971:
754:
992:
960:
1029:
1002:
982:
1050:
396:
203:
463:
162:
110:
361:
224:
287:
416:
310:
583:
1032:
341:
264:
244:
130:
77:
1120:
1091:
666:
1020:
1015:
611:
1010:
366:
912:
1115:
1084:
488:
80:
920:
167:
17:
421:
991:
719:
138:
1110:
1005:
1077:
940:
935:
861:
738:
726:
699:
659:
48:
782:
709:
545:
970:
930:
882:
856:
704:
513:
505:
981:
777:
550:
89:
975:
925:
846:
836:
714:
694:
574:
509:
500:
Complexes that do not have a free face cannot be collapsible. Two such interesting examples are
477:
44:
32:
945:
963:
829:
787:
652:
617:
607:
1061:
346:
209:
743:
689:
269:
401:
295:
802:
797:
892:
824:
326:
249:
229:
115:
62:
1104:
902:
812:
792:
995:
887:
807:
753:
517:
472:
A simplicial complex that has a sequence of collapses leading to a point is called
985:
897:
528:
841:
772:
731:
501:
484:
36:
1049:
866:
621:
1057:
851:
819:
768:
675:
40:
24:
606:. Mischaikow, Konstantin Michael, Mrozek, Marian. New York: Springer.
43:
subcomplex. Collapses, like CW complexes themselves, were invented by
531:
that is collapsible is in fact piecewise-linearly isomorphic to an
648:
644:
391:{\displaystyle \tau \subseteq \gamma \subseteq \sigma ,}
1065:
132:
such that the following two conditions are satisfied:
520:(homotopy equivalent to a point), but not collapsible.
424:
404:
369:
349:
329:
298:
272:
252:
232:
212:
170:
141:
118:
92:
65:
911:
875:
761:
682:
457:
410:
390:
355:
335:
304:
281:
258:
238:
218:
197:
156:
124:
104:
71:
584:Proceedings of the London Mathematical Society
1085:
660:
8:
569:
567:
198:{\displaystyle \dim \tau <\dim \sigma ;}
1092:
1078:
1028:
1001:
667:
653:
645:
458:{\displaystyle \dim \tau =\dim \sigma -1,}
423:
403:
368:
348:
328:
297:
271:
251:
231:
211:
169:
140:
117:
91:
64:
418:is a free face. If additionally we have
577:(1938). "Simplicial spaces, nuclei and
563:
157:{\displaystyle \tau \subseteq \sigma ,}
7:
1046:
1044:
487:and is the basis for the concept of
483:This definition can be extended to
1064:. You can help Knowledge (XXG) by
636:A Course in Simple-Homotopy Theory
14:
47:. Collapses find applications in
1121:Properties of topological spaces
1048:
1027:
1000:
990:
980:
969:
959:
958:
752:
480:, but the converse is not true.
343:is the removal of all simplices
476:. Every collapsible complex is
1:
246:and no other maximal face of
105:{\displaystyle \tau ,\sigma }
27:, a branch of mathematics, a
553: – Mathematical concept
489:simple-homotopy equivalence
81:abstract simplicial complex
1137:
1043:
921:Banach fixed-point theorem
638:, Springer-Verlag New York
634:Cohen, Marshall M. (1973)
602:Kaczynski, Tomasz (2004).
15:
954:
750:
465:then this is called an
356:{\displaystyle \gamma }
219:{\displaystyle \sigma }
976:Mathematics portal
876:Metrics and properties
862:Second-countable space
604:Computational homology
459:
412:
392:
357:
337:
306:
283:
282:{\displaystyle \tau ,}
260:
240:
220:
199:
158:
126:
106:
73:
49:computational homology
35:(or more generally, a
546:Discrete Morse theory
460:
413:
411:{\displaystyle \tau }
393:
358:
338:
307:
305:{\displaystyle \tau }
284:
261:
241:
226:is a maximal face of
221:
200:
159:
127:
112:are two simplices of
107:
74:
931:Invariance of domain
883:Euler characteristic
857:Bundle (mathematics)
506:house with two rooms
422:
402:
367:
347:
327:
296:
270:
250:
230:
210:
168:
139:
116:
90:
63:
16:For other uses, see
941:Tychonoff's theorem
936:Poincaré conjecture
690:General (point-set)
551:Shelling (topology)
467:elementary collapse
41:homotopy-equivalent
1116:Algebraic topology
926:De Rham cohomology
847:Polyhedral complex
837:Simplicial complex
510:Christopher Zeeman
455:
408:
388:
353:
333:
302:
279:
256:
236:
216:
195:
154:
122:
102:
69:
45:J. H. C. Whitehead
33:simplicial complex
1073:
1072:
1041:
1040:
830:fundamental group
575:Whitehead, J.H.C.
336:{\displaystyle K}
259:{\displaystyle K}
239:{\displaystyle K}
125:{\displaystyle K}
72:{\displaystyle K}
1128:
1094:
1087:
1080:
1058:topology-related
1052:
1045:
1031:
1030:
1004:
1003:
994:
984:
974:
973:
962:
961:
756:
669:
662:
655:
646:
639:
632:
626:
625:
599:
593:
592:
571:
464:
462:
461:
456:
417:
415:
414:
409:
397:
395:
394:
389:
362:
360:
359:
354:
342:
340:
339:
334:
311:
309:
308:
303:
288:
286:
285:
280:
265:
263:
262:
257:
245:
243:
242:
237:
225:
223:
222:
217:
204:
202:
201:
196:
163:
161:
160:
155:
131:
129:
128:
123:
111:
109:
108:
103:
78:
76:
75:
70:
1136:
1135:
1131:
1130:
1129:
1127:
1126:
1125:
1101:
1100:
1099:
1098:
1042:
1037:
968:
950:
946:Urysohn's lemma
907:
871:
757:
748:
720:low-dimensional
678:
673:
643:
642:
633:
629:
614:
601:
600:
596:
573:
572:
565:
560:
542:
497:
420:
419:
400:
399:
365:
364:
345:
344:
325:
324:
294:
293:
268:
267:
248:
247:
228:
227:
208:
207:
166:
165:
137:
136:
114:
113:
88:
87:
61:
60:
57:
21:
12:
11:
5:
1134:
1132:
1124:
1123:
1118:
1113:
1111:Topology stubs
1103:
1102:
1097:
1096:
1089:
1082:
1074:
1071:
1070:
1053:
1039:
1038:
1036:
1035:
1025:
1024:
1023:
1018:
1013:
998:
988:
978:
966:
955:
952:
951:
949:
948:
943:
938:
933:
928:
923:
917:
915:
909:
908:
906:
905:
900:
895:
893:Winding number
890:
885:
879:
877:
873:
872:
870:
869:
864:
859:
854:
849:
844:
839:
834:
833:
832:
827:
825:homotopy group
817:
816:
815:
810:
805:
800:
795:
785:
780:
775:
765:
763:
759:
758:
751:
749:
747:
746:
741:
736:
735:
734:
724:
723:
722:
712:
707:
702:
697:
692:
686:
684:
680:
679:
674:
672:
671:
664:
657:
649:
641:
640:
627:
612:
594:
562:
561:
559:
556:
555:
554:
548:
541:
538:
537:
536:
521:
496:
493:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
407:
387:
384:
381:
378:
375:
372:
352:
332:
301:
290:
289:
278:
275:
255:
235:
215:
205:
194:
191:
188:
185:
182:
179:
176:
173:
164:in particular
153:
150:
147:
144:
121:
101:
98:
95:
68:
56:
53:
13:
10:
9:
6:
4:
3:
2:
1133:
1122:
1119:
1117:
1114:
1112:
1109:
1108:
1106:
1095:
1090:
1088:
1083:
1081:
1076:
1075:
1069:
1067:
1063:
1060:article is a
1059:
1054:
1051:
1047:
1034:
1026:
1022:
1019:
1017:
1014:
1012:
1009:
1008:
1007:
999:
997:
993:
989:
987:
983:
979:
977:
972:
967:
965:
957:
956:
953:
947:
944:
942:
939:
937:
934:
932:
929:
927:
924:
922:
919:
918:
916:
914:
910:
904:
903:Orientability
901:
899:
896:
894:
891:
889:
886:
884:
881:
880:
878:
874:
868:
865:
863:
860:
858:
855:
853:
850:
848:
845:
843:
840:
838:
835:
831:
828:
826:
823:
822:
821:
818:
814:
811:
809:
806:
804:
801:
799:
796:
794:
791:
790:
789:
786:
784:
781:
779:
776:
774:
770:
767:
766:
764:
760:
755:
745:
742:
740:
739:Set-theoretic
737:
733:
730:
729:
728:
725:
721:
718:
717:
716:
713:
711:
708:
706:
703:
701:
700:Combinatorial
698:
696:
693:
691:
688:
687:
685:
681:
677:
670:
665:
663:
658:
656:
651:
650:
647:
637:
631:
628:
623:
619:
615:
613:9780387215976
609:
605:
598:
595:
590:
586:
585:
580:
576:
570:
568:
564:
557:
552:
549:
547:
544:
543:
539:
534:
530:
527:-dimensional
526:
522:
519:
515:
511:
507:
503:
499:
498:
494:
492:
490:
486:
481:
479:
475:
470:
468:
452:
449:
446:
443:
440:
437:
434:
431:
428:
425:
405:
385:
382:
379:
376:
373:
370:
350:
330:
322:
319:A simplicial
317:
315:
299:
276:
273:
253:
233:
213:
206:
192:
189:
186:
183:
180:
177:
174:
171:
151:
148:
145:
142:
135:
134:
133:
119:
99:
96:
93:
86:Suppose that
84:
82:
66:
54:
52:
50:
46:
42:
38:
34:
30:
26:
19:
1066:expanding it
1055:
1033:Publications
898:Chern number
888:Betti number
771: /
762:Key concepts
710:Differential
635:
630:
603:
597:
588:
582:
578:
532:
524:
518:contractible
485:CW-complexes
482:
478:contractible
473:
471:
466:
320:
318:
313:
312:is called a
291:
85:
58:
28:
22:
996:Wikiversity
913:Key results
529:PL manifold
516:; they are
474:collapsible
1105:Categories
842:CW complex
783:Continuity
773:Closed set
732:cohomology
591:: 243–327.
581:-groups".
558:References
502:R. H. Bing
363:such that
55:Definition
37:CW complex
31:reduces a
1021:geometric
1016:algebraic
867:Cobordism
803:Hausdorff
798:connected
715:Geometric
705:Continuum
695:Algebraic
514:dunce hat
447:−
444:σ
441:
432:τ
429:
406:τ
383:σ
380:⊆
377:γ
374:⊆
371:τ
351:γ
314:free face
300:τ
274:τ
266:contains
214:σ
190:σ
187:
178:τ
175:
149:σ
146:⊆
143:τ
100:σ
94:τ
986:Wikibook
964:Category
852:Manifold
820:Homotopy
778:Interior
769:Open set
727:Homology
676:Topology
622:55897585
540:See also
495:Examples
321:collapse
29:collapse
25:topology
18:Collapse
1011:general
813:uniform
793:compact
744:Digital
79:be an
39:) to a
1006:Topics
808:metric
683:Fields
620:
610:
535:-ball.
398:where
1056:This
788:Space
292:then
1062:stub
618:OCLC
608:ISBN
523:Any
508:and
181:<
59:Let
512:'s
504:'s
469:.
438:dim
426:dim
323:of
316:.
184:dim
172:dim
23:In
1107::
616:.
589:45
587:.
566:^
491:.
83:.
51:.
1093:e
1086:t
1079:v
1068:.
668:e
661:t
654:v
624:.
579:m
533:n
525:n
453:,
450:1
435:=
386:,
331:K
277:,
254:K
234:K
193:;
152:,
120:K
97:,
67:K
20:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.