559:
870:
1323:. That way, one can reduce it to the case where the assertion is known. These two theorems mean that it suffices to show that a given map is a quasifibration on certain subsets. Then one can patch these together in order to see that it holds on bigger subsets and finally, using a limiting argument, one sees that the map is a quasifibration on the whole space. This procedure has e.g. been used in the proof of the Dold-Thom theorem.
891:
499:
288:
799:
230:
293:
1151:
Checking whether a given map is a quasifibration tends to be quite tedious. The following two theorems are designed to make this problem easier. They will make use of the following notion: Let
494:{\displaystyle {\begin{aligned}\dots \to \pi _{i+1}(B,x)\to \pi _{i}(p^{-1}(x),y)\to \pi _{i}(E,y)&\to \pi _{i}(B,x)\to \dots \\&\to \pi _{0}(B,x)\to 0\end{aligned}}}
704:
86:). Equivalently, one can define a quasifibration to be a continuous map such that the inclusion of each fibre into its homotopy fibre is a
127:
909:
The projection of the letter L onto its base interval is a quasifibration, but not a fibration. More generally, the projection
1447:
1360:
1144:
A corollary of this theorem is that all fibres of a quasifibration are weakly homotopy equivalent if the base space is
87:
1479:
1495:
1370:
Dold, Albrecht; Lashof, Richard (1959), "Principal
Quasifibrations and Fibre Homotopy Equivalence of Bundles",
17:
1117:
The following is a direct consequence of the alternative definition of a fibration using the homotopy fibre:
1520:
1439:
903:
695:
1104:) consisting of two connected spaces. This is one of the main statements used in the proof of the
1306:
To see that the latter statement holds, one only needs to bear in mind that continuous images of
555:)) and therefore a morphism between the exact sequences of a quasifibration. Hence, the diagram
266:
By definition, quasifibrations share a key property of fibrations, namely that a quasifibration
1013:
1385:
1422:
1383:
Dold, Albrecht; Thom, RenΓ© (1958), "Quasifaserungen und unendliche symmetrische
Produkte",
1021:
8:
279:
1410:
1136:
factors through a fibration whose fibres are weakly homotopy equivalent to the ones of
31:
1433:
519:
This long exact sequence is also functorial in the following sense: Any fibrewise map
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1443:
1402:
1356:
1105:
91:
558:
1394:
923:
1418:
1263:
899:
67:
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1514:
1429:
1406:
1307:
869:
47:
1499:
119:
39:
51:
1504:
1414:
1289:
are moreover assumed to satisfy the first separation axiom. If all the
939:
43:
794:{\displaystyle E_{p}=\{(e,\gamma )\in E\times B^{I}:\gamma (0)=p(e)\}}
90:. One of the main applications of quasifibrations lies in proving the
1398:
816:, Ξ³) = Ξ³(1). Now consider the natural homotopy equivalence Ο :
225:{\displaystyle p_{*}\colon \pi _{i}(E,p^{-1}(x),y)\to \pi _{i}(B,x)}
66:
having the same behaviour as a fibration regarding the (relative)
1061:. This means that the projection is not a fibration in this case.
1457:
May, J. Peter (1990), "Weak
Equivalences and Quasifibrations",
1044:
starting at 0 cannot be lifted to paths starting at a point of
890:
504:
as follows directly from the long exact sequence for the pair (
531:
induces a morphism between the exact sequences of the pairs (
263:= 0,1 one can only speak of bijections between the two sets.
942:
onto the unit interval is a quasifibration if and only if Ο
853:) denotes the corresponding constant path. By definition,
1351:
Aguilar, Marcelo; Gitler, Samuel; Prieto, Carlos (2008).
615:
An equivalent definition is saying that a surjective map
1218:
are distinguished with respect to the continuous map
707:
291:
130:
1108:. In general, this map also fails to be a fibration.
1350:
1036:being a weak homotopy equivalence. Furthermore, if
793:
627:is a quasifibration if the inclusion of the fibre
493:
224:
102:A continuous surjective map of topological spaces
1512:
1353:Algebraic Topology from a Homotopical Viewpoint
995:. But by the long exact sequence of the pair (
788:
721:
902:is a quasifibration. This follows from the
54:. Roughly speaking, it is a continuous map
1369:
1469:
1355:. Springer Science & Business Media.
1040:is not surjective, non-constant paths in
866:such that one gets a commutative diagram
1382:
889:
1428:
14:
1513:
1496:Quasifibrations and homotopy pullbacks
1148:, as this is the case for fibrations.
1456:
1096:is a quasifibration for a CW pair (
882:yields the alternative definition.
24:
868:
557:
25:
1532:
1489:
1032:in general, it is equivalent to
1372:Illinois Journal of Mathematics
1338:Dold and Thom (1958), Satz 2.2
1332:
1163:be a continuous map. A subset
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736:
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652:is a weak equivalence for all
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588:being an element of the form
97:
1470:Piccinini, Renzo A. (1992).
1080:) induced by the projection
7:
1472:Lectures on Homotopy Theory
885:
696:path fibration construction
660:. To see this, recall that
10:
1537:
1507:from the Lehigh University
1440:Cambridge University Press
1258:be a continuous map where
635:) into the homotopy fibre
1024:. For topological spaces
904:Homotopy lifting property
572:being the restriction of
1326:
1298:are distinguished, then
1016:, this is equivalent to
27:Concept from mathematics
38:is a generalisation of
1459:Springer Lecture Notes
894:
873:
795:
562:
495:
226:
1386:Annals of Mathematics
1199:is a quasifibration.
1124:Every quasifibration
1048:outside the image of
893:
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796:
561:
496:
227:
1314:already lie in some
1302:is a quasifibration.
1206:If the open subsets
1022:homotopy equivalence
705:
289:
128:
1014:Whitehead's theorem
282:of homotopy groups
280:long exact sequence
1521:Algebraic topology
1435:Algebraic Topology
938:between connected
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874:
791:
563:
491:
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222:
32:algebraic topology
1449:978-0-521-79540-1
1389:, Second Series,
1362:978-0-387-22489-3
1179:(with respect to
1106:Dold-Thom theorem
92:Dold-Thom theorem
16:(Redirected from
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979:) holds for all
924:mapping cylinder
857:factors through
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698:. Thus, one has
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1200:
1198:
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1190:
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1177:distinguished
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1067:
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1019:
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969:
964:
960:
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946:
941:
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929:
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921:
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908:
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883:
880:
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867:
864:
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852:
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840:
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832:
829:, given by Ο(
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749:
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694:is the usual
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630:
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611:
607:
603:
599:
595:
591:
587:
583:
579:
575:
568:
560:
556:
554:
550:
546:
542:
538:
534:
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522:
517:
515:
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457:
451:
443:
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420:
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402:
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388:
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374:
371:
365:
357:
354:
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341:
337:
327:
324:
321:
313:
310:
307:
303:
296:
285:
284:
283:
281:
277:
273:
269:
264:
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258:
254:
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246:
242:
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216:
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164:
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57:
53:
49:
48:Albrecht Dold
45:
41:
40:fibre bundles
37:
33:
19:
1500:MathOverflow
1474:. Elsevier.
1471:
1462:
1458:
1434:
1390:
1384:
1378:(2): 285β305
1375:
1371:
1352:
1334:
1319:
1315:
1311:
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1274:
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1219:
1215:
1211:
1207:
1203:
1196:
1192:
1188:
1184:
1180:
1176:
1175:) is called
1172:
1168:
1164:
1160:
1156:
1152:
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1143:
1137:
1133:
1129:
1125:
1121:
1116:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1068:) : SP(
1065:
1057:
1053:
1049:
1045:
1041:
1037:
1033:
1029:
1025:
1017:
1009:
1005:
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940:CW complexes
935:
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927:
919:
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862:
858:
854:
850:
846:
842:
838:
834:
830:
825:
821:
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813:
809:
808:is given by
805:
803:
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682:
678:
674:
670:
665:
661:
657:
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649:
645:
640:
636:
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581:
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566:
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520:
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509:
505:
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275:
271:
267:
265:
260:
256:
252:
248:
244:
240:
236:
234:
120:isomorphisms
115:
114:is called a
111:
107:
103:
101:
83:
79:
75:
71:
63:
59:
55:
35:
29:
1064:The map SP(
965:)) = 0 = Ο
1345:References
1280:β ... All
1113:Properties
1012:)) and by
876:Applying Ο
845:)), where
600:)) for an
278:induces a
98:Definition
44:fibrations
1407:0003-486X
926:of a map
762:γ
746:×
740:∈
734:γ
482:→
458:π
454:→
444:…
441:→
417:π
413:→
385:π
381:→
355:−
338:π
334:→
304:π
300:→
297:⋯
259:β₯ 0. For
199:π
195:→
169:−
146:π
142::
137:∗
52:RenΓ© Thom
1515:Category
1465:: 91β101
1432:(2002).
1310:sets in
1244:Theorem.
1204:Theorem.
1122:Theorem.
1020:being a
886:Examples
543:)) and (
235:for all
1423:0097062
1415:1970005
1308:compact
1262:is the
1072:) β SP(
922:of the
1478:
1446:
1421:
1413:
1405:
1359:
898:Every
677:where
673:under
584:) and
255:) and
1411:JSTOR
1327:Notes
1183:) if
833:) = (
648:over
1476:ISBN
1463:1425
1444:ISBN
1403:ISSN
1357:ISBN
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1210:and
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1028:and
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1320:n
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429:B
426:(
421:i
406:)
403:y
400:,
397:E
394:(
389:i
378:)
375:y
372:,
369:)
366:x
363:(
358:1
351:p
347:(
342:i
331:)
328:x
325:,
322:B
319:(
314:1
311:+
308:i
276:B
272:E
268:p
261:i
257:i
253:x
251:(
249:p
245:y
241:B
237:x
220:)
217:x
214:,
211:B
208:(
203:i
192:)
189:y
186:,
183:)
180:x
177:(
172:1
165:p
161:,
158:E
155:(
150:i
133:p
112:B
108:E
104:p
84:x
82:(
80:p
76:B
72:E
64:B
60:E
56:p
20:)
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