753:
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822:
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1012:
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238:
1004:
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827:
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983:
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178:
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263:
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623:
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116:
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437:
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1033:
617:
424:
210:
47:
1029:
952:
17:
998:
155:
There is an analogous construction for topological spaces without basepoint. The
994:
234:
35:
922:
912:
892:
431:, modulo the quotients needed to convert the products to reduced products.
217:
with the circle, while the loop space construction is right adjoint to the
428:
230:
94:
31:
620:, and the aforementioned isomorphism is of those groups. Thus, setting
221:. This adjunction accounts for much of the importance of loop spaces in
1047:
206:
65:
824:
and the spheres can be obtained via suspensions of each-other, i.e.
1038:, Lecture Notes in Mathematics, vol. 271, Berlin, New York:
184:
984:
Topospaces wiki â Loop space of a based topological space
748:{\displaystyle \pi _{k}(X)\approxeq \pi _{k-1}(\Omega X)}
171:
with the compact-open topology. The free loop space of
830:
769:
694:
665:
626:
602:
582:
547:
512:
492:
472:
466:
does not have a group structure for arbitrary spaces
440:
406:
383:
357:
325:
266:
181:
253:
of the same space; this duality is sometimes called
865:
816:
747:
677:
651:
608:
588:
568:
533:
498:
478:
458:
412:
392:
369:
343:
308:
194:
233:, where the cartesian product is adjoint to the
1003:, Annals of Mathematics Studies, vol. 90,
427:. This homeomorphism is essentially that of
8:
152:are formed by applying Ω a number of times.
83:. With this operation, the loop space is an
851:
835:
829:
799:
774:
768:
721:
699:
693:
664:
637:
625:
601:
581:
546:
511:
491:
471:
439:
405:
382:
356:
324:
265:
183:
182:
180:
944:
351:is the set of homotopy classes of maps
958:A Concise Course in Algebraic Topology
576:do have natural group structures when
209:, the free loop space construction is
237:.) Informally this is referred to as
163:is the space of maps from the circle
7:
1035:The Geometry of Iterated Loop Spaces
866:{\displaystyle S^{k}=\Sigma S^{k-1}}
844:
736:
557:
516:
384:
297:
270:
25:
115:, i.e. the set of based-homotopy
93:. That is, the multiplication is
79:. Two loops can be multiplied by
56:is the space of (based) loops in
506:. However, it can be shown that
257:. The basic observation is that
933:Path space (algebraic topology)
685:sphere) gives the relationship
195:{\displaystyle {\mathcal {L}}X}
811:
792:
786:
780:
742:
733:
711:
705:
563:
548:
528:
513:
453:
441:
370:{\displaystyle A\rightarrow B}
361:
338:
326:
303:
288:
282:
267:
249:The loop space is dual to the
64:pointed maps from the pointed
1:
817:{\displaystyle \pi _{k}(X)=}
400:is the suspension of A, and
964:, U. Chicago Press, Chicago
225:. (A related phenomenon in
1108:
1005:Princeton University Press
973:(See chapter 8, section 2)
413:{\displaystyle \approxeq }
309:{\displaystyle \approxeq }
652:{\displaystyle Z=S^{k-1}}
393:{\displaystyle \Sigma A}
888:EilenbergâMacLane space
759:This follows since the
159:of a topological space
867:
818:
749:
679:
653:
610:
590:
570:
535:
500:
480:
460:
414:
394:
371:
345:
310:
255:EckmannâHilton duality
245:EckmannâHilton duality
239:EckmannâHilton duality
223:stable homotopy theory
196:
868:
819:
750:
680:
654:
611:
591:
571:
536:
501:
481:
461:
415:
395:
372:
346:
311:
197:
77:compact-open topology
1000:Infinite loop spaces
828:
767:
692:
663:
624:
600:
580:
545:
510:
490:
470:
438:
404:
381:
355:
323:
264:
179:
175:is often denoted by
146:iterated loop spaces
75:, equipped with the
928:Spectrum (topology)
678:{\displaystyle k-1}
117:equivalence classes
95:homotopy-coherently
1092:Topological spaces
1048:10.1007/BFb0067491
908:List of topologies
863:
814:
745:
675:
649:
606:
586:
566:
531:
496:
476:
456:
410:
390:
367:
341:
306:
219:reduced suspension
192:
119:of based loops in
18:Loop space functor
1057:978-3-540-05904-2
1014:978-0-691-08207-3
995:Adams, John Frank
903:Gray's conjecture
898:Fundamental group
609:{\displaystyle X}
589:{\displaystyle Z}
499:{\displaystyle B}
479:{\displaystyle A}
215:cartesian product
129:fundamental group
51:topological space
27:Topological space
16:(Redirected from
1099:
1068:
1025:
986:
981:
975:
971:
970:
969:
963:
949:
883:Bott periodicity
872:
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840:
839:
823:
821:
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746:
732:
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569:{\displaystyle }
567:
540:
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537:
534:{\displaystyle }
532:
505:
503:
502:
497:
485:
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482:
477:
465:
463:
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459:{\displaystyle }
457:
419:
417:
416:
411:
399:
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376:
374:
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344:{\displaystyle }
342:
315:
313:
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227:computer science
201:
199:
198:
193:
188:
187:
21:
1107:
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1102:
1101:
1100:
1098:
1097:
1096:
1087:Homotopy theory
1072:
1071:
1058:
1040:Springer-Verlag
1028:
1015:
993:
990:
989:
982:
978:
967:
965:
961:
951:
950:
946:
941:
918:Path (topology)
879:
847:
831:
826:
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795:
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765:
764:
717:
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690:
689:
661:
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402:
401:
379:
378:
353:
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321:
320:
262:
261:
247:
177:
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157:free loop space
136:
109:path components
90:
28:
23:
22:
15:
12:
11:
5:
1105:
1103:
1095:
1094:
1089:
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1070:
1069:
1056:
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988:
987:
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936:
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930:
925:
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910:
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838:
834:
813:
810:
807:
802:
798:
794:
791:
788:
785:
782:
777:
773:
763:is defined as
761:homotopy group
757:
756:
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741:
738:
735:
730:
727:
724:
720:
716:
713:
710:
707:
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636:
632:
629:
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585:
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562:
559:
556:
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530:
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524:
521:
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409:
389:
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366:
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340:
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331:
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317:
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305:
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278:
275:
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269:
246:
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191:
186:
134:
88:
34:, a branch of
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1104:
1093:
1090:
1088:
1085:
1083:
1080:
1079:
1077:
1067:
1063:
1059:
1053:
1049:
1045:
1041:
1037:
1036:
1031:
1030:May, J. Peter
1027:
1024:
1020:
1016:
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1002:
1001:
996:
992:
991:
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980:
977:
974:
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959:
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948:
945:
938:
934:
931:
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926:
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921:
919:
916:
914:
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880:
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874:
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855:
852:
848:
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832:
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805:
800:
796:
789:
783:
775:
771:
762:
739:
728:
725:
722:
718:
714:
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700:
696:
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672:
669:
666:
644:
641:
638:
634:
630:
627:
619:
603:
583:
560:
554:
551:
525:
522:
519:
493:
473:
450:
447:
444:
432:
430:
426:
425:homeomorphism
423:
407:
387:
364:
358:
335:
332:
329:
300:
294:
291:
285:
279:
276:
273:
260:
259:
258:
256:
252:
244:
242:
240:
236:
232:
228:
224:
220:
216:
212:
211:right adjoint
208:
203:
189:
174:
170:
166:
162:
158:
153:
151:
147:
142:
140:
133:
130:
126:
122:
118:
114:
110:
106:
101:
99:
96:
92:
87:
82:
81:concatenation
78:
74:
70:
67:
63:
59:
55:
52:
49:
45:
41:
37:
33:
19:
1034:
999:
979:
972:
966:, retrieved
957:
947:
758:
434:In general,
433:
420:denotes the
318:
248:
204:
172:
168:
164:
160:
156:
154:
149:
145:
143:
138:
131:
120:
112:
102:
85:
72:
68:
57:
53:
43:
39:
29:
235:hom functor
98:associative
36:mathematics
1076:Categories
968:2016-08-27
953:May, J. P.
939:References
923:Quasigroup
913:Loop group
251:suspension
62:continuous
40:loop space
893:Free loop
856:−
845:Σ
772:π
737:Ω
726:−
719:π
715:≊
697:π
670:−
642:−
558:Ω
517:Σ
408:≊
385:Σ
362:→
298:Ω
286:≊
271:Σ
1082:Topology
1032:(1972),
997:(1978),
955:(1999),
877:See also
429:currying
231:currying
32:topology
1066:0420610
1023:0505692
618:pointed
422:natural
207:functor
123:, is a
60:, i.e.
48:pointed
1064:
1054:
1021:
1011:
377:, and
319:where
132:π
127:, the
91:-space
66:circle
38:, the
962:(PDF)
659:(the
205:As a
125:group
46:of a
1052:ISBN
1009:ISBN
616:are
596:and
541:and
486:and
144:The
111:of Ω
103:The
1044:doi
229:is
213:to
202:.
167:to
148:of
141:).
107:of
105:set
71:to
30:In
1078::
1062:MR
1060:,
1050:,
1042:,
1019:MR
1017:,
1007:,
873:.
241:.
100:.
1046::
859:1
853:k
849:S
842:=
837:k
833:S
812:]
809:X
806:,
801:k
797:S
793:[
790:=
787:)
784:X
781:(
776:k
755:.
743:)
740:X
734:(
729:1
723:k
712:)
709:X
706:(
701:k
673:1
667:k
645:1
639:k
635:S
631:=
628:Z
604:X
584:Z
564:]
561:X
555:,
552:Z
549:[
529:]
526:X
523:,
520:Z
514:[
494:B
474:A
454:]
451:B
448:,
445:A
442:[
388:A
365:B
359:A
339:]
336:B
333:,
330:A
327:[
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301:X
295:,
292:Z
289:[
283:]
280:X
277:,
274:Z
268:[
190:X
185:L
173:X
169:X
165:S
161:X
150:X
139:X
137:(
135:1
121:X
113:X
89:â
86:A
73:X
69:S
58:X
54:X
44:X
42:Ω
20:)
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