
456:
435:
427:
423:
422:matrix traces
404:
390:
387:
379:
375:
358:
350:
347:
339:
336:
333:
329:
325:
320:
308:
307:
306:
292:
270:
258:
242:
235:
231:
211:
205:
202:
199:
192:
191:
190:
184:
182:
180:
176:
172:
167:
165:
149:
141:
137:
121:
114:
111:
107:
103:
99:
95:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
3257:
3214:
3208:
3170:
3164:
3115:
3106:
3067:
3064:
2827:
2801:
2709:
2327:
2323:
2288:
2157:
2130:and letting
2063:
1884:
1704:
1619:
1608:
1326:
1242:
1111:on a finite
1109:identity map
1106:
1071:denotes the
970:
850:
848:
739:
684:
562:
522:
419:
256:
227:
188:
178:
171:multiplicity
168:
102:fixed points
97:
91:
76:
67:
48:
3173:(1): 1–49.
2322:(often the
94:mathematics
62:introducing
3275:Categories
3150:References
2326:, or just
1961:cohomology
1113:CW complex
623:such that
496:groups of
70:March 2022
45:references
3264:EMS Press
3039:ℓ
3023:¯
2994:∗
2981:−
2972:Φ
2941:−
2929:∑
2920:
2894:#
2838:Φ
2767:ℓ
2747:ℓ
2724:¯
2684:ℓ
2668:¯
2634:∗
2597:−
2585:∑
2566:#
2429:…
2378:…
2307:¯
2232:¯
2154:Frobenius
1925:∗
1898:∗
1862:∗
1854:∘
1846:−
1833:∘
1828:∗
1820:∘
1783:−
1777:∑
1759:Λ
1617:of maps.
1579:→
1566::
1537:Λ
1482:→
1445::
1440:∗
1405:→
1392::
1208:χ
1191:Λ
1158:χ
1128:∗
947:Λ
892:∈
885:∑
694:Λ
660:homotopic
544:≠
535:Λ
348:−
337:≥
330:∑
317:Λ
267:Λ
209:→
203::
3159:(1926).
3114:(1980).
3076:See also
2017:are the
1662:manifold
518:rational
3266:, 2001
3239:1503373
3231:1968838
3197:1501331
3189:1989171
3138:0606196
1959:on the
426:induced
110:compact
108:from a
58:improve
3237:
3229:
3195:
3187:
3136:
3126:
2287:. The
1885:where
1223:
1028:, and
971:where
837:; see
492:, the
136:traces
96:, the
47:, but
3227:JSTOR
3185:JSTOR
3098:Notes
2198:with
1073:index
839:below
563:then
516:with
228:be a
104:of a
3124:ISBN
2158:Let
2084:and
2041:and
1990:and
1727:and
1640:and
3219:doi
3175:doi
2917:dim
2802:If
2293:of
1707:of
1297:to
812:of
662:to
603:in
448:on
428:by
305:by
285:of
179:any
142:of
92:In
3277::
3262:,
3256:,
3235:MR
3233:.
3225:.
3215:38
3213:.
3193:MR
3191:.
3183:.
3171:28
3169:.
3163:.
3134:MR
3132:.
3122:.
2799:.
1601:.
784:)
326::=
3241:.
3221::
3199:.
3177::
3140:.
3050:.
3047:)
3044:)
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3029:,
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3014:(
3009:i
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2965:(
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2906:)
2903:k
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2581:=
2578:)
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2572:(
2569:X
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2537:(
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2381:,
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2370:1
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2339:F
2304:X
2275:k
2255:X
2229:X
2206:q
2186:k
2166:X
2138:g
2118:Y
2115:=
2112:X
2092:g
2072:f
2049:Y
2029:X
2003:Y
1999:D
1976:X
1972:D
1947:g
1921:g
1894:f
1870:,
1867:)
1858:f
1849:1
1841:Y
1837:D
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1811:D
1807:(
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1800:t
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1786:1
1780:(
1774:=
1769:g
1766:,
1763:f
1735:g
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1671:X
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1570:D
1563:f
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1516:)
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1283:n
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1255:n
1220:.
1217:)
1214:X
1211:(
1205:=
1199:d
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1167:)
1164:X
1161:(
1124:f
1083:x
1059:)
1056:x
1053:,
1050:f
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1016:f
996:)
993:f
990:(
986:x
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980:F
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940:)
937:x
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924:d
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643:=
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54:.
20:)
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