511:
704:
631:
411:
284:
220:
734:
438:
558:
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362:
331:
240:
167:
147:
127:
99:
75:
49:
451:
1060:
1055:
636:
563:
370:
341:
181:
248:
706:
1001:
Kim, Raeyong (2015). "Every finite complex has the homology of some CAT(0) cubical duality group".
514:
1034:
989:
955:
934:
21:
1018:
973:
918:
881:
844:
798:
337:
291:
243:
185:
55:
1010:
965:
910:
873:
834:
790:
778:
441:
302:
190:
78:
1030:
985:
930:
893:
864:
Maunder, Charles
Richard Francis (1981). "A short proof of a theorem of Kan and Thurston".
856:
810:
712:
416:
1026:
981:
926:
901:
Hausmann, Jean-Claude (1986). "Every finite complex has the homology of a duality group".
889:
852:
806:
334:
170:
52:
543:
523:
347:
316:
225:
152:
132:
112:
84:
60:
34:
29:
1049:
1038:
993:
938:
839:
822:
794:
752:
174:
180:
More precisely, the theorem states that every path-connected topological space is
818:
287:
102:
17:
1014:
774:
298:
106:
1022:
977:
922:
885:
848:
802:
969:
877:
365:
914:
169:, and consequently the theorem is sometimes interpreted to mean that
960:
756:
781:(1976). "Every connected space has the homology of a K(π,1)".
149:
might then be regarded as a good approximation to the space
946:
Leary, Ian J. (2013). "A metric Kan-Thurston theorem".
715:
639:
566:
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526:
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419:
373:
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155:
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115:
87:
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37:
728:
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161:
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121:
93:
69:
43:
866:The Bulletin of the London Mathematical Society
823:"On the classifying spaces of discrete monoids"
506:{\displaystyle \pi _{1}(T_{X})\to \pi _{1}(X)}
242:, where homology-equivalent means there is a
8:
959:
838:
720:
714:
699:{\displaystyle H^{*}(TX;A)\to H^{*}(X;A)}
675:
644:
638:
626:{\displaystyle H_{*}(TX;A)\to H_{*}(X;A)}
602:
571:
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488:
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378:
372:
349:
318:
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227:
192:
154:
134:
114:
86:
62:
36:
745:
406:{\displaystyle t_{x}\colon T_{X}\to X}
309:Statement of the Kan-Thurston theorem
7:
305:who published their result in 1976.
520:for every local coefficient system
279:{\displaystyle K(G,1)\rightarrow X}
14:
693:
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620:
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592:
577:
500:
494:
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270:
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255:
209:
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1:
297:The theorem is attributed to
840:10.1016/0040-9383(79)90022-3
795:10.1016/0040-9383(76)90040-9
1077:
1015:10.1007/s10711-014-9956-4
173:can be viewed as part of
77:in such a way that the
730:
700:
627:
554:
534:
507:
434:
407:
358:
327:
280:
236:
216:
215:{\displaystyle K(G,1)}
163:
143:
123:
95:
71:
45:
970:10.1112/jtopol/jts035
903:Mathematische Annalen
878:10.1112/blms/13.4.325
731:
729:{\displaystyle t_{x}}
701:
628:
555:
535:
508:
435:
433:{\displaystyle T_{X}}
408:
359:
328:
281:
237:
217:
164:
144:
124:
96:
72:
46:
779:Thurston, William P.
753:Kan-Thurston theorem
713:
637:
564:
544:
524:
452:
417:
371:
348:
342:naturally associated
317:
249:
226:
222:of a discrete group
191:
153:
133:
113:
85:
61:
35:
26:Kan-Thurston theorem
1003:Geometriae Dedicata
948:Journal of Topology
184:-equivalent to the
915:10.1007/BF01458466
726:
696:
623:
550:
530:
503:
430:
403:
354:
323:
276:
232:
212:
159:
139:
119:
91:
67:
41:
22:algebraic topology
736:are isomorphisms.
553:{\displaystyle X}
533:{\displaystyle A}
357:{\displaystyle X}
338:topological space
326:{\displaystyle X}
235:{\displaystyle G}
186:classifying space
162:{\displaystyle X}
142:{\displaystyle G}
122:{\displaystyle X}
94:{\displaystyle G}
70:{\displaystyle X}
56:topological space
44:{\displaystyle G}
1068:
1042:
997:
963:
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531:
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477:
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448:the induced map
442:aspherical space
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303:William Thurston
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79:group cohomology
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50:
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42:
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1061:Homology theory
1056:Homotopy theory
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522:
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468:
455:
450:
449:
444:. Furthermore,
420:
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374:
369:
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366:Serre fibration
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189:
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171:homotopy theory
151:
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131:
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111:
110:
83:
82:
59:
58:
33:
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20:, particularly
12:
11:
5:
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1048:
1047:
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998:
954:(1): 251–284.
943:
909:(2): 327–336.
898:
872:(4): 325–327.
861:
833:(4): 313–320.
815:
789:(3): 253–258.
775:Kan, Daniel M.
769:
766:
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737:
723:
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335:path-connected
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231:
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158:
138:
118:
90:
66:
53:path-connected
40:
30:discrete group
13:
10:
9:
6:
4:
3:
2:
1073:
1062:
1059:
1057:
1054:
1053:
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1040:
1036:
1032:
1028:
1024:
1020:
1016:
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1008:
1004:
999:
995:
991:
987:
983:
979:
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971:
967:
962:
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953:
949:
944:
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936:
932:
928:
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920:
916:
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904:
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836:
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828:
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721:
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547:
527:
519:
516:
497:
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485:
473:
469:
460:
456:
447:
446:
445:
443:
425:
421:
400:
392:
388:
384:
379:
375:
367:
364:, there is a
351:
343:
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308:
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273:
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258:
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229:
206:
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178:
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156:
136:
129:. The group
116:
109:of the space
108:
104:
88:
80:
64:
57:
54:
38:
31:
28:associates a
27:
23:
19:
1006:
1002:
951:
947:
906:
902:
869:
865:
830:
826:
819:McDuff, Dusa
786:
782:
757:
748:
312:
296:
286:inducing an
179:
175:group theory
25:
15:
560:, the maps
288:isomorphism
103:the same as
18:mathematics
1050:Categories
768:References
515:surjective
299:Daniel Kan
107:cohomology
1039:119644662
1023:0046-5755
994:119162788
978:1753-8416
961:1009.1540
939:119913298
923:0025-5831
886:0024-6093
849:0040-9383
803:0040-9383
677:∗
669:→
646:∗
604:∗
596:→
573:∗
486:π
482:→
457:π
398:→
385::
340:. Then,
271:→
51:to every
827:Topology
821:(1979).
783:Topology
292:homology
182:homology
1031:3347570
1009:: 1–9.
986:3029427
931:0854015
894:0620046
857:0551013
811:1439159
755:at the
707:induced
1037:
1029:
1021:
992:
984:
976:
937:
929:
921:
892:
884:
855:
847:
809:
801:
440:is an
413:where
24:, the
1035:S2CID
990:S2CID
956:arXiv
935:S2CID
741:Notes
517:, and
333:be a
1019:ISSN
974:ISSN
919:ISSN
882:ISSN
845:ISSN
799:ISSN
633:and
313:Let
301:and
105:the
1011:doi
1007:176
966:doi
911:doi
907:275
874:doi
835:doi
791:doi
760:Lab
709:by
540:on
513:is
344:to
290:on
244:map
101:is
81:of
16:In
1052::
1033:.
1027:MR
1025:.
1017:.
1005:.
988:.
982:MR
980:.
972:.
964:.
950:.
933:.
927:MR
925:.
917:.
905:.
890:MR
888:.
880:.
870:13
868:.
853:MR
851:.
843:.
831:18
829:.
825:.
807:MR
805:.
797:.
787:15
785:.
777:;
294:.
177:.
1041:.
1013::
996:.
968::
958::
952:6
941:.
913::
896:.
876::
859:.
837::
813:.
793::
758:n
722:x
718:t
694:)
691:A
688:;
685:X
682:(
673:H
666:)
663:A
660:;
657:X
654:T
651:(
642:H
621:)
618:A
615:;
612:X
609:(
600:H
593:)
590:A
587:;
584:X
581:T
578:(
569:H
548:X
528:A
501:)
498:X
495:(
490:1
479:)
474:X
470:T
466:(
461:1
426:X
422:T
401:X
393:X
389:T
380:x
376:t
352:X
321:X
274:X
268:)
265:1
262:,
259:G
256:(
253:K
230:G
210:)
207:1
204:,
201:G
198:(
195:K
157:X
137:G
117:X
89:G
65:X
39:G
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