843:
365:
542:
653:
451:
1000:
942:
250:
245:
197:
696:
1211:
883:
1382:
1358:
1316:
849:
If the above diagram (without the dashed map) commutes (this is equivalent to the conditions above), then pair (X,A) has the homotopy extension property if there exists a map
456:
396:
745:
1244:
113:
1161:
1124:
807:
579:
147:
1406:
1284:
1090:
1068:
1042:
829:
773:
718:
83:
1408:. This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.
584:
401:
1470:
1445:
951:
892:
202:
156:
1508:
1010:
658:
44:
1166:
1417:
1006:
48:
360:{\displaystyle {\tilde {f}}_{0}\circ \iota =\left.{\tilde {f}}_{0}\right|_{A}=f_{0}=\pi _{0}\circ f_{\bullet },}
1503:
852:
1363:
945:
1331:
1289:
537:{\displaystyle {\tilde {f}}_{\bullet }\circ \iota =\left.{\tilde {f}}_{\bullet }\right|_{A}=f_{\bullet }}
842:
1214:
374:
24:
1220:
92:
1132:
1095:
778:
550:
118:
1466:
1441:
86:
36:
39:
can be extended to a homotopy defined on a larger space. The homotopy extension property of
1391:
723:
1257:
1073:
1051:
1025:
812:
756:
701:
66:
1460:
1497:
1456:
1480:
1385:
1045:
1326:
1319:
40:
20:
1485:
52:
886:
751:
32:
648:{\displaystyle G\colon ((X\times \{0\})\cup (A\times I))\rightarrow Y}
839:
The homotopy extension property is depicted in the following diagram
1286:
has the homotopy extension property, then the simple inclusion map
446:{\displaystyle {\tilde {f}}_{\bullet }\colon X\rightarrow Y^{I}}
1369:
1005:
Note that this diagram is dual to (opposite to) that of the
995:{\displaystyle {\tilde {f}}_{\bullet }\colon X\times I\to Y}
841:
490:
284:
937:{\displaystyle {\tilde {f}}_{\bullet }\colon X\to Y^{I}}
1394:
1366:
1334:
1292:
1260:
1223:
1169:
1135:
1098:
1076:
1054:
1028:
954:
895:
855:
815:
781:
759:
726:
704:
661:
587:
553:
459:
404:
377:
253:
240:{\displaystyle {\tilde {f}}_{0}\colon X\rightarrow Y}
205:
192:{\displaystyle f_{\bullet }\colon A\rightarrow Y^{I}}
159:
121:
95:
69:
809:
has the homotopy extension property with respect to
1163:has the homotopy extension property if and only if
1400:
1376:
1352:
1310:
1278:
1238:
1205:
1155:
1118:
1084:
1062:
1036:
994:
936:
877:
823:
801:
767:
739:
712:
690:
647:
573:
536:
445:
390:
359:
239:
191:
141:
107:
77:
1152:
1115:
1081:
1059:
1033:
820:
798:
764:
750:If the pair has this property only for a certain
736:
709:
570:
138:
74:
691:{\displaystyle G'\colon X\times I\rightarrow Y}
581:has the homotopy extension property if any map
1206:{\displaystyle (X\times \{0\}\cup A\times I)}
8:
1185:
1179:
612:
606:
1393:
1368:
1367:
1365:
1333:
1291:
1259:
1222:
1168:
1151:
1134:
1114:
1097:
1080:
1075:
1058:
1053:
1032:
1027:
1009:; this duality is loosely referred to as
968:
957:
956:
953:
928:
909:
898:
897:
894:
889:, note that homotopies expressed as maps
869:
858:
857:
854:
819:
814:
797:
780:
763:
758:
735:
725:
708:
703:
660:
586:
569:
552:
528:
515:
505:
494:
493:
473:
462:
461:
458:
437:
418:
407:
406:
403:
382:
376:
348:
335:
322:
309:
299:
288:
287:
267:
256:
255:
252:
219:
208:
207:
204:
183:
164:
158:
137:
120:
94:
73:
68:
1429:
878:{\displaystyle {\tilde {f}}_{\bullet }}
1377:{\displaystyle \mathbf {\mathit {Y}} }
7:
1126:has the homotopy extension property.
885:which makes the diagram commute. By
1353:{\displaystyle \iota \colon Y\to Z}
1311:{\displaystyle \iota \colon A\to X}
14:
747:agree on their common domain).
1465:. Cambridge University Press.
1438:Lectures on Algebraic Topology
1344:
1302:
1273:
1261:
1200:
1170:
1148:
1136:
1111:
1099:
986:
962:
921:
903:
863:
794:
782:
682:
639:
636:
633:
621:
615:
597:
594:
566:
554:
499:
467:
430:
412:
293:
261:
231:
213:
176:
134:
122:
16:Property in algebraic topology
1:
1481:"Homotopy extension property"
1325:In fact, if you consider any
391:{\displaystyle f_{\bullet }}
151:homotopy extension property
29:homotopy extension property
1525:
1239:{\displaystyle X\times I.}
108:{\displaystyle A\subset X}
1418:Homotopy lifting property
1156:{\displaystyle (X,A)\,\!}
1119:{\displaystyle (X,A)\,\!}
1007:homotopy lifting property
948:with expressions as maps
802:{\displaystyle (X,A)\,\!}
655:can be extended to a map
574:{\displaystyle (X,A)\,\!}
142:{\displaystyle (X,A)\,\!}
49:homotopy lifting property
115:. We say that the pair
51:that is used to define
1402:
1401:{\displaystyle \iota }
1378:
1354:
1312:
1280:
1240:
1207:
1157:
1120:
1086:
1064:
1038:
1011:Eckmann–Hilton duality
996:
938:
879:
846:
825:
803:
769:
741:
740:{\displaystyle G'\,\!}
714:
692:
649:
575:
538:
447:
392:
361:
241:
193:
143:
109:
79:
1403:
1379:
1355:
1313:
1281:
1279:{\displaystyle (X,A)}
1241:
1208:
1158:
1121:
1087:
1085:{\displaystyle X\,\!}
1065:
1063:{\displaystyle A\,\!}
1039:
1037:{\displaystyle X\,\!}
997:
939:
880:
845:
826:
824:{\displaystyle Y\,\!}
804:
770:
768:{\displaystyle Y\,\!}
742:
715:
713:{\displaystyle G\,\!}
693:
650:
576:
539:
448:
393:
367:then there exists an
362:
242:
194:
153:if, given a homotopy
144:
110:
80:
78:{\displaystyle X\,\!}
1392:
1364:
1360:, then we have that
1332:
1290:
1258:
1221:
1167:
1133:
1096:
1074:
1052:
1026:
952:
893:
853:
813:
779:
757:
724:
702:
659:
585:
551:
457:
402:
375:
251:
203:
157:
119:
93:
67:
1440:, pp. 84, Springer
1388:to its image under
1070:is a subcomplex of
1509:Algebraic topology
1462:Algebraic Topology
1398:
1374:
1350:
1308:
1276:
1236:
1203:
1153:
1116:
1082:
1060:
1034:
992:
934:
875:
847:
821:
799:
765:
737:
710:
688:
645:
571:
547:That is, the pair
534:
443:
388:
357:
237:
189:
139:
105:
75:
25:algebraic topology
965:
946:natural bijection
906:
866:
502:
470:
415:
296:
264:
216:
87:topological space
23:, in the area of
1516:
1490:
1476:
1448:
1434:
1407:
1405:
1404:
1399:
1383:
1381:
1380:
1375:
1373:
1372:
1359:
1357:
1356:
1351:
1317:
1315:
1314:
1309:
1285:
1283:
1282:
1277:
1245:
1243:
1242:
1237:
1212:
1210:
1209:
1204:
1162:
1160:
1159:
1154:
1125:
1123:
1122:
1117:
1092:, then the pair
1091:
1089:
1088:
1083:
1069:
1067:
1066:
1061:
1043:
1041:
1040:
1035:
1001:
999:
998:
993:
973:
972:
967:
966:
958:
943:
941:
940:
935:
933:
932:
914:
913:
908:
907:
899:
884:
882:
881:
876:
874:
873:
868:
867:
859:
830:
828:
827:
822:
808:
806:
805:
800:
774:
772:
771:
766:
746:
744:
743:
738:
734:
719:
717:
716:
711:
697:
695:
694:
689:
669:
654:
652:
651:
646:
580:
578:
577:
572:
543:
541:
540:
535:
533:
532:
520:
519:
514:
510:
509:
504:
503:
495:
478:
477:
472:
471:
463:
452:
450:
449:
444:
442:
441:
423:
422:
417:
416:
408:
397:
395:
394:
389:
387:
386:
366:
364:
363:
358:
353:
352:
340:
339:
327:
326:
314:
313:
308:
304:
303:
298:
297:
289:
272:
271:
266:
265:
257:
246:
244:
243:
238:
224:
223:
218:
217:
209:
198:
196:
195:
190:
188:
187:
169:
168:
148:
146:
145:
140:
114:
112:
111:
106:
84:
82:
81:
76:
31:indicates which
1524:
1523:
1519:
1518:
1517:
1515:
1514:
1513:
1504:Homotopy theory
1494:
1493:
1479:
1473:
1455:
1452:
1451:
1435:
1431:
1426:
1414:
1390:
1389:
1362:
1361:
1330:
1329:
1288:
1287:
1256:
1255:
1252:
1219:
1218:
1165:
1164:
1131:
1130:
1094:
1093:
1072:
1071:
1050:
1049:
1024:
1023:
1019:
955:
950:
949:
924:
896:
891:
890:
856:
851:
850:
837:
811:
810:
777:
776:
755:
754:
727:
722:
721:
700:
699:
662:
657:
656:
583:
582:
549:
548:
524:
492:
489:
488:
460:
455:
454:
433:
405:
400:
399:
378:
373:
372:
344:
331:
318:
286:
283:
282:
254:
249:
248:
206:
201:
200:
179:
160:
155:
154:
117:
116:
91:
90:
65:
64:
61:
17:
12:
11:
5:
1522:
1520:
1512:
1511:
1506:
1496:
1495:
1492:
1491:
1477:
1471:
1457:Hatcher, Allen
1450:
1449:
1428:
1427:
1425:
1422:
1421:
1420:
1413:
1410:
1397:
1371:
1349:
1346:
1343:
1340:
1337:
1307:
1304:
1301:
1298:
1295:
1275:
1272:
1269:
1266:
1263:
1251:
1248:
1247:
1246:
1235:
1232:
1229:
1226:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1150:
1147:
1144:
1141:
1138:
1127:
1113:
1110:
1107:
1104:
1101:
1079:
1057:
1031:
1018:
1015:
991:
988:
985:
982:
979:
976:
971:
964:
961:
931:
927:
923:
920:
917:
912:
905:
902:
872:
865:
862:
836:
833:
818:
796:
793:
790:
787:
784:
775:, we say that
762:
733:
730:
707:
687:
684:
681:
678:
675:
672:
668:
665:
644:
641:
638:
635:
632:
629:
626:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
568:
565:
562:
559:
556:
531:
527:
523:
518:
513:
508:
501:
498:
491:
487:
484:
481:
476:
469:
466:
440:
436:
432:
429:
426:
421:
414:
411:
398:to a homotopy
385:
381:
356:
351:
347:
343:
338:
334:
330:
325:
321:
317:
312:
307:
302:
295:
292:
285:
281:
278:
275:
270:
263:
260:
236:
233:
230:
227:
222:
215:
212:
186:
182:
178:
175:
172:
167:
163:
136:
133:
130:
127:
124:
104:
101:
98:
72:
60:
57:
15:
13:
10:
9:
6:
4:
3:
2:
1521:
1510:
1507:
1505:
1502:
1501:
1499:
1488:
1487:
1482:
1478:
1474:
1472:0-521-79540-0
1468:
1464:
1463:
1458:
1454:
1453:
1447:
1446:3-540-58660-1
1443:
1439:
1433:
1430:
1423:
1419:
1416:
1415:
1411:
1409:
1395:
1387:
1347:
1341:
1338:
1335:
1328:
1323:
1321:
1305:
1299:
1296:
1293:
1270:
1267:
1264:
1249:
1233:
1230:
1227:
1224:
1216:
1197:
1194:
1191:
1188:
1182:
1176:
1173:
1145:
1142:
1139:
1128:
1108:
1105:
1102:
1077:
1055:
1047:
1029:
1021:
1020:
1016:
1014:
1012:
1008:
1003:
989:
983:
980:
977:
974:
969:
959:
947:
929:
925:
918:
915:
910:
900:
888:
870:
860:
844:
840:
835:Visualisation
834:
832:
816:
791:
788:
785:
760:
753:
748:
731:
728:
705:
685:
679:
676:
673:
670:
666:
663:
642:
630:
627:
624:
618:
609:
603:
600:
591:
588:
563:
560:
557:
545:
529:
525:
521:
516:
511:
506:
496:
485:
482:
479:
474:
464:
438:
434:
427:
424:
419:
409:
383:
379:
370:
354:
349:
345:
341:
336:
332:
328:
323:
319:
315:
310:
305:
300:
290:
279:
276:
273:
268:
258:
234:
228:
225:
220:
210:
184:
180:
173:
170:
165:
161:
152:
131:
128:
125:
102:
99:
96:
88:
70:
58:
56:
54:
50:
46:
42:
38:
35:defined on a
34:
30:
26:
22:
1484:
1461:
1437:
1432:
1386:homeomorphic
1324:
1253:
1046:cell complex
1004:
848:
838:
749:
546:
368:
150:
62:
41:cofibrations
28:
18:
1327:cofibration
1320:cofibration
21:mathematics
1498:Categories
1486:PlanetMath
1424:References
1017:Properties
453:such that
247:such that
199:and a map
89:, and let
59:Definition
53:fibrations
33:homotopies
1436:A. Dold,
1396:ι
1345:→
1339::
1336:ι
1303:→
1297::
1294:ι
1228:×
1195:×
1189:∪
1177:×
987:→
981:×
975::
970:∙
963:~
922:→
916::
911:∙
904:~
871:∙
864:~
683:→
677:×
671::
640:→
628:×
619:∪
604:×
592::
530:∙
507:∙
500:~
483:ι
480:∘
475:∙
468:~
431:→
425::
420:∙
413:~
384:∙
369:extension
350:∙
342:∘
333:π
294:~
277:ι
274:∘
262:~
232:→
226::
214:~
177:→
171::
166:∙
100:⊂
1459:(2002).
1412:See also
887:currying
752:codomain
732:′
667:′
149:has the
37:subspace
1215:retract
1129:A pair
944:are in
45:dual to
1469:
1444:
698:(i.e.
27:, the
1318:is a
1250:Other
1213:is a
1044:is a
85:be a
1467:ISBN
1442:ISBN
1048:and
720:and
63:Let
47:the
1384:is
1254:If
1217:of
1022:If
371:of
43:is
19:In
1500::
1483:.
1322:.
1013:.
1002:.
831:.
544:.
55:.
1489:.
1475:.
1370:Y
1348:Z
1342:Y
1306:X
1300:A
1274:)
1271:A
1268:,
1265:X
1262:(
1234:.
1231:I
1225:X
1201:)
1198:I
1192:A
1186:}
1183:0
1180:{
1174:X
1171:(
1149:)
1146:A
1143:,
1140:X
1137:(
1112:)
1109:A
1106:,
1103:X
1100:(
1078:X
1056:A
1030:X
990:Y
984:I
978:X
960:f
930:I
926:Y
919:X
901:f
861:f
817:Y
795:)
792:A
789:,
786:X
783:(
761:Y
729:G
706:G
686:Y
680:I
674:X
664:G
643:Y
637:)
634:)
631:I
625:A
622:(
616:)
613:}
610:0
607:{
601:X
598:(
595:(
589:G
567:)
564:A
561:,
558:X
555:(
526:f
522:=
517:A
512:|
497:f
486:=
465:f
439:I
435:Y
428:X
410:f
380:f
355:,
346:f
337:0
329:=
324:0
320:f
316:=
311:A
306:|
301:0
291:f
280:=
269:0
259:f
235:Y
229:X
221:0
211:f
185:I
181:Y
174:A
162:f
135:)
132:A
129:,
126:X
123:(
103:X
97:A
71:X
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