713:
1027:
487:
704:
627:
545:
1446:
406:
291:
1301:
342:
1204:
757:
1160:
1359:
1059:
145:
1560:
1508:
1388:
927:
572:
1098:
1482:
1242:
198:
219:
1595:
1659:
1635:
1615:
857:
832:
788:
166:
1528:
1321:
889:
812:
239:
932:
411:
632:
1842:
1796:
1817:
1777:
1754:
577:
496:
1393:
1866:
347:
251:
1899:
1861:
1247:
900:
760:
764:
299:
1894:
1168:
726:
1835:
1115:
37:
1826:
Jean-Pierre
Marquis (2006) "A path to Epistemology of Mathematics: Homotopy theory", pages 239 to 260 in
1326:
1856:
1032:
712:
118:
91:
local lifting of paths to a given sheet; the uniqueness is because the fibers of a covering map are
1533:
46:
1487:
1364:
906:
550:
28:
1065:
1872:
1838:
1813:
1792:
1773:
1750:
1455:
1209:
759:
corresponds to a dotted arrow making the diagram commute. This diagram is dual to that of the
171:
50:
720:
203:
1568:
1718:
1644:
1620:
1600:
842:
817:
773:
719:
The outer square (without the dotted arrow) commutes if and only if the hypotheses of the
95:. The homotopy lifting property will hold in many situations, such as the projection in a
24:
150:
1743:
1670:
1513:
1306:
874:
797:
224:
92:
1807:
1888:
1766:
1700:
96:
100:
84:
115:
Assume all maps are continuous functions between topological spaces. Given a map
1831:
1695:
20:
1022:{\displaystyle T\mathrel {:=} (X\times \{0\})\cup (Y\times )\subseteq X\times }
869:
482:{\displaystyle f_{\bullet }\circ \iota _{0}=f_{0}=\pi \circ {\tilde {f}}_{0}}
1675:
865:
835:
104:
899:
There is a common generalization of the homotopy lifting property and the
699:{\displaystyle {\tilde {f}}_{0}=\left.{\tilde {f}}\right|_{Y\times \{0\}}}
1110:
246:
69:
1876:
1772:(Third Printing, 1965 ed.). New York: Academic Press Inc.
1641:
is trivially the lift of a constant map to the image point of
622:{\displaystyle f_{\bullet }=\pi \circ {\tilde {f}}_{\bullet }}
711:
540:{\displaystyle {\tilde {f}}_{\bullet }\colon Y\times I\to E}
1399:
660:
1441:{\displaystyle \left.{\tilde {f}}\right|_{T}={\tilde {g}}}
790:
satisfies the homotopy lifting property with respect to
868:, for which homotopy lifting is only required for all
1647:
1623:
1603:
1571:
1536:
1516:
1490:
1458:
1396:
1367:
1329:
1309:
1250:
1212:
1171:
1118:
1068:
1035:
935:
909:
877:
845:
820:
800:
776:
729:
635:
580:
553:
499:
414:
350:
302:
254:
227:
206:
174:
153:
121:
895:
Generalization: homotopy lifting extension property
401:{\displaystyle f_{0}=f_{\bullet }|_{Y\times \{0\}}}
1765:
1742:
1699:
1653:
1629:
1609:
1589:
1554:
1522:
1502:
1476:
1440:
1382:
1353:
1315:
1295:
1236:
1198:
1154:
1092:
1053:
1021:
921:
883:
851:
826:
806:
782:
751:
698:
621:
566:
539:
481:
400:
336:
285:
233:
221:has the homotopy lifting property with respect to
213:
192:
160:
139:
286:{\displaystyle f_{\bullet }\colon Y\times I\to B}
1296:{\displaystyle {\tilde {f}}\colon X\times \to E}
107:, where there need be no unique way of lifting.
709:The following diagram depicts this situation:
337:{\displaystyle {\tilde {f}}_{0}\colon Y\to E}
8:
1549:
1543:
959:
953:
691:
685:
393:
387:
200:has the homotopy lifting property, or that
60:. It is designed to support the picture of
1812:, Cambridge: Cambridge University Press,
1749:. Princeton: Princeton University Press.
1646:
1622:
1602:
1570:
1535:
1515:
1489:
1457:
1427:
1426:
1417:
1402:
1401:
1395:
1369:
1368:
1366:
1334:
1333:
1328:
1308:
1252:
1251:
1249:
1228:
1223:
1211:
1199:{\displaystyle {\tilde {g}}\colon T\to E}
1173:
1172:
1170:
1117:
1067:
1034:
939:
934:
908:
876:
844:
819:
799:
775:
763:; this duality is loosely referred to as
743:
732:
731:
728:
678:
663:
662:
649:
638:
637:
634:
613:
602:
601:
585:
579:
558:
552:
513:
502:
501:
498:
473:
462:
461:
445:
432:
419:
413:
380:
375:
368:
355:
349:
316:
305:
304:
301:
259:
253:
226:
210:
205:
173:
157:
152:
120:
1687:
752:{\displaystyle {\tilde {f}}_{\bullet }}
1828:The Architecture of Modern Mathematics
1791:(Third ed.). New York: Springer.
1155:{\displaystyle f\colon X\times \to B}
16:Homotopy theory in algebraic topology
7:
838:, or one sometimes simply says that
1637:is irrelevant in that every map to
1565:The homotopy extension property of
1102:homotopy lifting extension property
1497:
1354:{\displaystyle \pi {\tilde {f}}=f}
35:(also known as an instance of the
14:
1452:The homotopy lifting property of
1054:{\displaystyle \pi \colon E\to B}
859:has the homotopy lifting property
140:{\displaystyle \pi \colon E\to B}
864:A weaker notion of fibration is
45:) is a technical condition on a
1617:to be a constant map, so that
1584:
1572:
1471:
1459:
1432:
1407:
1374:
1339:
1287:
1284:
1272:
1257:
1224:
1190:
1178:
1146:
1143:
1131:
1087:
1069:
1045:
1016:
1004:
992:
989:
977:
968:
962:
944:
737:
668:
643:
607:
531:
507:
467:
376:
328:
310:
277:
187:
175:
131:
1:
1745:The Topology of Fibre Bundles
1555:{\displaystyle X\times \{0\}}
1503:{\displaystyle Y=\emptyset }
1383:{\displaystyle {\tilde {g}}}
1029:. Given additionally a map
922:{\displaystyle X\supseteq Y}
567:{\displaystyle f_{\bullet }}
1862:Encyclopedia of Mathematics
929:, for simplicity we denote
901:homotopy extension property
761:homotopy extension property
1916:
1855:A.V. Chernavskii (2001) ,
1244:, there exists a homotopy
1093:{\displaystyle (X,Y,\pi )}
903:. Given a pair of spaces
76:to be moved "upstairs" to
1873:homotopy lifting property
1787:Husemoller, Dale (1994).
1741:Steenrod, Norman (1951).
33:homotopy lifting property
1477:{\displaystyle (X,\pi )}
1237:{\displaystyle g=f|_{T}}
493:there exists a homotopy
193:{\displaystyle (Y,\pi )}
1836:Oxford University Press
1806:Hatcher, Allen (2002),
629:) which also satisfies
43:covering homotopy axiom
1655:
1631:
1611:
1597:is obtained by taking
1591:
1556:
1524:
1504:
1484:is obtained by taking
1478:
1442:
1384:
1355:
1317:
1297:
1238:
1200:
1156:
1094:
1055:
1023:
923:
885:
853:
828:
808:
784:
765:Eckmann–Hilton duality
753:
716:
700:
623:
568:
541:
483:
402:
338:
287:
235:
215:
214:{\displaystyle \pi \,}
194:
162:
141:
38:right lifting property
1830:, J. Ferreiros &
1764:Hu, Sze-Tsen (1959).
1656:
1632:
1612:
1592:
1590:{\displaystyle (X,Y)}
1557:
1525:
1505:
1479:
1443:
1385:
1356:
1318:
1298:
1239:
1201:
1157:
1095:
1056:
1024:
924:
886:
854:
829:
809:
785:
754:
715:
701:
624:
569:
542:
484:
403:
339:
288:
236:
216:
195:
163:
142:
1654:{\displaystyle \pi }
1645:
1630:{\displaystyle \pi }
1621:
1610:{\displaystyle \pi }
1601:
1569:
1534:
1514:
1488:
1456:
1394:
1365:
1327:
1307:
1248:
1210:
1169:
1116:
1066:
1033:
933:
907:
875:
852:{\displaystyle \pi }
843:
827:{\displaystyle \pi }
818:
798:
783:{\displaystyle \pi }
774:
727:
723:are true. A lifting
633:
578:
551:
497:
412:
348:
300:
252:
225:
204:
172:
151:
119:
1857:"Covering homotopy"
161:{\displaystyle Y\,}
47:continuous function
23:, in particular in
1900:Algebraic topology
1809:Algebraic Topology
1651:
1627:
1607:
1587:
1552:
1520:
1500:
1474:
1438:
1380:
1351:
1313:
1293:
1234:
1196:
1152:
1090:
1051:
1019:
919:
881:
849:
824:
804:
780:
749:
717:
696:
619:
564:
537:
479:
398:
334:
283:
231:
211:
190:
158:
137:
87:has a property of
29:algebraic topology
1843:978-0-19-856793-6
1798:978-0-387-94087-8
1523:{\displaystyle T}
1435:
1410:
1390:(i.e., such that
1377:
1342:
1323:(i.e., such that
1316:{\displaystyle f}
1260:
1181:
884:{\displaystyle Y}
807:{\displaystyle Y}
740:
671:
646:
610:
510:
470:
313:
234:{\displaystyle Y}
111:Formal definition
51:topological space
1907:
1869:
1822:
1802:
1783:
1771:
1760:
1748:
1728:
1726:
1719:Husemoller, Dale
1715:
1709:
1707:
1705:
1692:
1660:
1658:
1657:
1652:
1636:
1634:
1633:
1628:
1616:
1614:
1613:
1608:
1596:
1594:
1593:
1588:
1561:
1559:
1558:
1553:
1530:above is simply
1529:
1527:
1526:
1521:
1509:
1507:
1506:
1501:
1483:
1481:
1480:
1475:
1447:
1445:
1444:
1439:
1437:
1436:
1428:
1422:
1421:
1416:
1412:
1411:
1403:
1389:
1387:
1386:
1381:
1379:
1378:
1370:
1360:
1358:
1357:
1352:
1344:
1343:
1335:
1322:
1320:
1319:
1314:
1302:
1300:
1299:
1294:
1262:
1261:
1253:
1243:
1241:
1240:
1235:
1233:
1232:
1227:
1205:
1203:
1202:
1197:
1183:
1182:
1174:
1165:For any lifting
1161:
1159:
1158:
1153:
1099:
1097:
1096:
1091:
1061:, one says that
1060:
1058:
1057:
1052:
1028:
1026:
1025:
1020:
943:
928:
926:
925:
920:
890:
888:
887:
882:
858:
856:
855:
850:
833:
831:
830:
825:
813:
811:
810:
805:
789:
787:
786:
781:
758:
756:
755:
750:
748:
747:
742:
741:
733:
721:lifting property
705:
703:
702:
697:
695:
694:
677:
673:
672:
664:
654:
653:
648:
647:
639:
628:
626:
625:
620:
618:
617:
612:
611:
603:
590:
589:
573:
571:
570:
565:
563:
562:
546:
544:
543:
538:
518:
517:
512:
511:
503:
488:
486:
485:
480:
478:
477:
472:
471:
463:
450:
449:
437:
436:
424:
423:
407:
405:
404:
399:
397:
396:
379:
373:
372:
360:
359:
343:
341:
340:
335:
321:
320:
315:
314:
306:
292:
290:
289:
284:
264:
263:
240:
238:
237:
232:
220:
218:
217:
212:
199:
197:
196:
191:
168:, one says that
167:
165:
164:
159:
146:
144:
143:
138:
72:taking place in
56:to another one,
1915:
1914:
1910:
1909:
1908:
1906:
1905:
1904:
1895:Homotopy theory
1885:
1884:
1854:
1851:
1820:
1805:
1799:
1786:
1780:
1768:Homotopy Theory
1763:
1757:
1740:
1737:
1732:
1731:
1717:
1716:
1712:
1702:Homotopy Theory
1694:
1693:
1689:
1684:
1667:
1643:
1642:
1619:
1618:
1599:
1598:
1567:
1566:
1532:
1531:
1512:
1511:
1486:
1485:
1454:
1453:
1398:
1397:
1392:
1391:
1363:
1362:
1325:
1324:
1305:
1304:
1246:
1245:
1222:
1208:
1207:
1167:
1166:
1114:
1113:
1064:
1063:
1031:
1030:
931:
930:
905:
904:
897:
873:
872:
866:Serre fibration
841:
840:
816:
815:
796:
795:
772:
771:
730:
725:
724:
659:
658:
636:
631:
630:
600:
581:
576:
575:
574:(i.e., so that
554:
549:
548:
500:
495:
494:
460:
441:
428:
415:
410:
409:
408:(i.e., so that
374:
364:
351:
346:
345:
303:
298:
297:
255:
250:
249:
223:
222:
202:
201:
170:
169:
149:
148:
117:
116:
113:
93:discrete spaces
83:For example, a
25:homotopy theory
17:
12:
11:
5:
1913:
1911:
1903:
1902:
1897:
1887:
1886:
1883:
1882:
1870:
1850:
1849:External links
1847:
1846:
1845:
1824:
1818:
1803:
1797:
1784:
1778:
1761:
1755:
1736:
1733:
1730:
1729:
1710:
1686:
1685:
1683:
1680:
1679:
1678:
1673:
1671:Covering space
1666:
1663:
1650:
1626:
1606:
1586:
1583:
1580:
1577:
1574:
1551:
1548:
1545:
1542:
1539:
1519:
1499:
1496:
1493:
1473:
1470:
1467:
1464:
1461:
1450:
1449:
1434:
1431:
1425:
1420:
1415:
1409:
1406:
1400:
1376:
1373:
1361:) and extends
1350:
1347:
1341:
1338:
1332:
1312:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1259:
1256:
1231:
1226:
1221:
1218:
1215:
1195:
1192:
1189:
1186:
1180:
1177:
1163:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1050:
1047:
1044:
1041:
1038:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
942:
938:
918:
915:
912:
896:
893:
880:
848:
823:
803:
779:
746:
739:
736:
693:
690:
687:
684:
681:
676:
670:
667:
661:
657:
652:
645:
642:
616:
609:
606:
599:
596:
593:
588:
584:
561:
557:
536:
533:
530:
527:
524:
521:
516:
509:
506:
491:
490:
476:
469:
466:
459:
456:
453:
448:
444:
440:
435:
431:
427:
422:
418:
395:
392:
389:
386:
383:
378:
371:
367:
363:
358:
354:
333:
330:
327:
324:
319:
312:
309:
294:
282:
279:
276:
273:
270:
267:
262:
258:
230:
209:
189:
186:
183:
180:
177:
156:
147:, and a space
136:
133:
130:
127:
124:
112:
109:
68:by allowing a
15:
13:
10:
9:
6:
4:
3:
2:
1912:
1901:
1898:
1896:
1893:
1892:
1890:
1881:
1879:
1874:
1871:
1868:
1864:
1863:
1858:
1853:
1852:
1848:
1844:
1840:
1837:
1833:
1829:
1825:
1821:
1819:0-521-79540-0
1815:
1811:
1810:
1804:
1800:
1794:
1790:
1789:Fibre Bundles
1785:
1781:
1779:0-12-358450-7
1775:
1770:
1769:
1762:
1758:
1756:0-691-00548-6
1752:
1747:
1746:
1739:
1738:
1734:
1724:
1723:Fibre Bundles
1720:
1714:
1711:
1704:
1703:
1697:
1691:
1688:
1681:
1677:
1674:
1672:
1669:
1668:
1664:
1662:
1648:
1640:
1624:
1604:
1581:
1578:
1575:
1563:
1546:
1540:
1537:
1517:
1494:
1491:
1468:
1465:
1462:
1429:
1423:
1418:
1413:
1404:
1371:
1348:
1345:
1336:
1330:
1310:
1303:which covers
1290:
1281:
1278:
1275:
1269:
1266:
1263:
1254:
1229:
1219:
1216:
1213:
1193:
1187:
1184:
1175:
1164:
1149:
1140:
1137:
1134:
1128:
1125:
1122:
1119:
1112:
1108:
1107:
1106:
1104:
1103:
1084:
1081:
1078:
1075:
1072:
1048:
1042:
1039:
1036:
1013:
1010:
1007:
1001:
998:
995:
986:
983:
980:
974:
971:
965:
956:
950:
947:
940:
936:
916:
913:
910:
902:
894:
892:
878:
871:
867:
862:
860:
846:
837:
821:
801:
793:
777:
768:
766:
762:
744:
734:
722:
714:
710:
707:
688:
682:
679:
674:
665:
655:
650:
640:
614:
604:
597:
594:
591:
586:
582:
559:
555:
534:
528:
525:
522:
519:
514:
504:
474:
464:
457:
454:
451:
446:
442:
438:
433:
429:
425:
420:
416:
390:
384:
381:
369:
365:
361:
356:
352:
331:
325:
322:
317:
307:
295:
280:
274:
271:
268:
265:
260:
256:
248:
244:
243:
242:
228:
207:
184:
181:
178:
154:
134:
128:
125:
122:
110:
108:
106:
102:
98:
97:vector bundle
94:
90:
86:
81:
79:
75:
71:
67:
63:
59:
55:
52:
48:
44:
40:
39:
34:
30:
26:
22:
1877:
1860:
1827:
1808:
1788:
1767:
1744:
1722:
1713:
1701:
1696:Hu, Sze-Tsen
1690:
1638:
1564:
1451:
1101:
1062:
898:
870:CW complexes
863:
839:
834:is called a
791:
769:
718:
708:
492:
296:for any map
114:
101:fiber bundle
88:
85:covering map
82:
77:
73:
65:
61:
57:
53:
42:
36:
32:
18:
1834:, editors,
770:If the map
21:mathematics
1889:Categories
1735:References
1510:, so that
1867:EMS Press
1832:J.J. Gray
1676:Fibration
1649:π
1625:π
1605:π
1541:×
1498:∅
1469:π
1433:~
1408:~
1375:~
1340:~
1331:π
1288:→
1270:×
1264::
1258:~
1191:→
1185::
1179:~
1147:→
1129:×
1123::
1085:π
1046:→
1040::
1037:π
1002:×
996:⊆
975:×
966:∪
951:×
914:⊇
847:π
836:fibration
822:π
778:π
745:∙
738:~
683:×
669:~
644:~
615:∙
608:~
598:∘
595:π
587:∙
560:∙
532:→
526:×
520::
515:∙
508:~
468:~
458:∘
455:π
430:ι
426:∘
421:∙
385:×
370:∙
329:→
323::
311:~
278:→
272:×
266::
261:∙
208:π
185:π
132:→
126::
123:π
105:fibration
1721:(1994).
1698:(1959).
1665:See also
1111:homotopy
1109:For any
1100:has the
547:lifting
344:lifting
247:homotopy
245:for any
70:homotopy
64:"above"
1875:at the
1708:page 24
814:, then
794:spaces
49:from a
41:or the
27:within
1841:
1816:
1795:
1776:
1753:
1727:page 7
241:, if:
89:unique
31:, the
1682:Notes
1162:, and
293:, and
1839:ISBN
1814:ISBN
1793:ISBN
1774:ISBN
1751:ISBN
1105:if:
1880:Lab
1206:of
792:all
103:or
19:In
1891::
1865:,
1859:,
1661:.
1562:.
1448:).
941::=
891:.
861:.
767:.
706:.
489:),
99:,
80:.
1878:n
1823:.
1801:.
1782:.
1759:.
1725:.
1706:.
1639:E
1585:)
1582:Y
1579:,
1576:X
1573:(
1550:}
1547:0
1544:{
1538:X
1518:T
1495:=
1492:Y
1472:)
1466:,
1463:X
1460:(
1430:g
1424:=
1419:T
1414:|
1405:f
1372:g
1349:f
1346:=
1337:f
1311:f
1291:E
1285:]
1282:1
1279:,
1276:0
1273:[
1267:X
1255:f
1230:T
1225:|
1220:f
1217:=
1214:g
1194:E
1188:T
1176:g
1150:B
1144:]
1141:1
1138:,
1135:0
1132:[
1126:X
1120:f
1088:)
1082:,
1079:Y
1076:,
1073:X
1070:(
1049:B
1043:E
1017:]
1014:1
1011:,
1008:0
1005:[
999:X
993:)
990:]
987:1
984:,
981:0
978:[
972:Y
969:(
963:)
960:}
957:0
954:{
948:X
945:(
937:T
917:Y
911:X
879:Y
802:Y
735:f
692:}
689:0
686:{
680:Y
675:|
666:f
656:=
651:0
641:f
605:f
592:=
583:f
556:f
535:E
529:I
523:Y
505:f
475:0
465:f
452:=
447:0
443:f
439:=
434:0
417:f
394:}
391:0
388:{
382:Y
377:|
366:f
362:=
357:0
353:f
332:E
326:Y
318:0
308:f
281:B
275:I
269:Y
257:f
229:Y
188:)
182:,
179:Y
176:(
155:Y
135:B
129:E
78:E
74:B
66:B
62:E
58:B
54:E
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
