2776:
2341:
2771:{\displaystyle {\begin{aligned}\Omega (\mathbb {Z} \times BO)&\simeq O=(O\times O)/O&\Omega (\mathbb {Z} \times \operatorname {BSp} )&\simeq \operatorname {Sp} =(\operatorname {Sp} \times \operatorname {Sp} )/\operatorname {Sp} \\\Omega O&\simeq O/U&\Omega \operatorname {Sp} &\simeq \operatorname {Sp} /U\\\Omega (O/U)&\simeq U/\operatorname {Sp} &\Omega (\operatorname {Sp} /U)&\simeq U/O\\\Omega (U/\operatorname {Sp} )&\simeq \mathbb {Z} \times \operatorname {BSp} =\mathbb {Z} \times \operatorname {Sp} /(\operatorname {Sp} \times \operatorname {Sp} )&\Omega (U/O)&\simeq \mathbb {Z} \times BO=\mathbb {Z} \times O/(O\times O)\end{aligned}}}
632:
2121:
258:
2112:
627:{\displaystyle {\begin{aligned}\pi _{0}(O(\infty ))&\simeq \mathbb {Z} _{2}\\\pi _{1}(O(\infty ))&\simeq \mathbb {Z} _{2}\\\pi _{2}(O(\infty ))&\simeq 0\\\pi _{3}(O(\infty ))&\simeq \mathbb {Z} \\\pi _{4}(O(\infty ))&\simeq 0\\\pi _{5}(O(\infty ))&\simeq 0\\\pi _{6}(O(\infty ))&\simeq 0\\\pi _{7}(O(\infty ))&\simeq \mathbb {Z} \end{aligned}}}
1140:
1922:
2326:
2004:
1307:
935:
1838:
1993:
2182:
2107:{\displaystyle \mathbb {R} \oplus \mathbb {R} \subset \mathbb {R} \subset \mathbb {C} \subset \mathbb {H} \subset \mathbb {H} \oplus \mathbb {H} \subset \mathbb {H} \subset \mathbb {C} \subset \mathbb {R} \subset \mathbb {R} \oplus \mathbb {R} ,}
1154:
1700:
1763:
One elegant formulation of Bott periodicity makes use of the observation that there are natural embeddings (as closed subgroups) between the classical groups. The loop spaces in Bott periodicity are then homotopy equivalent to the
806:
3212:
674:) operation, and seeing what (roughly speaking) remained of homotopy theory once one was allowed to suspend both sides of an equation, as many times as one wished. The stable theory was still hard to compute with, in practice.
3318:
1739:
247:
1596:
1470:
1135:{\displaystyle {\begin{aligned}\pi _{k}(U)&=\pi _{k+2}(U)\\\pi _{k}(O)&=\pi _{k+4}(\operatorname {Sp} )\\\pi _{k}(\operatorname {Sp} )&=\pi _{k+4}(O)&&k=0,1,\ldots \end{aligned}}}
1827:
2346:
2187:
1159:
940:
263:
2885:
72:. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the
1917:{\displaystyle O\times O\subset O\subset U\subset \operatorname {Sp} \subset \operatorname {Sp} \times \operatorname {Sp} \subset \operatorname {Sp} \subset U\subset O\subset O\times O.}
1658:
1635:
1517:
2321:{\displaystyle {\begin{aligned}\Omega U&\simeq \mathbb {Z} \times BU=\mathbb {Z} \times U/(U\times U)\\\Omega (\mathbb {Z} \times BU)&\simeq U=(U\times U)/U\end{aligned}}}
2124:
Animation of the Bott periodicity clock using a Mod 8 clock face with second hand mnemonics taken from the I-Ching with the real
Clifford algebra of signature (p,q) denoted as Cl
1941:
3394:
3112:
2971:
1372:
2148:
902:
863:
3433:
3360:
3251:
3148:
3078:
3013:
2924:
2836:
150:
1302:{\displaystyle {\begin{aligned}\pi _{k}(O)&=\pi _{k+8}(O)\\\pi _{k}(\operatorname {Sp} )&=\pi _{k+8}(\operatorname {Sp} ),&&k=0,1,\ldots \end{aligned}}}
1396:
3462:
3042:
1705:
719:
3154:
3257:
677:
What Bott periodicity offered was an insight into some highly non-trivial spaces, with central status in topology because of the connection of their
2785:, and are the successive quotients of the terms of the Bott periodicity clock. These equivalences immediately yield the Bott periodicity theorems.
3611:
1551:
1425:
96:
170:
3785:
3848:
1601:
1483:
1785:
1932:
2842:
3811:
831:
112:
69:
17:
3843:
1988:{\displaystyle \mathbb {C} \oplus \mathbb {C} \subset \mathbb {C} \subset \mathbb {C} \oplus \mathbb {C} .}
2789:
3366:
3084:
2930:
872:
from the (unstable) homotopy groups of the (stable) classical groups to these stable homotopy groups
115:, which causes the period 8 Bott periodicity to be visible in the stable homotopy groups of spheres.
1344:
663:
3529:/Sp is reductive. As these show, the difference can be interpreted as whether or not one includes
2131:
875:
836:
3747:
3469:
3411:
3402:
3338:
3229:
3126:
3056:
2991:
2902:
2814:
1341:
in infinite dimensions). One formulation of Bott periodicity describes the twofold loop space,
126:
2154:
As they are 2-periodic/8-periodic, they can be arranged in a circle, where they are called the
824:
659:
153:
60:), which proved to be of foundational significance for much further research, in particular in
3771:
1695:{\displaystyle \Omega ^{8}\operatorname {BSp} \simeq \mathbb {Z} \times \operatorname {BSp} ;}
1148:
The second and third of these isomorphisms intertwine to give the 8-fold periodicity results:
3680:
1407:
1381:
3780:, Contemporary Mathematics, vol. 199, American Mathematical Society, pp. 107–124,
3795:
3717:
3650:
3620:
3599:
2166:
1473:
682:
77:
8:
3745:(1970), "The periodicity theorem for the classical groups and some of its applications",
3439:
3019:
905:
3624:
3705:
3638:
651:
3726:
3659:
3807:
3781:
3761:
3731:
3697:
3664:
3587:
3552:
1648:
1541:
1415:
1330:
801:{\displaystyle U(1)\subset U(2)\subset \cdots \subset U=\bigcup _{k=1}^{\infty }U(k)}
157:
3207:{\displaystyle \mathbb {Z} \times \mathrm {Sp} /(\mathrm {Sp} \times \mathrm {Sp} )}
3756:
3721:
3689:
3654:
3628:
3577:
3328:
3324:
2981:
2977:
1928:
1644:
1529:
926:
909:
820:
104:
100:
3791:
3713:
3646:
3595:
2782:
1765:
1418:
construction. Bott periodicity states that this double loop space is essentially
655:
161:
45:
3612:
Proceedings of the
National Academy of Sciences of the United States of America
3313:{\displaystyle \mathrm {Sp} =(\mathrm {Sp} \times \mathrm {Sp} )/\mathrm {Sp} }
1776:
913:
866:
643:
108:
41:
1521:
Either of these has the immediate effect of showing why (complex) topological
3837:
3822:
3701:
3591:
1652:
1545:
1403:
1334:
1318:
667:
73:
65:
3568:(1956), "An application of the Morse theory to the topology of Lie-groups",
3735:
3668:
3633:
3487:
3220:
3119:
space of quaternionic structures compatible with a given complex structure
2893:
1411:
1338:
710:
689:) homotopy groups could be calculated. These spaces are the (infinite, or
3766:. An expository account of the theorem and the mathematics surrounding it.
3049:
space of complex structures compatible with a given orthogonal structure
1734:{\displaystyle \Omega ^{8}\operatorname {Sp} \simeq \operatorname {Sp} .}
1548:. In this case, Bott periodicity states that, for the 8-fold loop space,
84:
33:
2781:
The resulting spaces are homotopy equivalent to the classical reductive
3742:
3709:
3675:
3606:
3582:
3565:
2120:
1399:
678:
92:
83:
There are corresponding period-8 phenomena for the matching theories, (
49:
3775:
658:, have proved elusive (and the theory is complicated). The subject of
3642:
3513:
The interpretation and labeling is slightly incorrect, and refers to
3495:
88:
3693:
111:
is a homomorphism from the homotopy groups of orthogonal groups to
2117:
where the division algebras indicate "matrices over that algebra".
61:
27:
Describes a periodicity in the homotopy groups of classical groups
1755:-theory (also known as KSp-theory) are 8-fold periodic theories.
242:{\displaystyle \pi _{n}(O(\infty ))\simeq \pi _{n+8}(O(\infty ))}
3774:, in Banaszak, Grzegorz; Gajda, Wojciech; Krasoń, Piotr (eds.),
671:
647:
1768:
of successive quotients, with additional discrete factors of
1643:-theory is an 8-fold periodic theory. Also, for the infinite
2165:
The Bott periodicity results then refine to a sequence of
1591:{\displaystyle \Omega ^{8}BO\simeq \mathbb {Z} \times BO}
1465:{\displaystyle \Omega ^{2}BU\simeq \mathbb {Z} \times BU}
919:
Bott's original results may be succinctly summarized in:
819:
in the title of his seminal paper refers to these stable
912:(1963) which was finally resolved in the affirmative by
3678:(1959), "The stable homotopy of the classical groups",
3609:(1957), "The stable homotopy of the classical groups",
830:
The important connection of Bott periodicity with the
662:
was conceived as a simplification, by introducing the
3442:
3414:
3369:
3341:
3260:
3232:
3157:
3129:
3087:
3059:
3022:
2994:
2933:
2905:
2845:
2817:
2344:
2185:
2134:
2007:
1944:
1841:
1788:
1708:
1661:
1604:
1554:
1486:
1428:
1384:
1347:
1312:
1157:
938:
878:
839:
722:
261:
173:
129:
1822:{\displaystyle U\times U\subset U\subset U\times U.}
650:, which would be expected to play the basic part in
3517:symmetric spaces, while these are the more general
3456:
3427:
3388:
3354:
3312:
3245:
3206:
3142:
3106:
3072:
3036:
3007:
2965:
2918:
2879:
2830:
2770:
2320:
2142:
2106:
1987:
1916:
1821:
1733:
1694:
1629:
1590:
1511:
1464:
1390:
1366:
1301:
1134:
896:
857:
800:
626:
241:
144:
1758:
1476:to) the union of a countable number of copies of
925:The (unstable) homotopy groups of the (infinite)
3835:
2880:{\displaystyle \mathbb {Z} \times O/(O\times O)}
252:and the first 8 homotopy groups are as follows:
815:and Sp. Note that Bott's use of the word
3570:Bulletin de la Société Mathématique de France
1528:In the corresponding theory for the infinite
1927:These sequences corresponds to sequences in
693:) unitary, orthogonal and symplectic groups
642:The context of Bott periodicity is that the
637:
3494:had used earlier to study the homology of
2788:The specific spaces are, (for groups, the
1317:For the theory associated to the infinite
3827:This Week's Finds in Mathematical Physics
3772:"Bott periodicity and the Q-construction"
3760:
3725:
3658:
3632:
3581:
3498:. Many different proofs have been given.
3159:
2847:
2734:
2717:
2654:
2640:
2417:
2356:
2262:
2221:
2204:
2136:
2097:
2089:
2081:
2073:
2065:
2057:
2049:
2041:
2033:
2025:
2017:
2009:
1978:
1970:
1962:
1954:
1946:
1679:
1575:
1449:
616:
446:
354:
303:
2119:
1998:Over the real numbers and quaternions:
1832:Over the real numbers and quaternions:
14:
3836:
3801:
3769:
1751:-theory) and topological quaternionic
908:, it became the subject of the famous
118:
1630:{\displaystyle \Omega ^{8}O\simeq O,}
1525:-theory is a 2-fold periodic theory.
1512:{\displaystyle \Omega ^{2}U\simeq U.}
3820:
3741:
3674:
3605:
3564:
3491:
3483:
57:
53:
1933:classification of Clifford algebras
1655:, and Bott periodicity states that
24:
3416:
3374:
3371:
3343:
3306:
3303:
3290:
3287:
3279:
3276:
3265:
3262:
3234:
3197:
3194:
3186:
3183:
3170:
3167:
3131:
3100:
3097:
3061:
2996:
2907:
2819:
2689:
2612:
2567:
2527:
2496:
2470:
2410:
2349:
2255:
2190:
1710:
1663:
1639:which yields the consequence that
1606:
1556:
1488:
1430:
1385:
1349:
1313:Loop spaces and classifying spaces
781:
599:
557:
515:
473:
429:
387:
336:
285:
230:
193:
136:
25:
3860:
832:stable homotopy groups of spheres
113:stable homotopy groups of spheres
70:stable homotopy groups of spheres
1480:. An equivalent formulation is
713:) of the sequence of inclusions
3389:{\displaystyle \mathrm {Sp} /U}
3107:{\displaystyle U/\mathrm {Sp} }
2966:{\displaystyle O=(O\times O)/O}
865:comes via the so-called stable
701:and Sp. In this context,
40:describes a periodicity in the
3806:. Princeton University Press.
3545:
3507:
3294:
3272:
3201:
3179:
2952:
2940:
2874:
2862:
2761:
2749:
2706:
2692:
2684:
2672:
2629:
2615:
2587:
2570:
2544:
2530:
2455:
2443:
2427:
2413:
2397:
2385:
2369:
2352:
2303:
2291:
2275:
2258:
2248:
2236:
1759:Geometric model of loop spaces
1265:
1259:
1233:
1227:
1210:
1204:
1178:
1172:
1101:
1095:
1069:
1063:
1046:
1040:
1014:
1008:
991:
985:
959:
953:
795:
789:
747:
741:
732:
726:
605:
602:
596:
590:
563:
560:
554:
548:
521:
518:
512:
506:
479:
476:
470:
464:
435:
432:
426:
420:
393:
390:
384:
378:
342:
339:
333:
327:
291:
288:
282:
276:
236:
233:
227:
221:
199:
196:
190:
184:
139:
133:
13:
1:
3538:
1367:{\displaystyle \Omega ^{2}BU}
3762:10.1016/0001-8708(70)90030-7
2143:{\displaystyle \mathbb {R} }
1935:; over the complex numbers:
897:{\displaystyle \pi _{n}^{S}}
858:{\displaystyle \pi _{n}^{S}}
7:
3849:Theorems in homotopy theory
3821:Baez, John (21 June 1997).
3428:{\displaystyle \Omega ^{7}}
3355:{\displaystyle \Omega ^{6}}
3246:{\displaystyle \Omega ^{5}}
3143:{\displaystyle \Omega ^{4}}
3073:{\displaystyle \Omega ^{3}}
3008:{\displaystyle \Omega ^{2}}
2919:{\displaystyle \Omega ^{1}}
2831:{\displaystyle \Omega ^{0}}
2790:principal homogeneous space
1743:Thus both topological real
1647:, Sp, the space BSp is the
904:. Originally described by
705:refers to taking the union
10:
3865:
3525:/Sp is irreducible, while
2331:For real and quaternionic
145:{\displaystyle O(\infty )}
3477:
1472:is essentially (that is,
99:, associated to the real
3501:
1651:for stable quaternionic
638:Context and significance
38:Bott periodicity theorem
3748:Advances in Mathematics
3482:Bott's original proof (
3470:Lagrangian Grassmannian
3403:Lagrangian Grassmannian
1747:-theory (also known as
1422:again; more precisely,
1391:{\displaystyle \Omega }
3844:Topology of Lie groups
3634:10.1073/pnas.43.10.933
3458:
3429:
3390:
3356:
3314:
3247:
3208:
3144:
3108:
3074:
3038:
3009:
2967:
2920:
2881:
2832:
2772:
2322:
2160:Clifford algebra clock
2156:Bott periodicity clock
2151:
2144:
2108:
1989:
1918:
1823:
1735:
1696:
1631:
1592:
1513:
1466:
1392:
1368:
1303:
1136:
898:
859:
802:
785:
683:characteristic classes
660:stable homotopy theory
628:
243:
146:
3770:Giffen, C.H. (1996),
3681:Annals of Mathematics
3521:spaces. For example,
3459:
3430:
3391:
3357:
3315:
3248:
3209:
3145:
3109:
3075:
3039:
3010:
2968:
2921:
2882:
2833:
2773:
2323:
2167:homotopy equivalences
2145:
2123:
2109:
1990:
1919:
1824:
1736:
1697:
1632:
1593:
1514:
1467:
1393:
1369:
1304:
1137:
899:
860:
803:
765:
685:, for which all the (
629:
244:
147:
103:and the quaternionic
3440:
3412:
3367:
3339:
3258:
3230:
3155:
3127:
3085:
3057:
3020:
2992:
2931:
2903:
2843:
2815:
2342:
2335:- and KSp-theories:
2183:
2132:
2005:
1942:
1839:
1786:
1706:
1659:
1602:
1552:
1484:
1426:
1382:
1345:
1155:
936:
876:
837:
720:
259:
171:
127:
123:Bott showed that if
107:, respectively. The
78:topological K-theory
3802:Milnor, J. (1969).
3625:1957PNAS...43..933B
3457:{\displaystyle U/O}
3037:{\displaystyle O/U}
1474:homotopy equivalent
1333:for stable complex
906:George W. Whitehead
893:
854:
709:(also known as the
119:Statement of result
3777:Algebraic K-Theory
3583:10.24033/bsmf.1472
3454:
3425:
3386:
3352:
3310:
3243:
3204:
3140:
3104:
3070:
3034:
3005:
2963:
2916:
2877:
2828:
2768:
2766:
2318:
2316:
2152:
2140:
2104:
1985:
1914:
1819:
1731:
1692:
1627:
1588:
1509:
1462:
1388:
1364:
1299:
1297:
1132:
1130:
894:
879:
855:
840:
811:and similarly for
798:
652:algebraic topology
624:
622:
239:
152:is defined as the
142:
76:. See for example
64:of stable complex
3787:978-0-8218-0511-4
3684:, Second Series,
3475:
3474:
2792:is also listed):
1929:Clifford algebras
1649:classifying space
1598:or equivalently,
1542:classifying space
1416:classifying space
1331:classifying space
158:orthogonal groups
68:, as well as the
16:(Redirected from
3856:
3830:
3817:
3798:
3765:
3764:
3738:
3729:
3671:
3662:
3636:
3602:
3585:
3557:
3556:
3549:
3533:
3511:
3463:
3461:
3460:
3455:
3450:
3434:
3432:
3431:
3426:
3424:
3423:
3395:
3393:
3392:
3387:
3382:
3377:
3361:
3359:
3358:
3353:
3351:
3350:
3329:Stiefel manifold
3325:Symplectic group
3319:
3317:
3316:
3311:
3309:
3301:
3293:
3282:
3268:
3252:
3250:
3249:
3244:
3242:
3241:
3213:
3211:
3210:
3205:
3200:
3189:
3178:
3173:
3162:
3149:
3147:
3146:
3141:
3139:
3138:
3113:
3111:
3110:
3105:
3103:
3095:
3079:
3077:
3076:
3071:
3069:
3068:
3043:
3041:
3040:
3035:
3030:
3014:
3012:
3011:
3006:
3004:
3003:
2982:Stiefel manifold
2978:Orthogonal group
2972:
2970:
2969:
2964:
2959:
2925:
2923:
2922:
2917:
2915:
2914:
2886:
2884:
2883:
2878:
2861:
2850:
2837:
2835:
2834:
2829:
2827:
2826:
2795:
2794:
2783:symmetric spaces
2777:
2775:
2774:
2769:
2767:
2748:
2737:
2720:
2702:
2671:
2657:
2643:
2625:
2604:
2583:
2561:
2540:
2519:
2490:
2462:
2420:
2404:
2359:
2327:
2325:
2324:
2319:
2317:
2310:
2265:
2235:
2224:
2207:
2149:
2147:
2146:
2141:
2139:
2113:
2111:
2110:
2105:
2100:
2092:
2084:
2076:
2068:
2060:
2052:
2044:
2036:
2028:
2020:
2012:
1994:
1992:
1991:
1986:
1981:
1973:
1965:
1957:
1949:
1923:
1921:
1920:
1915:
1828:
1826:
1825:
1820:
1766:symmetric spaces
1740:
1738:
1737:
1732:
1718:
1717:
1702:or equivalently
1701:
1699:
1698:
1693:
1682:
1671:
1670:
1645:symplectic group
1636:
1634:
1633:
1628:
1614:
1613:
1597:
1595:
1594:
1589:
1578:
1564:
1563:
1544:for stable real
1530:orthogonal group
1518:
1516:
1515:
1510:
1496:
1495:
1471:
1469:
1468:
1463:
1452:
1438:
1437:
1397:
1395:
1394:
1389:
1373:
1371:
1370:
1365:
1357:
1356:
1308:
1306:
1305:
1300:
1298:
1272:
1258:
1257:
1226:
1225:
1203:
1202:
1171:
1170:
1141:
1139:
1138:
1133:
1131:
1105:
1094:
1093:
1062:
1061:
1039:
1038:
1007:
1006:
984:
983:
952:
951:
927:classical groups
910:Adams conjecture
903:
901:
900:
895:
892:
887:
864:
862:
861:
856:
853:
848:
821:classical groups
807:
805:
804:
799:
784:
779:
654:by analogy with
633:
631:
630:
625:
623:
619:
589:
588:
547:
546:
505:
504:
463:
462:
449:
419:
418:
377:
376:
363:
362:
357:
326:
325:
312:
311:
306:
275:
274:
248:
246:
245:
240:
220:
219:
183:
182:
151:
149:
148:
143:
105:symplectic group
101:orthogonal group
48:, discovered by
46:classical groups
21:
18:Bott periodicity
3864:
3863:
3859:
3858:
3857:
3855:
3854:
3853:
3834:
3833:
3814:
3788:
3694:10.2307/1970106
3561:
3560:
3551:
3550:
3546:
3541:
3536:
3512:
3508:
3504:
3480:
3446:
3441:
3438:
3437:
3419:
3415:
3413:
3410:
3409:
3378:
3370:
3368:
3365:
3364:
3346:
3342:
3340:
3337:
3336:
3302:
3297:
3286:
3275:
3261:
3259:
3256:
3255:
3237:
3233:
3231:
3228:
3227:
3193:
3182:
3174:
3166:
3158:
3156:
3153:
3152:
3134:
3130:
3128:
3125:
3124:
3096:
3091:
3086:
3083:
3082:
3064:
3060:
3058:
3055:
3054:
3026:
3021:
3018:
3017:
2999:
2995:
2993:
2990:
2989:
2955:
2932:
2929:
2928:
2910:
2906:
2904:
2901:
2900:
2857:
2846:
2844:
2841:
2840:
2822:
2818:
2816:
2813:
2812:
2765:
2764:
2744:
2733:
2716:
2709:
2698:
2687:
2667:
2653:
2639:
2632:
2621:
2609:
2608:
2600:
2590:
2579:
2565:
2557:
2547:
2536:
2524:
2523:
2515:
2502:
2494:
2486:
2476:
2467:
2466:
2458:
2430:
2416:
2408:
2400:
2372:
2355:
2345:
2343:
2340:
2339:
2315:
2314:
2306:
2278:
2261:
2252:
2251:
2231:
2220:
2203:
2196:
2186:
2184:
2181:
2180:
2135:
2133:
2130:
2129:
2127:
2096:
2088:
2080:
2072:
2064:
2056:
2048:
2040:
2032:
2024:
2016:
2008:
2006:
2003:
2002:
1977:
1969:
1961:
1953:
1945:
1943:
1940:
1939:
1840:
1837:
1836:
1787:
1784:
1783:
1777:complex numbers
1761:
1713:
1709:
1707:
1704:
1703:
1678:
1666:
1662:
1660:
1657:
1656:
1609:
1605:
1603:
1600:
1599:
1574:
1559:
1555:
1553:
1550:
1549:
1491:
1487:
1485:
1482:
1481:
1448:
1433:
1429:
1427:
1424:
1423:
1383:
1380:
1379:
1352:
1348:
1346:
1343:
1342:
1315:
1296:
1295:
1271:
1247:
1243:
1236:
1221:
1217:
1214:
1213:
1192:
1188:
1181:
1166:
1162:
1158:
1156:
1153:
1152:
1129:
1128:
1104:
1083:
1079:
1072:
1057:
1053:
1050:
1049:
1028:
1024:
1017:
1002:
998:
995:
994:
973:
969:
962:
947:
943:
939:
937:
934:
933:
888:
883:
877:
874:
873:
849:
844:
838:
835:
834:
825:stable homotopy
780:
769:
721:
718:
717:
656:homology theory
644:homotopy groups
640:
621:
620:
615:
608:
584:
580:
577:
576:
566:
542:
538:
535:
534:
524:
500:
496:
493:
492:
482:
458:
454:
451:
450:
445:
438:
414:
410:
407:
406:
396:
372:
368:
365:
364:
358:
353:
352:
345:
321:
317:
314:
313:
307:
302:
301:
294:
270:
266:
262:
260:
257:
256:
209:
205:
178:
174:
172:
169:
168:
162:homotopy groups
154:inductive limit
128:
125:
124:
121:
42:homotopy groups
28:
23:
22:
15:
12:
11:
5:
3862:
3852:
3851:
3846:
3832:
3831:
3818:
3812:
3799:
3786:
3767:
3755:(3): 353–411,
3739:
3688:(2): 313–337,
3672:
3603:
3559:
3558:
3553:"Introduction"
3543:
3542:
3540:
3537:
3535:
3534:
3505:
3503:
3500:
3479:
3476:
3473:
3472:
3467:
3464:
3453:
3449:
3445:
3435:
3422:
3418:
3406:
3405:
3399:
3396:
3385:
3381:
3376:
3373:
3362:
3349:
3345:
3333:
3332:
3327:(quaternionic
3322:
3320:
3308:
3305:
3300:
3296:
3292:
3289:
3285:
3281:
3278:
3274:
3271:
3267:
3264:
3253:
3240:
3236:
3224:
3223:
3219:Quaternionic
3217:
3214:
3203:
3199:
3196:
3192:
3188:
3185:
3181:
3177:
3172:
3169:
3165:
3161:
3150:
3137:
3133:
3121:
3120:
3117:
3114:
3102:
3099:
3094:
3090:
3080:
3067:
3063:
3051:
3050:
3047:
3044:
3033:
3029:
3025:
3015:
3002:
2998:
2986:
2985:
2975:
2973:
2962:
2958:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2926:
2913:
2909:
2897:
2896:
2890:
2887:
2876:
2873:
2870:
2867:
2864:
2860:
2856:
2853:
2849:
2838:
2825:
2821:
2809:
2808:
2805:
2804:Cartan's label
2802:
2799:
2779:
2778:
2763:
2760:
2757:
2754:
2751:
2747:
2743:
2740:
2736:
2732:
2729:
2726:
2723:
2719:
2715:
2712:
2710:
2708:
2705:
2701:
2697:
2694:
2691:
2688:
2686:
2683:
2680:
2677:
2674:
2670:
2666:
2663:
2660:
2656:
2652:
2649:
2646:
2642:
2638:
2635:
2633:
2631:
2628:
2624:
2620:
2617:
2614:
2611:
2610:
2607:
2603:
2599:
2596:
2593:
2591:
2589:
2586:
2582:
2578:
2575:
2572:
2569:
2566:
2564:
2560:
2556:
2553:
2550:
2548:
2546:
2543:
2539:
2535:
2532:
2529:
2526:
2525:
2522:
2518:
2514:
2511:
2508:
2505:
2503:
2501:
2498:
2495:
2493:
2489:
2485:
2482:
2479:
2477:
2475:
2472:
2469:
2468:
2465:
2461:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2431:
2429:
2426:
2423:
2419:
2415:
2412:
2409:
2407:
2403:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2373:
2371:
2368:
2365:
2362:
2358:
2354:
2351:
2348:
2347:
2329:
2328:
2313:
2309:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2279:
2277:
2274:
2271:
2268:
2264:
2260:
2257:
2254:
2253:
2250:
2247:
2244:
2241:
2238:
2234:
2230:
2227:
2223:
2219:
2216:
2213:
2210:
2206:
2202:
2199:
2197:
2195:
2192:
2189:
2188:
2138:
2125:
2115:
2114:
2103:
2099:
2095:
2091:
2087:
2083:
2079:
2075:
2071:
2067:
2063:
2059:
2055:
2051:
2047:
2043:
2039:
2035:
2031:
2027:
2023:
2019:
2015:
2011:
1996:
1995:
1984:
1980:
1976:
1972:
1968:
1964:
1960:
1956:
1952:
1948:
1925:
1924:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1830:
1829:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1760:
1757:
1730:
1727:
1724:
1721:
1716:
1712:
1691:
1688:
1685:
1681:
1677:
1674:
1669:
1665:
1653:vector bundles
1626:
1623:
1620:
1617:
1612:
1608:
1587:
1584:
1581:
1577:
1573:
1570:
1567:
1562:
1558:
1546:vector bundles
1508:
1505:
1502:
1499:
1494:
1490:
1461:
1458:
1455:
1451:
1447:
1444:
1441:
1436:
1432:
1387:
1363:
1360:
1355:
1351:
1335:vector bundles
1314:
1311:
1310:
1309:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1256:
1253:
1250:
1246:
1242:
1239:
1237:
1235:
1232:
1229:
1224:
1220:
1216:
1215:
1212:
1209:
1206:
1201:
1198:
1195:
1191:
1187:
1184:
1182:
1180:
1177:
1174:
1169:
1165:
1161:
1160:
1143:
1142:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1092:
1089:
1086:
1082:
1078:
1075:
1073:
1071:
1068:
1065:
1060:
1056:
1052:
1051:
1048:
1045:
1042:
1037:
1034:
1031:
1027:
1023:
1020:
1018:
1016:
1013:
1010:
1005:
1001:
997:
996:
993:
990:
987:
982:
979:
976:
972:
968:
965:
963:
961:
958:
955:
950:
946:
942:
941:
929:are periodic:
914:Daniel Quillen
891:
886:
882:
852:
847:
843:
809:
808:
797:
794:
791:
788:
783:
778:
775:
772:
768:
764:
761:
758:
755:
752:
749:
746:
743:
740:
737:
734:
731:
728:
725:
639:
636:
635:
634:
618:
614:
611:
609:
607:
604:
601:
598:
595:
592:
587:
583:
579:
578:
575:
572:
569:
567:
565:
562:
559:
556:
553:
550:
545:
541:
537:
536:
533:
530:
527:
525:
523:
520:
517:
514:
511:
508:
503:
499:
495:
494:
491:
488:
485:
483:
481:
478:
475:
472:
469:
466:
461:
457:
453:
452:
448:
444:
441:
439:
437:
434:
431:
428:
425:
422:
417:
413:
409:
408:
405:
402:
399:
397:
395:
392:
389:
386:
383:
380:
375:
371:
367:
366:
361:
356:
351:
348:
346:
344:
341:
338:
335:
332:
329:
324:
320:
316:
315:
310:
305:
300:
297:
295:
293:
290:
287:
284:
281:
278:
273:
269:
265:
264:
250:
249:
238:
235:
232:
229:
226:
223:
218:
215:
212:
208:
204:
201:
198:
195:
192:
189:
186:
181:
177:
164:are periodic:
141:
138:
135:
132:
120:
117:
109:J-homomorphism
66:vector bundles
50:Raoul Bott
26:
9:
6:
4:
3:
2:
3861:
3850:
3847:
3845:
3842:
3841:
3839:
3828:
3824:
3819:
3815:
3813:0-691-08008-9
3809:
3805:
3800:
3797:
3793:
3789:
3783:
3779:
3778:
3773:
3768:
3763:
3758:
3754:
3750:
3749:
3744:
3740:
3737:
3733:
3728:
3723:
3719:
3715:
3711:
3707:
3703:
3699:
3695:
3691:
3687:
3683:
3682:
3677:
3673:
3670:
3666:
3661:
3656:
3652:
3648:
3644:
3640:
3635:
3630:
3626:
3622:
3619:(10): 933–5,
3618:
3614:
3613:
3608:
3604:
3601:
3597:
3593:
3589:
3584:
3579:
3575:
3571:
3567:
3563:
3562:
3554:
3548:
3544:
3532:
3528:
3524:
3520:
3516:
3510:
3506:
3499:
3497:
3493:
3489:
3485:
3471:
3468:
3465:
3451:
3447:
3443:
3436:
3420:
3408:
3407:
3404:
3400:
3397:
3383:
3379:
3363:
3347:
3335:
3334:
3330:
3326:
3323:
3321:
3298:
3283:
3269:
3254:
3238:
3226:
3225:
3222:
3218:
3215:
3190:
3175:
3163:
3151:
3135:
3123:
3122:
3118:
3115:
3092:
3088:
3081:
3065:
3053:
3052:
3048:
3045:
3031:
3027:
3023:
3016:
3000:
2988:
2987:
2983:
2979:
2976:
2974:
2960:
2956:
2949:
2946:
2943:
2937:
2934:
2927:
2911:
2899:
2898:
2895:
2891:
2888:
2871:
2868:
2865:
2858:
2854:
2851:
2839:
2823:
2811:
2810:
2806:
2803:
2800:
2797:
2796:
2793:
2791:
2786:
2784:
2758:
2755:
2752:
2745:
2741:
2738:
2730:
2727:
2724:
2721:
2713:
2711:
2703:
2699:
2695:
2681:
2678:
2675:
2668:
2664:
2661:
2658:
2650:
2647:
2644:
2636:
2634:
2626:
2622:
2618:
2605:
2601:
2597:
2594:
2592:
2584:
2580:
2576:
2573:
2562:
2558:
2554:
2551:
2549:
2541:
2537:
2533:
2520:
2516:
2512:
2509:
2506:
2504:
2499:
2491:
2487:
2483:
2480:
2478:
2473:
2463:
2459:
2452:
2449:
2446:
2440:
2437:
2434:
2432:
2424:
2421:
2405:
2401:
2394:
2391:
2388:
2382:
2379:
2376:
2374:
2366:
2363:
2360:
2338:
2337:
2336:
2334:
2311:
2307:
2300:
2297:
2294:
2288:
2285:
2282:
2280:
2272:
2269:
2266:
2245:
2242:
2239:
2232:
2228:
2225:
2217:
2214:
2211:
2208:
2200:
2198:
2193:
2179:
2178:
2177:
2175:
2170:
2168:
2163:
2161:
2157:
2122:
2118:
2101:
2093:
2085:
2077:
2069:
2061:
2053:
2045:
2037:
2029:
2021:
2013:
2001:
2000:
1999:
1982:
1974:
1966:
1958:
1950:
1938:
1937:
1936:
1934:
1930:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1835:
1834:
1833:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1782:
1781:
1780:
1778:
1773:
1771:
1767:
1756:
1754:
1750:
1746:
1741:
1728:
1725:
1722:
1719:
1714:
1689:
1686:
1683:
1675:
1672:
1667:
1654:
1650:
1646:
1642:
1637:
1624:
1621:
1618:
1615:
1610:
1585:
1582:
1579:
1571:
1568:
1565:
1560:
1547:
1543:
1539:
1535:
1531:
1526:
1524:
1519:
1506:
1503:
1500:
1497:
1492:
1479:
1475:
1459:
1456:
1453:
1445:
1442:
1439:
1434:
1421:
1417:
1413:
1409:
1405:
1404:right adjoint
1401:
1377:
1361:
1358:
1353:
1340:
1336:
1332:
1328:
1324:
1320:
1319:unitary group
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1268:
1262:
1254:
1251:
1248:
1244:
1240:
1238:
1230:
1222:
1218:
1207:
1199:
1196:
1193:
1189:
1185:
1183:
1175:
1167:
1163:
1151:
1150:
1149:
1147:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1098:
1090:
1087:
1084:
1080:
1076:
1074:
1066:
1058:
1054:
1043:
1035:
1032:
1029:
1025:
1021:
1019:
1011:
1003:
999:
988:
980:
977:
974:
970:
966:
964:
956:
948:
944:
932:
931:
930:
928:
924:
920:
917:
915:
911:
907:
889:
884:
880:
871:
870:-homomorphism
869:
850:
845:
841:
833:
828:
826:
822:
818:
814:
792:
786:
776:
773:
770:
766:
762:
759:
756:
753:
750:
744:
738:
735:
729:
723:
716:
715:
714:
712:
708:
704:
700:
696:
692:
688:
684:
680:
675:
673:
669:
668:smash product
665:
661:
657:
653:
649:
645:
612:
610:
593:
585:
581:
573:
570:
568:
551:
543:
539:
531:
528:
526:
509:
501:
497:
489:
486:
484:
467:
459:
455:
442:
440:
423:
415:
411:
403:
400:
398:
381:
373:
369:
359:
349:
347:
330:
322:
318:
308:
298:
296:
279:
271:
267:
255:
254:
253:
224:
216:
213:
210:
206:
202:
187:
179:
175:
167:
166:
165:
163:
159:
155:
130:
116:
114:
110:
106:
102:
98:
94:
90:
86:
81:
79:
75:
74:unitary group
71:
67:
63:
59:
55:
51:
47:
43:
39:
35:
30:
19:
3826:
3804:Morse Theory
3803:
3776:
3752:
3746:
3685:
3679:
3616:
3610:
3573:
3569:
3547:
3531:orientation.
3530:
3526:
3522:
3518:
3514:
3509:
3488:Morse theory
3481:
3221:Grassmannian
2894:Grassmannian
2807:Description
2787:
2780:
2332:
2330:
2173:
2172:For complex
2171:
2164:
2159:
2155:
2153:
2116:
1997:
1926:
1831:
1774:
1769:
1762:
1752:
1748:
1744:
1742:
1640:
1638:
1537:
1536:, the space
1533:
1527:
1522:
1520:
1477:
1419:
1412:left adjoint
1375:
1339:Grassmannian
1326:
1325:, the space
1322:
1316:
1145:
1144:
922:
921:
918:
867:
829:
816:
812:
810:
711:direct limit
706:
702:
698:
694:
690:
686:
676:
641:
251:
122:
93:quaternionic
82:
37:
31:
29:
3743:Bott, Raoul
3676:Bott, Raoul
3607:Bott, Raoul
3576:: 251–281,
3566:Bott, Raoul
3515:irreducible
3492:Bott (1956)
823:and not to
160:, then its
34:mathematics
3838:Categories
3823:"Week 105"
3539:References
3496:Lie groups
2798:Loop space
2150:)=Cl(p,q).
1408:suspension
1400:loop space
923:Corollary:
679:cohomology
664:suspension
97:KSp-theory
3702:0003-486X
3592:0037-9484
3519:reductive
3484:Bott 1959
3417:Ω
3344:Ω
3284:×
3235:Ω
3191:×
3164:×
3132:Ω
3062:Ω
2997:Ω
2947:×
2908:Ω
2869:×
2852:×
2820:Ω
2756:×
2739:×
2722:×
2714:≃
2690:Ω
2679:×
2665:
2659:×
2645:×
2637:≃
2613:Ω
2595:≃
2577:
2568:Ω
2552:≃
2528:Ω
2513:
2507:≃
2497:Ω
2481:≃
2471:Ω
2450:×
2435:≃
2422:×
2411:Ω
2392:×
2377:≃
2361:×
2350:Ω
2298:×
2283:≃
2267:×
2256:Ω
2243:×
2226:×
2209:×
2201:≃
2191:Ω
2176:-theory:
2094:⊕
2086:⊂
2078:⊂
2070:⊂
2062:⊂
2054:⊕
2046:⊂
2038:⊂
2030:⊂
2022:⊂
2014:⊕
1975:⊕
1967:⊂
1959:⊂
1951:⊕
1906:×
1900:⊂
1894:⊂
1888:⊂
1882:⊂
1876:×
1870:⊂
1864:⊂
1858:⊂
1852:⊂
1846:×
1811:×
1805:⊂
1799:⊂
1793:×
1775:Over the
1723:≃
1711:Ω
1684:×
1676:≃
1664:Ω
1619:≃
1607:Ω
1580:×
1572:≃
1557:Ω
1501:≃
1489:Ω
1454:×
1446:≃
1431:Ω
1402:functor,
1386:Ω
1378:. Here,
1350:Ω
1293:…
1245:π
1219:π
1190:π
1164:π
1126:…
1081:π
1055:π
1026:π
1000:π
971:π
945:π
881:π
842:π
782:∞
767:⋃
757:⊂
754:⋯
751:⊂
736:⊂
613:≃
600:∞
582:π
571:≃
558:∞
540:π
529:≃
516:∞
498:π
487:≃
474:∞
456:π
443:≃
430:∞
412:π
401:≃
388:∞
370:π
350:≃
337:∞
319:π
299:≃
286:∞
268:π
231:∞
207:π
203:≃
194:∞
176:π
137:∞
89:KO-theory
3736:16590113
3669:16590113
3490:, which
3401:complex
2801:Quotient
916:(1971).
827:groups.
687:unstable
62:K-theory
3796:1409620
3718:0110104
3710:1970106
3651:0102802
3621:Bibcode
3600:0087035
3486:) used
1540:is the
1414:to the
1398:is the
1329:is the
670:with a
648:spheres
156:of the
52: (
3810:
3794:
3784:
3734:
3727:528555
3724:
3716:
3708:
3700:
3667:
3660:528555
3657:
3649:
3641:
3598:
3590:
3478:Proofs
2980:(real
1931:– see
817:stable
703:stable
691:stable
672:circle
36:, the
3706:JSTOR
3643:89403
3639:JSTOR
3502:Notes
2892:Real
1146:Note:
681:with
91:and (
3808:ISBN
3782:ISBN
3732:PMID
3698:ISSN
3665:PMID
3588:ISSN
3046:DIII
2158:and
1410:and
85:real
58:1959
54:1957
3757:doi
3722:PMC
3690:doi
3655:PMC
3629:doi
3578:doi
3216:CII
3116:AII
2889:BDI
2648:BSp
2425:BSp
2126:p,q
1687:BSp
1673:BSp
1406:to
1374:of
1337:(a
646:of
44:of
32:In
3840::
3825:.
3792:MR
3790:,
3751:,
3730:,
3720:,
3714:MR
3712:,
3704:,
3696:,
3686:70
3663:,
3653:,
3647:MR
3645:,
3637:,
3627:,
3617:43
3615:,
3596:MR
3594:,
3586:,
3574:84
3572:,
3523:SU
3466:AI
3398:CI
3331:)
2984:)
2682:Sp
2676:Sp
2662:Sp
2627:Sp
2574:Sp
2563:Sp
2510:Sp
2500:Sp
2464:Sp
2453:Sp
2447:Sp
2438:Sp
2333:KO
2169::
2162:.
1885:Sp
1879:Sp
1873:Sp
1867:Sp
1779::
1772:.
1749:KO
1726:Sp
1720:Sp
1641:KO
1538:BO
1532:,
1478:BU
1420:BU
1376:BU
1327:BU
1321:,
1263:Sp
1231:Sp
1067:Sp
1044:Sp
697:,
95:)
87:)
80:.
56:,
3829:.
3816:.
3759::
3753:4
3692::
3631::
3623::
3580::
3555:.
3527:U
3452:O
3448:/
3444:U
3421:7
3384:U
3380:/
3375:p
3372:S
3348:6
3307:p
3304:S
3299:/
3295:)
3291:p
3288:S
3280:p
3277:S
3273:(
3270:=
3266:p
3263:S
3239:5
3202:)
3198:p
3195:S
3187:p
3184:S
3180:(
3176:/
3171:p
3168:S
3160:Z
3136:4
3101:p
3098:S
3093:/
3089:U
3066:3
3032:U
3028:/
3024:O
3001:2
2961:O
2957:/
2953:)
2950:O
2944:O
2941:(
2938:=
2935:O
2912:1
2875:)
2872:O
2866:O
2863:(
2859:/
2855:O
2848:Z
2824:0
2762:)
2759:O
2753:O
2750:(
2746:/
2742:O
2735:Z
2731:=
2728:O
2725:B
2718:Z
2707:)
2704:O
2700:/
2696:U
2693:(
2685:)
2673:(
2669:/
2655:Z
2651:=
2641:Z
2630:)
2623:/
2619:U
2616:(
2606:O
2602:/
2598:U
2588:)
2585:U
2581:/
2571:(
2559:/
2555:U
2545:)
2542:U
2538:/
2534:O
2531:(
2521:U
2517:/
2492:U
2488:/
2484:O
2474:O
2460:/
2456:)
2444:(
2441:=
2428:)
2418:Z
2414:(
2406:O
2402:/
2398:)
2395:O
2389:O
2386:(
2383:=
2380:O
2370:)
2367:O
2364:B
2357:Z
2353:(
2312:U
2308:/
2304:)
2301:U
2295:U
2292:(
2289:=
2286:U
2276:)
2273:U
2270:B
2263:Z
2259:(
2249:)
2246:U
2240:U
2237:(
2233:/
2229:U
2222:Z
2218:=
2215:U
2212:B
2205:Z
2194:U
2174:K
2137:R
2128:(
2102:,
2098:R
2090:R
2082:R
2074:C
2066:H
2058:H
2050:H
2042:H
2034:C
2026:R
2018:R
2010:R
1983:.
1979:C
1971:C
1963:C
1955:C
1947:C
1912:.
1909:O
1903:O
1897:O
1891:U
1861:U
1855:O
1849:O
1843:O
1817:.
1814:U
1808:U
1802:U
1796:U
1790:U
1770:Z
1753:K
1745:K
1729:.
1715:8
1690:;
1680:Z
1668:8
1625:,
1622:O
1616:O
1611:8
1586:O
1583:B
1576:Z
1569:O
1566:B
1561:8
1534:O
1523:K
1507:.
1504:U
1498:U
1493:2
1460:U
1457:B
1450:Z
1443:U
1440:B
1435:2
1362:U
1359:B
1354:2
1323:U
1290:,
1287:1
1284:,
1281:0
1278:=
1275:k
1269:,
1266:)
1260:(
1255:8
1252:+
1249:k
1241:=
1234:)
1228:(
1223:k
1211:)
1208:O
1205:(
1200:8
1197:+
1194:k
1186:=
1179:)
1176:O
1173:(
1168:k
1123:,
1120:1
1117:,
1114:0
1111:=
1108:k
1102:)
1099:O
1096:(
1091:4
1088:+
1085:k
1077:=
1070:)
1064:(
1059:k
1047:)
1041:(
1036:4
1033:+
1030:k
1022:=
1015:)
1012:O
1009:(
1004:k
992:)
989:U
986:(
981:2
978:+
975:k
967:=
960:)
957:U
954:(
949:k
890:S
885:n
868:J
851:S
846:n
813:O
796:)
793:k
790:(
787:U
777:1
774:=
771:k
763:=
760:U
748:)
745:2
742:(
739:U
733:)
730:1
727:(
724:U
707:U
699:O
695:U
666:(
617:Z
606:)
603:)
597:(
594:O
591:(
586:7
574:0
564:)
561:)
555:(
552:O
549:(
544:6
532:0
522:)
519:)
513:(
510:O
507:(
502:5
490:0
480:)
477:)
471:(
468:O
465:(
460:4
447:Z
436:)
433:)
427:(
424:O
421:(
416:3
404:0
394:)
391:)
385:(
382:O
379:(
374:2
360:2
355:Z
343:)
340:)
334:(
331:O
328:(
323:1
309:2
304:Z
292:)
289:)
283:(
280:O
277:(
272:0
237:)
234:)
228:(
225:O
222:(
217:8
214:+
211:n
200:)
197:)
191:(
188:O
185:(
180:n
140:)
134:(
131:O
20:)
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