25:
648:
706:
447:
260:
217:
842:
742:
809:
506:
334:
1059:
592:
412:
286:
780:
562:
542:
382:
354:
170:
898:
89:
751:
61:
42:
68:
845:
75:
336:
determine the maximum number of linearly independent sections of the tangent bundle of any homotopy sphere. The case of
357:
1040:
997:
108:
57:
1086:
46:
1091:
864:
asks for multiplicative identities between quadratic forms. The classical results were revisited in 1952 by
600:
989:
660:
82:
461:
35:
1032:
417:
230:
187:
821:
715:
785:
482:
310:
938:
130:
815:
570:
224:
1050:
1007:
967:
8:
873:
391:
265:
145:
1025:
955:
915:
765:
547:
527:
367:
361:
339:
155:
134:
262:
is the exact number of pointwise linearly independent vector fields that exist on an (
1036:
993:
919:
849:
465:
16:
How many linearly independent smooth nowhere-zero vector fields can be on an n-sphere
818:, the Radon–Hurwitz number counts the maximum size of a linear subspace of the real
1046:
1003:
963:
947:
907:
464:, it is the same to ask for (pointwise) linear independence or fields that give an
453:
384:
even is an extension of that. Adams showed that the maximum number of continuous (
181:
138:
1020:
861:
517:
388:
would be no different here) pointwise linearly-independent vector fields on the (
220:
173:
857:
521:
297:
911:
1080:
869:
853:
513:
301:
865:
565:
509:
933:
177:
149:
122:
959:
893:
227:
to prove that no more independent vector fields could be found. Hence
951:
296:
In detail, the question applies to the 'round spheres' and to their
24:
988:. London Mathematical Society Lecture Note Series. Vol. 171.
896:(1957). "Whitehead products and vector-fields on spheres".
746:
148:
smooth nowhere-zero vector fields can be constructed on a
452:
The construction of the fields is related to the real
824:
814:
These numbers occur also in other, related areas. In
788:
768:
718:
663:
603:
573:
550:
530:
485:
420:
394:
370:
342:
313:
268:
233:
190:
180:. It was already known, by direct construction using
158:
758:
2, 4, 2, 8, 2, 4, 2, 9, 2, 4, 2, 8, 2, 4, 2, 10, ...
49:. Unsourced material may be challenged and removed.
1024:
899:Proceedings of the Cambridge Philosophical Society
836:
803:
774:
736:
700:
642:
586:
556:
536:
500:
441:
406:
376:
348:
328:
280:
254:
219:such fields (see definition below). Adams applied
211:
164:
1078:
1060:"Vector fields on spheres, etc. (course notes)"
844:matrices, for which each non-zero matrix is a
176:. A definitive answer was provided in 1962 by
868:. They are now applied in areas including
137:, and early work on the classification of
109:Learn how and when to remove this message
1019:
544:written as the product of an odd number
471:
460: 8 that also shows up here. By the
983:
456:, which is a theory with a periodicity
144:Specifically, the question is how many
1079:
979:
977:
932:
892:
304:have isomorphic tangent bundles, the
936:(1962). "Vector Fields on Spheres".
291:
47:adding citations to reliable sources
18:
974:
926:
643:{\displaystyle B=c+4d,0\leq c<4}
13:
1057:
1031:. Van Nostrand Reinhold. pp.
14:
1103:
701:{\displaystyle \rho (n)=2^{c}+8d}
23:
34:needs additional citations for
886:
798:
792:
731:
722:
673:
667:
495:
489:
430:
424:
323:
317:
243:
237:
200:
194:
1:
879:
782:, the value of the function
356:odd is taken care of by the
7:
358:Poincaré–Hopf index theorem
184:, that there were at least
129:was a classical problem of
10:
1108:
990:Cambridge University Press
442:{\displaystyle \rho (n)-1}
255:{\displaystyle \rho (n)-1}
212:{\displaystyle \rho (n)-1}
58:"Vector fields on spheres"
912:10.1017/S0305004100032928
846:similarity transformation
837:{\displaystyle n\times n}
737:{\displaystyle \rho (2n)}
508:occur in earlier work of
804:{\displaystyle \rho (n)}
712:The first few values of
501:{\displaystyle \rho (n)}
329:{\displaystyle \rho (n)}
127:vector fields on spheres
984:Rajwade, A. R. (1993).
848:, i.e. a product of an
838:
805:
776:
738:
702:
644:
588:
558:
538:
502:
443:
408:
378:
350:
330:
288:)-dimensional sphere.
282:
256:
213:
166:
1087:Differential topology
939:Annals of Mathematics
839:
806:
777:
739:
703:
645:
589:
587:{\displaystyle 2^{B}}
559:
539:
503:
478:Radon–Hurwitz numbers
472:Radon–Hurwitz numbers
444:
409:
379:
351:
331:
306:Radon–Hurwitz numbers
283:
257:
214:
167:
133:, beginning with the
131:differential topology
1092:Theorems in topology
1027:Topological Geometry
822:
786:
766:
744:are (from (sequence
716:
661:
601:
571:
548:
528:
483:
462:Gram–Schmidt process
418:
414:)-sphere is exactly
392:
368:
340:
311:
300:: in fact since all
266:
231:
225:topological K-theory
188:
156:
146:linearly independent
125:, the discussion of
43:improve this article
874:theoretical physics
407:{\displaystyle n-1}
281:{\displaystyle n-1}
834:
801:
772:
734:
698:
640:
584:
554:
534:
498:
439:
404:
374:
362:hairy ball theorem
346:
326:
278:
252:
209:
162:
135:hairy ball theorem
850:orthogonal matrix
775:{\displaystyle n}
557:{\displaystyle A}
537:{\displaystyle n}
466:orthonormal basis
454:Clifford algebras
377:{\displaystyle n}
349:{\displaystyle n}
292:Technical details
182:Clifford algebras
165:{\displaystyle n}
139:division algebras
119:
118:
111:
93:
1099:
1073:
1071:
1069:
1064:
1054:
1030:
1012:
1011:
981:
972:
971:
930:
924:
923:
890:
843:
841:
840:
835:
810:
808:
807:
802:
781:
779:
778:
773:
749:
743:
741:
740:
735:
707:
705:
704:
699:
688:
687:
649:
647:
646:
641:
593:
591:
590:
585:
583:
582:
563:
561:
560:
555:
543:
541:
540:
535:
507:
505:
504:
499:
448:
446:
445:
440:
413:
411:
410:
405:
383:
381:
380:
375:
355:
353:
352:
347:
335:
333:
332:
327:
287:
285:
284:
279:
261:
259:
258:
253:
218:
216:
215:
210:
171:
169:
168:
163:
114:
107:
103:
100:
94:
92:
51:
27:
19:
1107:
1106:
1102:
1101:
1100:
1098:
1097:
1096:
1077:
1076:
1067:
1065:
1062:
1043:
1016:
1015:
1000:
992:. p. 127.
982:
975:
952:10.2307/1970213
931:
927:
891:
887:
882:
862:Hurwitz problem
858:quadratic forms
823:
820:
819:
787:
784:
783:
767:
764:
763:
745:
717:
714:
713:
683:
679:
662:
659:
658:
602:
599:
598:
578:
574:
572:
569:
568:
549:
546:
545:
529:
526:
525:
522:quadratic forms
518:Hurwitz problem
484:
481:
480:
474:
468:at each point.
419:
416:
415:
393:
390:
389:
369:
366:
365:
364:), so the case
341:
338:
337:
312:
309:
308:
298:tangent bundles
294:
267:
264:
263:
232:
229:
228:
221:homotopy theory
189:
186:
185:
174:Euclidean space
157:
154:
153:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
1105:
1095:
1094:
1089:
1075:
1074:
1055:
1041:
1021:Porteous, I.R.
1014:
1013:
998:
973:
946:(3): 603–632.
925:
906:(4): 817–820.
884:
883:
881:
878:
833:
830:
827:
800:
797:
794:
791:
771:
760:
759:
733:
730:
727:
724:
721:
710:
709:
697:
694:
691:
686:
682:
678:
675:
672:
669:
666:
652:
651:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
609:
606:
581:
577:
553:
533:
516:(1923) on the
497:
494:
491:
488:
473:
470:
438:
435:
432:
429:
426:
423:
403:
400:
397:
373:
345:
325:
322:
319:
316:
302:exotic spheres
293:
290:
277:
274:
271:
251:
248:
245:
242:
239:
236:
208:
205:
202:
199:
196:
193:
161:
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
1104:
1093:
1090:
1088:
1085:
1084:
1082:
1061:
1058:Miller, H.R.
1056:
1052:
1048:
1044:
1042:0-442-06606-6
1038:
1034:
1029:
1028:
1022:
1018:
1017:
1009:
1005:
1001:
999:0-521-42668-5
995:
991:
987:
980:
978:
969:
965:
961:
957:
953:
949:
945:
941:
940:
935:
929:
921:
917:
913:
909:
905:
901:
900:
895:
889:
885:
877:
875:
871:
870:coding theory
867:
863:
859:
855:
854:scalar matrix
851:
847:
831:
828:
825:
817:
816:matrix theory
812:
795:
789:
769:
757:
756:
755:
753:
748:
728:
725:
719:
695:
692:
689:
684:
680:
676:
670:
664:
657:
656:
655:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
604:
597:
596:
595:
579:
575:
567:
551:
531:
523:
519:
515:
514:Adolf Hurwitz
511:
492:
486:
479:
469:
467:
463:
459:
455:
450:
436:
433:
427:
421:
401:
398:
395:
387:
371:
363:
359:
343:
320:
314:
307:
303:
299:
289:
275:
272:
269:
249:
246:
240:
234:
226:
222:
206:
203:
197:
191:
183:
179:
175:
172:-dimensional
159:
151:
147:
142:
140:
136:
132:
128:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
1066:. Retrieved
1026:
985:
943:
937:
934:Adams, J. F.
928:
903:
897:
894:James, I. M.
888:
866:Beno Eckmann
813:
761:
711:
653:
566:power of two
510:Johann Radon
477:
475:
457:
451:
385:
305:
295:
143:
126:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
1068:10 November
512:(1922) and
178:Frank Adams
123:mathematics
1081:Categories
1051:0186.06304
1008:0785.11022
968:0112.38102
880:References
69:newspapers
920:119646042
829:×
790:ρ
720:ρ
665:ρ
629:≤
487:ρ
434:−
422:ρ
399:−
315:ρ
273:−
247:−
235:ρ
204:−
192:ρ
1023:(1969).
811:is one.
762:For odd
594:, write
99:May 2012
1033:336–352
986:Squares
960:1970213
750:in the
747:A053381
524:. For
83:scholar
1049:
1039:
1006:
996:
966:
958:
918:
860:, the
856:. In
852:and a
564:and a
458:modulo
386:smooth
150:sphere
85:
78:
71:
64:
56:
1063:(PDF)
956:JSTOR
916:S2CID
654:Then
360:(see
90:JSTOR
76:books
1070:2018
1037:ISBN
994:ISBN
872:and
754:)):
752:OEIS
635:<
476:The
223:and
62:news
1047:Zbl
1004:Zbl
964:Zbl
948:doi
908:doi
520:on
152:in
121:In
45:by
1083::
1045:.
1035:.
1002:.
976:^
962:.
954:.
944:75
942:.
914:.
904:53
902:.
876:.
449:.
141:.
1072:.
1053:.
1010:.
970:.
950::
922:.
910::
832:n
826:n
799:)
796:n
793:(
770:n
732:)
729:n
726:2
723:(
708:.
696:d
693:8
690:+
685:c
681:2
677:=
674:)
671:n
668:(
650:.
638:4
632:c
626:0
623:,
620:d
617:4
614:+
611:c
608:=
605:B
580:B
576:2
552:A
532:n
496:)
493:n
490:(
437:1
431:)
428:n
425:(
402:1
396:n
372:n
344:n
324:)
321:n
318:(
276:1
270:n
250:1
244:)
241:n
238:(
207:1
201:)
198:n
195:(
160:n
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
