558:-skeleton, then one can also prove that the resulting cocycle would differ from the first by a coboundary. Therefore we have a well-defined element of the cohomology group
134:, given the mapping already defined on its boundary edges. Likewise, then extending the mapping to the 3-skeleton involves extending the mapping to each solid 3-simplex of
729:
In dimension at most 2 (Rado), and 3 (Moise), the notions of topological manifolds and piecewise linear manifolds coincide. In dimension 4 they are not the same.
215:. If the class is not equal to zero, it is called the obstruction to extending the mapping over the n-skeleton, given its homotopy class on the (n-1)-skeleton.
531:
The fact that this cochain came from a partially defined section (as opposed to an arbitrary collection of maps from all the boundaries of all the
207:. When this cohomology class is equal to 0, it turns out that the mapping may be modified within its homotopy class on the (n-1)-skeleton of
883:
865:
844:
822:
907:
539:-simplices) can be used to prove that this cochain is a cocycle. If one started with a different partially defined section
672:, this construction can be used to see if there are obstructions to the existence of a lift (up to homotopy) of a map into
163:
of the mapping already defined on its boundary, (at least one of which will be non-zero). These assignments define an
817:
719:
122:), an extension to the 1-skeleton will be possible whenever the image of the 0-skeleton will belong to the same
902:
912:
32:
723:
259:
55:
54:. Obstruction theory turns out to be an application of cohomology theory to the problem of constructing a
732:
In dimensions at most 6 the notions of piecewise linear manifolds and differentiable manifolds coincide.
130:. Extending from the 1-skeleton to the 2-skeleton means defining the mapping on each solid triangle from
149:, this procedure might be impossible. In that case, one can assign to each n-simplex the homotopy class
805:
785:
765:
715:
47:
753:
749:
459:
409:
25:
711:
394:
79:
75:
524:-skeleton by using the homotopy between (the partially defined section on the boundary of each
917:
879:
861:
840:
793:
698:
662:
633:. Thus it provides a complete invariant of the existence of sections up to homotopy on the
233:
102:
to define obstructions to extensions. For example, with a mapping from a simplicial complex
95:
91:
853:
71:
43:
797:
777:
741:
669:
418:
123:
99:
39:
50:
were defined as obstructions to the existence of certain fields of linear independent
896:
789:
29:
516:
are trivial, which means that our partially defined section can be extended to the
59:
51:
17:
74:
relates to the procedure, inductive with respect to dimension, for extending a
462:
286:
186:
164:
111:
83:
773:
251:
801:
761:
333:
141:
At some point, say extending the mapping from the (n-1)-skeleton of
389:. Because fibrations satisfy the homotopy lifting property, and
658:
as the first of the above cohomology classes that is non-zero.
500:
because it being the zero means that all of these elements of
661:
This can be used to find obstructions to trivializations of
629:
is homotopic (as opposed to equal) to the identity map on
590:-skeleton exists that agrees with the given choice on the
416:. So this partially defined section assigns an element of
211:
so that the mapping may be extended to the n-skeleton of
258:. Furthermore, assume that we have a partially defined
219:
Obstruction to extending a section of a principal bundle
601:
The converse is also true if one allows such things as
598:-skeleton, then this cohomology class must be trivial.
138:, given the mapping already defined on its boundary.
804:
to performing surgery on the normal map to obtain a
582:such that if a partially defined section on the
780:. In both cases there are two obstructions for
714:, obstruction theory is concerned with when a
722:, and when a piecewise linear manifold has a
8:
181:. Amazingly, this cochain turns out to be a
70:The older meaning for obstruction theory in
796:, allowing the definition of the secondary
442:-simplex. This is precisely the data of a
670:any map can be turned into a fibration
744:are whether a topological space with
697:It is crucial to the construction of
550:that agreed with the original on the
189:class in the nth cohomology group of
7:
792:: if this vanishes there exists a
788:obstruction to the existence of a
326:can be restricted to the boundary
14:
878:. American Mathematical Society.
24:is a name given to two different
860:, Princeton University Press,
656:first obstruction to a section
496:. This cochain is called the
1:
876:The wild world of 4-manifolds
858:The Topology of Fibre Bundles
823:Wall's finiteness obstruction
86:. It is traditionally called
236:simplicial complex and that
88:Eilenberg obstruction theory
874:Scorpan, Alexandru (2005).
740:The two basic questions of
110:, defined initially on the
934:
772:-dimensional manifolds is
720:piecewise linear structure
835:Husemöller, Dale (1994),
528:) and the constant map.
332:(which is a topological
38:In the original work of
374:defines a map from the
818:Kirby–Siebenmann class
724:differential structure
654:, one can construct a
48:characteristic classes
28:, both of which yield
908:Differential topology
764:, and also whether a
706:In geometric topology
476:, i.e. an element of
193:with coefficients in
167:with coefficients in
145:to the n-skeleton of
98:with coefficients in
26:mathematical theories
806:homotopy equivalence
786:topological K-theory
766:homotopy equivalence
716:topological manifold
839:, Springer Verlag,
798:surgery obstruction
754:homotopy equivalent
694:is not a fibration.
498:obstruction cochain
410:homotopy equivalent
802:algebraic L-theory
712:geometric topology
650:By inducting over
80:simplicial complex
76:continuous mapping
66:In homotopy theory
22:obstruction theory
736:In surgery theory
699:Postnikov systems
663:principal bundles
603:homotopy sections
185:and so defines a
118:(the vertices of
96:cohomology groups
925:
889:
870:
854:Steenrod, Norman
849:
750:Poincaré duality
693:
679:
675:
653:
640:
632:
628:
618:
597:
589:
581:
557:
549:
538:
527:
523:
515:
495:
475:
471:
457:
441:
433:
415:
407:
392:
388:
377:
373:
362:
358:
342:
336:
331:
325:
314:
310:
306:
295:
289:
284:
257:
249:
234:simply connected
231:
206:
180:
162:
92:Samuel Eilenberg
933:
932:
928:
927:
926:
924:
923:
922:
903:Homotopy theory
893:
892:
886:
873:
868:
852:
847:
834:
831:
814:
738:
708:
681:
677:
673:
651:
647:
634:
630:
620:
606:
591:
583:
575:
559:
551:
548:
540:
532:
525:
517:
509:
501:
489:
477:
473:
466:
451:
443:
435:
426:
417:
413:
398:
390:
379:
375:
372:
364:
360:
352:
344:
340:
334:
327:
324:
316:
312:
308:
300:
293:
287:
279:
270:
262:
255:
237:
229:
226:
221:
200:
194:
174:
168:
156:
150:
100:homotopy groups
72:homotopy theory
68:
12:
11:
5:
931:
929:
921:
920:
915:
913:Surgery theory
910:
905:
895:
894:
891:
890:
884:
871:
866:
850:
845:
830:
827:
826:
825:
820:
813:
810:
778:diffeomorphism
742:surgery theory
737:
734:
707:
704:
703:
702:
695:
676:to a map into
666:
659:
646:
643:
571:
544:
505:
485:
447:
422:
368:
348:
320:
275:
266:
225:
222:
220:
217:
198:
172:
154:
124:path-connected
94:. It involves
67:
64:
13:
10:
9:
6:
4:
3:
2:
930:
919:
916:
914:
911:
909:
906:
904:
901:
900:
898:
887:
885:0-8218-3749-4
881:
877:
872:
869:
867:0-691-08055-0
863:
859:
855:
851:
848:
846:0-387-94087-1
842:
838:
837:Fibre Bundles
833:
832:
828:
824:
821:
819:
816:
815:
811:
809:
807:
803:
799:
795:
791:
790:vector bundle
787:
783:
779:
775:
771:
767:
763:
760:-dimensional
759:
755:
751:
748:-dimensional
747:
743:
735:
733:
730:
727:
725:
721:
717:
713:
705:
700:
696:
692:
688:
684:
671:
667:
664:
660:
657:
649:
648:
644:
642:
638:
627:
623:
617:
613:
609:
605:, i.e. a map
604:
599:
595:
587:
579:
574:
570:
566:
562:
555:
547:
543:
536:
529:
521:
513:
508:
504:
499:
493:
488:
484:
480:
469:
464:
461:
455:
450:
446:
439:
432:
430:
425:
421:
411:
405:
401:
396:
386:
382:
371:
367:
356:
351:
347:
338:
330:
323:
319:
304:
297:
291:
283:
278:
274:
269:
265:
261:
253:
248:
244:
240:
235:
228:Suppose that
223:
218:
216:
214:
210:
204:
197:
192:
188:
184:
178:
171:
166:
160:
153:
148:
144:
139:
137:
133:
129:
126:component of
125:
121:
117:
113:
109:
105:
101:
97:
93:
89:
85:
81:
78:defined on a
77:
73:
65:
63:
61:
57:
56:cross-section
53:
49:
45:
41:
36:
34:
31:
30:cohomological
27:
23:
19:
875:
857:
836:
784:, a primary
781:
769:
757:
745:
739:
731:
728:
709:
690:
686:
682:
655:
645:Applications
636:
625:
621:
615:
611:
607:
602:
600:
593:
585:
577:
572:
568:
564:
560:
553:
545:
541:
534:
530:
519:
511:
506:
502:
497:
491:
486:
482:
478:
467:
453:
448:
444:
437:
428:
423:
419:
403:
399:
395:contractible
384:
380:
369:
365:
354:
349:
345:
328:
321:
317:
302:
298:
281:
276:
272:
267:
263:
246:
242:
238:
227:
224:Construction
212:
208:
202:
195:
190:
182:
176:
169:
158:
151:
146:
142:
140:
135:
131:
127:
119:
115:
107:
106:to another,
103:
87:
69:
37:
21:
15:
641:-skeleton.
378:-sphere to
343:sends each
339:). Because
254:with fiber
18:mathematics
897:Categories
829:References
794:normal map
619:such that
596:− 1)
556:− 1)
465:of degree
460:simplicial
299:For every
187:cohomology
112:0-skeleton
84:CW complex
33:invariants
774:homotopic
434:to every
307:-simplex
290:-skeleton
252:fibration
165:n-cochain
918:Theories
856:(1951),
812:See also
762:manifold
685: :
680:even if
668:Because
610: :
458:-valued
359:back to
271: :
241: :
90:, after
463:cochain
337:-sphere
285:on the
260:section
183:cocycle
52:vectors
44:Whitney
40:Stiefel
882:
864:
843:
782:n>9
756:to an
718:has a
60:bundle
776:to a
250:is a
232:is a
82:, or
58:of a
880:ISBN
862:ISBN
841:ISBN
639:+ 1)
588:+ 1)
537:+ 1)
522:+ 1)
481:(B;
440:+ 1)
305:+ 1)
42:and
800:in
768:of
752:is
710:In
472:on
470:+ 1
412:to
408:is
393:is
311:in
292:of
199:n-1
173:n-1
155:n-1
114:of
16:In
899::
808:.
726:.
689:→
624:∘
614:→
580:))
567:;
494:))
397:;
363:,
361:∂Δ
355:∂Δ
329:∂Δ
315:,
296:.
280:→
245:→
62:.
46:,
35:.
20:,
888:.
770:n
758:n
746:n
701:.
691:B
687:E
683:p
678:E
674:B
665:.
652:n
637:n
635:(
631:B
626:σ
622:p
616:E
612:B
608:σ
594:n
592:(
586:n
584:(
578:F
576:(
573:n
569:π
565:B
563:(
561:H
554:n
552:(
546:n
542:σ
535:n
533:(
526:Δ
520:n
518:(
514:)
512:F
510:(
507:n
503:π
492:F
490:(
487:n
483:π
479:C
474:B
468:n
456:)
454:F
452:(
449:n
445:π
438:n
436:(
431:)
429:F
427:(
424:n
420:π
414:F
406:)
404:Δ
402:(
400:p
391:Δ
387:)
385:Δ
383:(
381:p
376:n
370:n
366:σ
357:)
353:(
350:n
346:σ
341:p
335:n
322:n
318:σ
313:B
309:Δ
303:n
301:(
294:B
288:n
282:E
277:n
273:B
268:n
264:σ
256:F
247:B
243:E
239:p
230:B
213:X
209:X
205:)
203:Y
201:(
196:π
191:X
179:)
177:Y
175:(
170:π
161:)
159:Y
157:(
152:π
147:X
143:X
136:X
132:X
128:Y
120:X
116:X
108:Y
104:X
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.