742:
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407:
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672:
603:
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563:
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338:
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105:
93:
73:
39:
1032:
871:
215:
1050:
1028:
1007:
987:
906:
144:
There is an analogous construction for topological spaces without basepoint. The
749:
97:
1064:
606:
413:
199:
36:
1018:
941:
420:, modulo the quotients needed to convert the products to reduced products.
206:
with the circle, while the loop space construction is right adjoint to the
983:
223:
24:
609:, and the aforementioned isomorphism is of those groups. Thus, setting
210:. This adjunction accounts for much of the importance of loop spaces in
1036:
911:
901:
881:
813:
and the spheres can be obtained via suspensions of each-other, i.e.
417:
219:
83:
20:
1027:, Lecture Notes in Mathematics, vol. 271, Berlin, New York:
195:
54:
973:
Topospaces wiki â Loop space of a based topological space
737:{\displaystyle \pi _{k}(X)\approxeq \pi _{k-1}(\Omega X)}
160:
with the compact-open topology. The free loop space of
819:
758:
683:
654:
615:
591:
571:
536:
501:
481:
461:
455:
does not have a group structure for arbitrary spaces
429:
395:
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242:
of the same space; this duality is sometimes called
854:
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640:
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467:
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381:
358:
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297:
183:
1062:
222:, where the cartesian product is adjoint to the
992:, Annals of Mathematics Studies, vol. 90,
416:. This homeomorphism is essentially that of
141:are formed by applying Ω a number of times.
72:. With this operation, the loop space is an
233:
340:is the set of homotopy classes of maps
1063:
947:A Concise Course in Algebraic Topology
565:do have natural group structures when
198:, the free loop space construction is
982:
226:.) Informally this is referred to as
152:is the space of maps from the circle
1024:The Geometry of Iterated Loop Spaces
855:{\displaystyle S^{k}=\Sigma S^{k-1}}
1017:
940:
13:
934:
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725:
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373:
286:
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14:
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104:, i.e. the set of based-homotopy
82:. That is, the multiplication is
68:. Two loops can be multiplied by
45:is the space of (based) loops in
495:. However, it can be shown that
246:. The basic observation is that
922:Path space (algebraic topology)
674:sphere) gives the relationship
184:{\displaystyle {\mathcal {L}}X}
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359:{\displaystyle A\rightarrow B}
350:
327:
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277:
271:
256:
238:The loop space is dual to the
53:pointed maps from the pointed
1:
927:
806:{\displaystyle \pi _{k}(X)=}
389:is the suspension of A, and
7:
953:, U. Chicago Press, Chicago
865:
214:. (A related phenomenon in
10:
1097:
994:Princeton University Press
962:(See chapter 8, section 2)
402:{\displaystyle \approxeq }
298:{\displaystyle \approxeq }
641:{\displaystyle Z=S^{k-1}}
382:{\displaystyle \Sigma A}
877:EilenbergâMacLane space
748:This follows since the
148:of a topological space
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807:
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642:
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469:
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244:EckmannâHilton duality
234:EckmannâHilton duality
228:EckmannâHilton duality
212:stable homotopy theory
185:
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66:compact-open topology
989:Infinite loop spaces
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479:
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393:
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168:
164:is often denoted by
135:iterated loop spaces
64:, equipped with the
917:Spectrum (topology)
667:{\displaystyle k-1}
106:equivalence classes
84:homotopy-coherently
1081:Topological spaces
1037:10.1007/BFb0067491
897:List of topologies
852:
803:
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208:reduced suspension
181:
108:of based loops in
1046:978-3-540-05904-2
1003:978-0-691-08207-3
984:Adams, John Frank
892:Gray's conjecture
887:Fundamental group
598:{\displaystyle X}
578:{\displaystyle Z}
488:{\displaystyle B}
468:{\displaystyle A}
204:cartesian product
118:fundamental group
40:topological space
16:Topological space
1088:
1057:
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872:Bott periodicity
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558:{\displaystyle }
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523:{\displaystyle }
521:
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448:{\displaystyle }
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365:
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333:{\displaystyle }
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216:computer science
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1076:Homotopy theory
1061:
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1029:Springer-Verlag
1004:
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978:
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956:
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907:Path (topology)
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146:free loop space
125:
98:path components
79:
17:
12:
11:
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799:
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787:
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780:
777:
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766:
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752:is defined as
750:homotopy group
746:
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727:
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329:
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23:, a branch of
15:
9:
6:
4:
3:
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1093:
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1079:
1077:
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1072:
1069:
1068:
1066:
1056:
1052:
1048:
1042:
1038:
1034:
1030:
1026:
1025:
1020:
1019:May, J. Peter
1016:
1013:
1009:
1005:
999:
995:
991:
990:
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981:
980:
974:
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963:
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948:
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789:
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572:
549:
543:
540:
514:
511:
508:
482:
462:
439:
436:
433:
421:
419:
415:
414:homeomorphism
412:
396:
376:
353:
347:
324:
321:
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289:
283:
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274:
268:
265:
262:
249:
248:
247:
245:
241:
231:
229:
225:
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217:
213:
209:
205:
201:
200:right adjoint
197:
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163:
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155:
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147:
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136:
131:
129:
122:
119:
115:
111:
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103:
99:
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71:
70:concatenation
67:
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38:
34:
30:
26:
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1023:
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968:
961:
955:, retrieved
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936:
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423:In general,
422:
409:denotes the
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157:
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149:
145:
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138:
134:
132:
127:
120:
109:
101:
91:
74:
61:
57:
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42:
32:
28:
18:
224:hom functor
87:associative
25:mathematics
1065:Categories
957:2016-08-27
942:May, J. P.
928:References
912:Quasigroup
902:Loop group
240:suspension
51:continuous
29:loop space
882:Free loop
845:−
834:Σ
761:π
726:Ω
715:−
708:π
704:≊
686:π
659:−
631:−
547:Ω
506:Σ
397:≊
374:Σ
351:→
287:Ω
275:≊
260:Σ
1071:Topology
1021:(1972),
986:(1978),
944:(1999),
866:See also
418:currying
220:currying
21:topology
1055:0420610
1012:0505692
607:pointed
411:natural
196:functor
112:, is a
49:, i.e.
37:pointed
1053:
1043:
1010:
1000:
366:, and
308:where
121:π
116:, the
80:-space
55:circle
27:, the
951:(PDF)
648:(the
194:As a
114:group
35:of a
1041:ISBN
998:ISBN
605:are
585:and
530:and
475:and
133:The
100:of Ω
92:The
1033:doi
218:is
202:to
191:.
156:to
137:of
130:).
96:of
94:set
60:to
19:In
1067::
1051:MR
1049:,
1039:,
1031:,
1008:MR
1006:,
996:,
862:.
230:.
89:.
1035::
848:1
842:k
838:S
831:=
826:k
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798:X
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790:k
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782:[
779:=
776:)
773:X
770:(
765:k
744:.
732:)
729:X
723:(
718:1
712:k
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698:X
695:(
690:k
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624:S
620:=
617:Z
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573:Z
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550:X
544:,
541:Z
538:[
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515:X
512:,
509:Z
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316:[
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31:Ω
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