1090:
1279:
919:
819:
581:
236:
714:
453:
393:
330:
908:
864:
741:
637:
282:
1135:
663:
497:
153:
114:
1149:
Some facts about homology groups can be derived directly from the axioms, such as the fact that homotopically equivalent spaces have isomorphic homology groups.
1085:{\displaystyle \cdots \to H_{n}(A)\,{\xrightarrow {i_{*}}}\,H_{n}(X)\,{\xrightarrow {j_{*}}}\,H_{n}(X,A)\,{\xrightarrow {\partial }}\,H_{n-1}(A)\to \cdots .}
746:
62:
satisfying the
Eilenberg–Steenrod axioms. The axiomatic approach, which was developed in 1945, allows one to prove results, such as the
1384:
1246:
518:
1188:
165:
1182:
70:
1416:
1141:. For example, singular homology (taken with integer coefficients, as is most common) has as coefficients the integers.
674:
63:
1169:
402:
342:
1348:
1180:
A "homology-like" theory satisfying all of the
Eilenberg–Steenrod axioms except the dimension axiom is called an
1411:
287:
160:
117:
869:
69:
If one omits the dimension axiom (described below), then the remaining axioms define what is called an
1288:
1193:
837:
831:
156:
1161:
719:
24:
600:
245:
1380:
1324:
1242:
1234:
1104:
40:
36:
642:
1372:
1336:
1314:
1296:
1266:
1197:
463:
121:
78:
44:
1394:
1356:
1310:
470:
126:
92:
1390:
1352:
1340:
1306:
1270:
48:
32:
1156:, can be calculated directly from the axioms. From this it can be easily shown that the (
39:
have in common. The quintessential example of a homology theory satisfying the axioms is
1292:
1319:
1274:
1405:
1213:
456:
1364:
584:
20:
1280:
Proceedings of the
National Academy of Sciences of the United States of America
1376:
396:
1371:. Graduate Texts in Mathematics. Vol. 139. New York: Springer-Verlag.
1328:
1301:
1153:
74:
55:
59:
16:
Properties that homology theories of topological spaces have in common
814:{\displaystyle H_{n}(X)\cong \bigoplus _{\alpha }H_{n}(X_{\alpha }).}
1040:
994:
954:
66:, that are common to all homology theories satisfying the axioms.
1204:
homology theories, and come with homology theories dual to them.
1192:). Important examples of these were found in the 1950s, such as
339:: Homotopic maps induce the same map in homology. That is, if
89:
The
Eilenberg–Steenrod axioms apply to a sequence of functors
576:{\displaystyle i\colon (X\setminus U,A\setminus U)\to (X,A)}
231:{\displaystyle \partial \colon H_{i}(X,A)\to H_{i-1}(A)}
1152:
The homology of some relatively simple spaces, such as
716:, the disjoint union of a family of topological spaces
1107:
922:
872:
840:
749:
722:
677:
645:
603:
521:
473:
405:
345:
290:
248:
168:
129:
95:
73:. Extraordinary cohomology theories first arose in
1129:
1084:
902:
858:
813:
735:
708:
657:
631:
575:
491:
447:
387:
324:
276:
230:
147:
108:
709:{\displaystyle X=\coprod _{\alpha }{X_{\alpha }}}
155:of topological spaces to the category of abelian
448:{\displaystyle h\colon (X,A)\rightarrow (Y,B)}
388:{\displaystyle g\colon (X,A)\rightarrow (Y,B)}
8:
1318:
1300:
1241:. Amsterdam: Elsevier. pp. 797–836.
1112:
1106:
1052:
1047:
1035:
1034:
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1008:
999:
989:
988:
973:
968:
959:
949:
948:
933:
921:
871:
839:
799:
786:
776:
754:
748:
727:
721:
699:
694:
688:
676:
644:
608:
602:
520:
472:
404:
344:
295:
289:
253:
247:
207:
179:
167:
128:
100:
94:
1275:"Axiomatic approach to homology theory"
1225:
546:
534:
325:{\displaystyle H_{i-1}(A,\varnothing )}
316:
1168:-disk. This is used in a proof of the
1160: − 1)-sphere is not a
54:One can define a homology theory as a
7:
1041:
903:{\displaystyle j\colon X\to (X,A)}
169:
14:
1345:Foundations of algebraic topology
1235:"History of homological algebra"
834:in homology, via the inclusions
511:is contained in the interior of
1189:extraordinary cohomology theory
1124:
1118:
1073:
1070:
1064:
1031:
1019:
985:
979:
945:
939:
926:
897:
885:
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859:{\displaystyle i\colon A\to X}
850:
805:
792:
766:
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570:
558:
555:
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528:
486:
474:
442:
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352:
319:
307:
271:
265:
225:
219:
200:
197:
185:
142:
130:
1:
1183:extraordinary homology theory
1101:is the one point space, then
597:be the one-point space; then
71:extraordinary homology theory
1233:Weibel, Charles A. (1999).
1170:Brouwer fixed point theorem
736:{\displaystyle X_{\alpha }}
1433:
1349:Princeton University Press
1200:, which are extraordinary
632:{\displaystyle H_{n}(P)=0}
277:{\displaystyle H_{i-1}(A)}
1377:10.1007/978-1-4757-6848-0
1347:. Princeton, New Jersey:
1237:. In James, I. M. (ed.).
515:, then the inclusion map
507:such that the closure of
29:Eilenberg–Steenrod axioms
1130:{\displaystyle H_{0}(P)}
658:{\displaystyle n\neq 0}
64:Mayer–Vietoris sequence
1131:
1086:
904:
860:
815:
737:
710:
659:
633:
577:
493:
449:
389:
326:
278:
232:
161:natural transformation
149:
110:
1369:Topology and Geometry
1302:10.1073/pnas.31.4.117
1132:
1087:
905:
861:
816:
738:
711:
660:
634:
578:
494:
492:{\displaystyle (X,A)}
455:, then their induced
450:
390:
327:
279:
233:
150:
148:{\displaystyle (X,A)}
111:
109:{\displaystyle H_{n}}
1194:topological K-theory
1105:
920:
870:
838:
747:
720:
675:
643:
601:
519:
471:
403:
343:
288:
246:
166:
127:
93:
31:are properties that
1417:Mathematical axioms
1341:Steenrod, Norman E.
1293:1945PNAS...31..117E
1271:Steenrod, Norman E.
1239:History of Topology
1044:
1005:
965:
832:long exact sequence
332:). The axioms are:
284:is a shorthand for
1127:
1082:
900:
856:
811:
781:
733:
706:
693:
655:
629:
573:
489:
445:
385:
322:
274:
228:
159:, together with a
145:
106:
37:topological spaces
25:algebraic topology
23:, specifically in
1337:Eilenberg, Samuel
1267:Eilenberg, Samuel
1139:coefficient group
1045:
1006:
966:
772:
684:
85:Formal definition
41:singular homology
33:homology theories
1424:
1398:
1360:
1332:
1322:
1304:
1253:
1252:
1230:
1198:cobordism theory
1136:
1134:
1133:
1128:
1117:
1116:
1091:
1089:
1088:
1083:
1063:
1062:
1046:
1036:
1018:
1017:
1007:
1004:
1003:
990:
978:
977:
967:
964:
963:
950:
938:
937:
909:
907:
906:
901:
865:
863:
862:
857:
820:
818:
817:
812:
804:
803:
791:
790:
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759:
758:
742:
740:
739:
734:
732:
731:
715:
713:
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707:
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703:
692:
664:
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661:
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638:
636:
635:
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613:
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582:
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579:
574:
498:
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394:
392:
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331:
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306:
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264:
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107:
105:
104:
45:Samuel Eilenberg
1432:
1431:
1427:
1426:
1425:
1423:
1422:
1421:
1412:Homology theory
1402:
1401:
1387:
1363:
1335:
1265:
1262:
1257:
1256:
1249:
1232:
1231:
1227:
1222:
1210:
1178:
1176:Dimension axiom
1147:
1108:
1103:
1102:
1048:
1009:
995:
969:
955:
929:
918:
917:
868:
867:
836:
835:
795:
782:
750:
745:
744:
723:
718:
717:
695:
673:
672:
641:
640:
604:
599:
598:
517:
516:
503:is a subset of
469:
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401:
400:
341:
340:
291:
286:
285:
249:
244:
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203:
175:
164:
163:
125:
124:
96:
91:
90:
87:
49:Norman Steenrod
43:, developed by
17:
12:
11:
5:
1430:
1428:
1420:
1419:
1414:
1404:
1403:
1400:
1399:
1385:
1361:
1333:
1287:(4): 117–120.
1261:
1258:
1255:
1254:
1247:
1224:
1223:
1221:
1218:
1217:
1216:
1209:
1206:
1177:
1174:
1146:
1143:
1137:is called the
1126:
1123:
1120:
1115:
1111:
1095:
1094:
1093:
1092:
1081:
1078:
1075:
1072:
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1043:
1039:
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1027:
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998:
993:
987:
984:
981:
976:
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962:
958:
953:
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855:
852:
849:
846:
843:
821:
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798:
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789:
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771:
768:
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762:
757:
753:
730:
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702:
698:
691:
687:
683:
680:
666:
654:
651:
648:
628:
625:
622:
619:
616:
611:
607:
588:
572:
569:
566:
563:
560:
557:
554:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
499:is a pair and
488:
485:
482:
479:
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460:
444:
441:
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435:
432:
429:
426:
423:
420:
417:
414:
411:
408:
384:
381:
378:
375:
372:
369:
366:
363:
360:
357:
354:
351:
348:
321:
318:
315:
312:
309:
304:
301:
298:
294:
273:
270:
267:
262:
259:
256:
252:
227:
224:
221:
216:
213:
210:
206:
202:
199:
196:
193:
190:
187:
182:
178:
174:
171:
144:
141:
138:
135:
132:
103:
99:
86:
83:
15:
13:
10:
9:
6:
4:
3:
2:
1429:
1418:
1415:
1413:
1410:
1409:
1407:
1396:
1392:
1388:
1386:0-387-97926-3
1382:
1378:
1374:
1370:
1366:
1362:
1358:
1354:
1350:
1346:
1342:
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1334:
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1326:
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1312:
1308:
1303:
1298:
1294:
1290:
1286:
1282:
1281:
1276:
1272:
1268:
1264:
1263:
1259:
1250:
1248:0-444-82375-1
1244:
1240:
1236:
1229:
1226:
1219:
1215:
1214:Zig-zag lemma
1212:
1211:
1207:
1205:
1203:
1199:
1195:
1191:
1190:
1185:
1184:
1175:
1173:
1171:
1167:
1163:
1159:
1155:
1150:
1144:
1142:
1140:
1121:
1113:
1109:
1100:
1079:
1076:
1067:
1059:
1056:
1053:
1049:
1037:
1028:
1025:
1022:
1014:
1010:
1000:
996:
991:
982:
974:
970:
960:
956:
951:
942:
934:
930:
923:
916:
915:
914:
913:
894:
891:
888:
879:
876:
873:
853:
847:
844:
841:
833:
829:
825:
822:
808:
800:
796:
787:
783:
777:
773:
769:
763:
755:
751:
728:
724:
700:
696:
689:
685:
681:
678:
670:
667:
652:
649:
646:
626:
623:
617:
609:
605:
596:
592:
589:
586:
567:
564:
561:
549:
543:
540:
537:
531:
525:
522:
514:
510:
506:
502:
483:
480:
477:
466:
465:
461:
459:are the same.
458:
457:homomorphisms
439:
436:
433:
421:
418:
415:
409:
406:
398:
379:
376:
373:
361:
358:
355:
349:
346:
338:
335:
334:
333:
313:
310:
302:
299:
296:
292:
268:
260:
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254:
250:
241:
222:
214:
211:
208:
204:
194:
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188:
180:
176:
172:
162:
158:
139:
136:
133:
123:
119:
101:
97:
84:
82:
80:
76:
72:
67:
65:
61:
57:
52:
50:
46:
42:
38:
34:
30:
26:
22:
1368:
1365:Bredon, Glen
1344:
1284:
1278:
1238:
1228:
1201:
1187:
1181:
1179:
1165:
1157:
1151:
1148:
1145:Consequences
1138:
1098:
1096:
827:
826:: Each pair
823:
668:
594:
590:
587:in homology.
512:
508:
504:
500:
462:
336:
240:boundary map
239:
88:
68:
53:
28:
18:
585:isomorphism
583:induces an
238:called the
21:mathematics
1406:Categories
1260:References
830:induces a
669:Additivity
1186:(dually,
1154:n-spheres
1077:⋯
1074:→
1057:−
1042:∂
1001:∗
961:∗
927:→
924:⋯
883:→
877::
851:→
845::
824:Exactness
801:α
778:α
774:⨁
770:≅
729:α
701:α
690:α
686:∐
650:≠
591:Dimension
556:→
547:∖
535:∖
526::
428:→
410::
397:homotopic
368:→
350::
317:∅
300:−
258:−
212:−
201:→
173::
170:∂
116:from the
79:cobordism
1367:(1993).
1343:(1952).
1329:16578143
1273:(1945).
1208:See also
1038:→
992:→
952:→
639:for all
464:Excision
337:Homotopy
118:category
75:K-theory
60:functors
56:sequence
1395:1224675
1357:0050886
1320:1078770
1311:0012228
1289:Bibcode
1164:of the
1162:retract
743:, then
1393:
1383:
1355:
1327:
1317:
1309:
1245:
828:(X, A)
593:: Let
242:(here
157:groups
27:, the
1220:Notes
671:: If
467:: If
122:pairs
1381:ISBN
1325:PMID
1243:ISBN
1196:and
866:and
77:and
47:and
1373:doi
1315:PMC
1297:doi
1097:If
399:to
395:is
120:of
58:of
35:of
19:In
1408::
1391:MR
1389:.
1379:.
1353:MR
1351:.
1339:;
1323:.
1313:.
1307:MR
1305:.
1295:.
1285:31
1283:.
1277:.
1269:;
1202:co
1172:.
81:.
51:.
1397:.
1375::
1359:.
1331:.
1299::
1291::
1251:.
1166:n
1158:n
1125:)
1122:P
1119:(
1114:0
1110:H
1099:P
1080:.
1071:)
1068:A
1065:(
1060:1
1054:n
1050:H
1032:)
1029:A
1026:,
1023:X
1020:(
1015:n
1011:H
997:j
986:)
983:X
980:(
975:n
971:H
957:i
946:)
943:A
940:(
935:n
931:H
910::
898:)
895:A
892:,
889:X
886:(
880:X
874:j
854:X
848:A
842:i
809:.
806:)
797:X
793:(
788:n
784:H
767:)
764:X
761:(
756:n
752:H
725:X
697:X
682:=
679:X
665:.
653:0
647:n
627:0
624:=
621:)
618:P
615:(
610:n
606:H
595:P
571:)
568:A
565:,
562:X
559:(
553:)
550:U
544:A
541:,
538:U
532:X
529:(
523:i
513:A
509:U
505:A
501:U
487:)
484:A
481:,
478:X
475:(
443:)
440:B
437:,
434:Y
431:(
425:)
422:A
419:,
416:X
413:(
407:h
383:)
380:B
377:,
374:Y
371:(
365:)
362:A
359:,
356:X
353:(
347:g
320:)
314:,
311:A
308:(
303:1
297:i
293:H
272:)
269:A
266:(
261:1
255:i
251:H
226:)
223:A
220:(
215:1
209:i
205:H
198:)
195:A
192:,
189:X
186:(
181:i
177:H
143:)
140:A
137:,
134:X
131:(
102:n
98:H
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