20:
643:
1005:
359:
857:
1121:
218:
1210:
486:
875:
497:
232:
734:
661:, we see that two fiber bundles with the same base, fiber, structure group, trivializing neighborhoods, and transition functions are isomorphic.
99:
construction where one starts with a given bundle and surgically replaces the fiber with a new space while keeping all other data the same.
405:′ be another fiber bundle with the same base space, fiber, structure group, and trivializing neighborhoods, but transition functions
1326:
1303:
1048:
153:
1351:
710:
161:
714:
128:
1132:
1346:
424:
445:
867:
709:. That is, it actually constructs a fiber bundle with the given properties. One starts by taking the
706:
1322:
1299:
863:
124:
116:
96:
19:
1000:{\displaystyle (j,x,y)\sim (i,x,t_{ij}(x)\cdot y)\qquad \forall x\in U_{i}\cap U_{j},y\in F.}
638:{\displaystyle t'_{ij}(x)=t_{i}(x)^{-1}t_{ij}(x)t_{j}(x)\qquad \forall x\in U_{i}\cap U_{j}.}
1281:
673:
354:{\displaystyle t_{ik}(x)=t_{ij}(x)t_{jk}(x)\qquad \forall x\in U_{i}\cap U_{j}\cap U_{k}}
24:
1315:
436:
88:
67:, the Möbius strip. This can be visualised as a "twisting" of one of the local charts.
1340:
852:{\displaystyle T=\coprod _{i\in I}U_{i}\times F=\{(i,x,y):i\in I,x\in U_{i},y\in F\}}
16:
Constructs a fiber bundle from a base space, fiber and a set of transition functions
84:
27:
can be constructed by a non-trivial gluing of two trivial bundles on open subsets
72:
136:
92:
681:
1296:
Differential
Geometry: Cartan's Generalization of Klein's Erlangen Program
1280:
with the action of left multiplication then one obtains the associated
80:
91:. The theorem also gives conditions under which two such bundles are
47:) one obtains the trivial bundle, but with the non-trivial gluing of
1116:{\displaystyle \phi _{i}:\pi ^{-1}(U_{i})\to U_{i}\times F}
63:
on the second overlap, one obtains the non-trivial bundle
664:
A similar theorem holds in the smooth category, where
1135:
1051:
878:
737:
500:
448:
235:
164:
87:
from a given base space, fiber and a suitable set of
1314:
1272:′ in the construction theorem. If one takes
1204:
1115:
999:
851:
637:
480:
353:
212:
1026:is the map which sends the equivalence class of (
223:defined on each nonempty overlap, such that the
8:
846:
779:
657:to be constant functions to the identity in
213:{\displaystyle t_{ij}:U_{i}\cap U_{j}\to G}
1244:-space. One can form an associated bundle
1321:. Princeton: Princeton University Press.
1145:
1140:
1134:
1101:
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18:
1260:by taking any local trivialization of
1205:{\displaystyle \phi _{i}^{-1}(x,y)=.}
7:
364:holds, there exists a fiber bundle
950:
600:
306:
95:. The theorem is important in the
14:
77:fiber bundle construction theorem
481:{\displaystyle t_{i}:U_{i}\to G}
1018:/~ and the projection π :
949:
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1196:
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1:
1317:The Topology of Fibre Bundles
684:with a smooth left action on
39:. When glued trivially (with
1256:′ and structure group
1042:. The local trivializations
705:The proof of the theorem is
389:} with transition functions
380:that is trivializable over {
1368:
1228:a fiber bundle with fiber
1333:See Part I, §2.10 and §3.
1313:Steenrod, Norman (1951).
1240:′ be another left
1298:. New York: Springer.
1294:Sharpe, R. W. (1997).
1206:
1117:
1001:
853:
639:
482:
439:there exist functions
355:
214:
129:continuous left action
68:
1207:
1118:
1002:
854:
640:
483:
356:
215:
22:
1352:Theorems in topology
1232:and structure group
1133:
1126:are then defined by
1049:
876:
868:equivalence relation
735:
498:
446:
376:and structure group
233:
162:
154:continuous functions
89:transition functions
1153:
862:and then forms the
516:
415:. If the action of
83:which constructs a
55:on one overlap and
1202:
1136:
1113:
997:
849:
759:
635:
501:
478:
351:
210:
117:topological spaces
69:
1216:Associated bundle
1014:of the bundle is
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225:cocycle condition
125:topological group
97:associated bundle
1359:
1332:
1320:
1309:
1282:principal bundle
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1209:
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1010:The total space
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929:
858:
856:
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850:
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769:
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697:are all smooth.
674:smooth manifolds
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103:Formal statement
1367:
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703:
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496:
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462:
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435:are isomorphic
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1264:and replacing
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715:product spaces
711:disjoint union
702:
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489:
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437:if and only if
410:
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35:of the circle
15:
13:
10:
9:
6:
4:
3:
2:
1364:
1353:
1350:
1348:
1347:Fiber bundles
1345:
1344:
1342:
1330:
1328:0-691-00548-6
1324:
1319:
1318:
1311:
1307:
1305:0-387-94732-9
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1252:with a fiber
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1199:
1190:
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837:
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755:
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749:
745:
741:
738:
731:
730:
729:
728:
723:
719:
716:
712:
708:
700:
698:
695:
691:
688:and the maps
687:
683:
679:
675:
671:
667:
662:
660:
655:
651:
632:
627:
623:
619:
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610:
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563:
559:
553:
550:
542:
534:
530:
526:
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513:
509:
506:
502:
494:
493:
492:
475:
467:
463:
459:
454:
450:
442:
441:
440:
438:
434:
430:
426:
422:
418:
413:
408:
404:
399:
396:
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387:
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375:
371:
367:
346:
342:
338:
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329:
325:
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316:
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299:
291:
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284:
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207:
199:
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173:
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158:
157:
156:
155:
152:and a set of
151:
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94:
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21:
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1125:
1039:
1035:
1031:
1027:
1023:
1019:
1015:
1011:
1009:
861:
726:
721:
717:
707:constructive
704:
701:Construction
693:
689:
685:
677:
669:
665:
663:
658:
653:
649:
647:
490:
432:
431:′ and
428:
420:
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411:
406:
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385:
381:
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373:
369:
365:
363:
224:
222:
149:
144:
140:
132:
120:
112:
108:
106:
85:fiber bundle
76:
70:
64:
56:
48:
40:
36:
32:
28:
25:Möbius strip
372:with fiber
135:. Given an
73:mathematics
1341:Categories
1288:References
1248:′ →
1236:, and let
491:such that
137:open cover
93:isomorphic
1147:−
1138:ϕ
1108:×
1095:→
1071:−
1067:π
1054:ϕ
989:∈
970:∩
957:∈
951:∀
941:⋅
901:∼
841:∈
822:∈
810:∈
771:×
753:∈
746:∐
682:Lie group
620:∩
607:∈
601:∀
551:−
473:→
339:∩
326:∩
313:∈
307:∀
205:→
192:∩
864:quotient
514:′
425:faithful
119:and let
866:by the
725:×
713:of the
648:Taking
427:, then
409:′
127:with a
81:theorem
1325:
1302:
75:, the
1038:) to
680:is a
148:} of
123:be a
79:is a
1323:ISBN
1300:ISBN
1220:Let
672:are
668:and
401:Let
111:and
107:Let
31:and
23:The
1268:by
423:is
419:on
131:on
115:be
71:In
61:=-1
1343::
1284:.
1224:→
1034:,
1030:,
1022:→
694:ij
676:,
412:ij
398:.
395:ij
368:→
59:UV
53:=1
51:UV
45:=1
43:UV
1331:.
1308:.
1278:G
1274:F
1270:F
1266:F
1262:E
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1254:F
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1200:.
1197:]
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774:F
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521:x
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57:g
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29:U
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