5798:
2621:
3304:
5605:
2315:
6195:
theories where additional interactions imply that other fields may cancel the third
StiefelâWhitney class. The mathematical description of spinors in supergravity and string theory is a particularly subtle open problem, which was recently addressed in references. It turns out that the standard notion
3168:
5793:{\displaystyle \dots \longrightarrow {\textrm {H}}^{2}(M;\mathbf {Z} ){\stackrel {2}{\longrightarrow }}{\textrm {H}}^{2}(M;\mathbf {Z} )\longrightarrow {\textrm {H}}^{2}(M;\mathbf {Z} _{2}){\stackrel {\beta }{\longrightarrow }}{\textrm {H}}^{3}(M;\mathbf {Z} )\longrightarrow \dots ,}
5010:
4414:
5272:
3130:
2616:{\displaystyle {\begin{aligned}E_{2}^{0,1}&=H^{0}(M,H^{1}(\operatorname {SO} (n),\mathbb {Z} /2))=H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)\\E_{2}^{2,0}&=H^{2}(M,H^{0}(\operatorname {SO} (n),\mathbb {Z} /2))=H^{2}(M,\mathbb {Z} /2)\end{aligned}}}
5144:
2304:
2832:
3960:
4208:
727:
3299:{\displaystyle {\begin{matrix}\operatorname {Spin} (n)&\to &{\tilde {P}}_{E}&\to &M\\\downarrow &&\downarrow &&\downarrow \\\operatorname {SO} (n)&\to &P_{E}&\to &M\end{matrix}}}
3691:
500:
6196:
of spin structure is too restrictive for applications to supergravity and string theory, and that the correct notion of spinorial structure for the mathematical formulation of these theories is a "Lipschitz structure".
4911:
3605:
4539:
643:
591:
2977:
3460:
1369:
444:
838:
6735:
2158:
2320:
124:
have a spin structure. This is not always possible since there is potentially a topological obstruction to the existence of spin structures. Spin structures will exist if and only if the second
4266:
539:
1111:
5904:). Identify this class with the first element in the above exact sequence. The next arrow doubles this Chern class, and so legitimate bundles will correspond to even elements in the second
4258:
871:
777:
5159:
2988:
2686:
1234:
909:
4022:
3523:
2891:
1714:
1061:
4469:
4074:
3743:
2036:
1163:
3555:
2068:
6028:
4887:â a Dirac operator is a square root of a second order operator, and exists due to the spin structure being a "square root". This was a motivating example for the index theorem.
3828:
1007:
961:
5054:
3397:
1891:
4682:
5916:, while odd elements will correspond to bundles that fail the triple overlap condition. The obstruction then is classified by the failure of an element in the second H(
5482:
When a manifold carries a spin structure at all, the set of spin structures forms an affine space. Moreover, the set of spin structures has a free transitive action of
2173:
2101:
1196:
6077:
cohomology but also of integral cohomology in one higher degree. In fact this is the case for all even
StiefelâWhitney classes. It is traditional to use an uppercase
5856:
lift of each transition function, which is a choice of sign. The lift does not exist when the product of these three signs on a triple overlap is â1, which yields the
1292:
1265:
1950:
4765:
2691:
4100:
1984:
5393:
that extends over the 3-skeleton. Similarly to the case of spin structures, one takes a
Whitney sum with a trivial line bundle if the manifold is odd-dimensional.
3770:
3334:
3160:
1629:
1327:
375:
3856:
4654:
3848:
3354:
1911:
667:
395:
5474:
of the square of the U(1) part of any obtained spin bundle. By a theorem of Hopf and
Hirzebruch, closed orientable 4-manifolds always admit a spin structure.
4108:
677:
6152:. Therefore, the choice of spin structure is part of the data needed to define the wavefunction, and one often needs to sum over these choices in the
6081:
for the resulting classes in odd degree, which are called the integral
StiefelâWhitney classes, and are labeled by their degree (which is always odd).
3610:
449:
4549:
When spin structures exist, the inequivalent spin structures on a manifold have a one-to-one correspondence (not canonical) with the elements of H(
5535:
This failure occurs at precisely the same intersections as an identical failure in the triple products of transition functions of the obstructed
6424:
5005:{\displaystyle 1\to \mathbb {Z} _{2}\to \operatorname {Spin} ^{\mathbf {C} }(n)\to \operatorname {SO} (n)\times \operatorname {U} (1)\to 1.}
3560:
4474:
6153:
602:
5528:. In particular, the product of transition functions on a three-way intersection is not always equal to one, as is required for a
548:
2899:
1840:
that extends over the 2-skeleton. If the dimension is lower than 3, one first takes a
Whitney sum with a trivial line bundle.
17:
3402:
1383:
Haefliger found necessary and sufficient conditions for the existence of a spin structure on an oriented
Riemannian manifold (
6838:
6819:
6796:
6709:
6580:
6484:
1332:
407:
5524:. When the spin structure is nonzero this square root bundle has a non-integral Chern class, which means that it fails the
782:
4409:{\displaystyle 0\to H^{1}(M,\mathbb {Z} /2)\to H^{1}(P_{E},\mathbb {Z} /2)\to H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)}
2112:
505:
322:(1956) found the topological obstruction to the existence of a spin structure on an orientable Riemannian manifold and
5267:{\displaystyle {\mathrm {Spin} }^{\mathbb {C} }(n)={\mathrm {Spin} }(n)\times _{\mathbb {Z} _{2}}{\mathrm {U} }(1)\,,}
5149:
This will always have the element (â1,â1) in the kernel. Taking the quotient modulo this element gives the group Spin(
3125:{\displaystyle H^{1}(P_{E},\mathbb {Z} /2)\to H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)\to H^{2}(M,\mathbb {Z} /2)}
1069:
6892:
6869:
6617:
4216:
843:
742:
6933:
2629:
1201:
876:
6381:
5809:
is induced by multiplication by 2, the third is induced by restriction modulo 2 and the fourth is the associated
5286:
4876:
3968:
3469:
2837:
1660:
1020:
6948:
6928:
6811:
6605:
6568:
5970:
is in the image of the preceding arrow only if it is in the kernel of the next arrow, which we recall is the
4561:
4422:
4027:
3696:
1989:
1116:
3528:
2041:
105:
6938:
6125:
5984:
5139:{\displaystyle \kappa \times i\colon {\mathrm {Spin} }(n)\times {\mathrm {U} }(1)\to {\mathrm {U} }(N).}
6788:
6472:
5889:
bundle nor the U(1) bundle satisfies the triple overlap condition and so neither is actually a bundle.
3775:
966:
920:
5956:
is not in the image of the arrow, then there does not exist any U(1) bundle with obstruction equal to
6417:
5428:
3359:
2038:. These results can be easily proven using a spectral sequence argument for the associated principal
1914:
1857:
1403:
125:
5555:
and so the spin bundle satisfies the triple overlap condition and is therefore a legitimate bundle.
4662:
6210:
6137:
5525:
5382:
4807:
4793:
4779:
4076:, showing this latter cohomology group classifies the various spin structures on the vector bundle
2299:{\displaystyle 0\to E_{3}^{0,1}\to E_{2}^{0,1}\xrightarrow {d_{2}} E_{2}^{2,0}\to E_{3}^{2,0}\to 0}
1829:
1558:
6733:
Friedrich, Thomas; Trautman, Andrzej (2000). "Spin spaces, Lipschitz groups, and spinor bundles".
6609:
6572:
4857:
of a spin manifold is an integer, and is an even integer if in addition the dimension is 4 mod 8.
6460:
6101:
5466:(in other words, the third integral StiefelâWhitney class vanishes). In this case one says that
4836:
2164:
1481:
2827:{\displaystyle E_{\infty }^{0,1}=H^{1}(P_{E},\mathbb {Z} /2)/F^{1}(H^{1}(P_{E},\mathbb {Z} /2))}
2073:
1168:
6682:
Lazaroiu, C.; Shahbazi, C.S. (2019). "On the spin geometry of supergravity and string theory".
6506:
6070:
This argument also demonstrates that second
StiefelâWhitney class defines elements not only of
5971:
5810:
1832:, a spin structure can equivalently be thought of as a homotopy-class of trivialization of the
1520:. There are topological obstructions to being able to do it, and consequently, a given bundle
1270:
1243:
6881:"4.5 Notes Spin structures, the structure group definition; Equivalence of the definitions of"
6880:
1919:
6205:
4744:
264:
77:
28:
6597:
6560:
6464:
6281:
4102:. This can be done by looking at the long exact sequence of homotopy groups of the fibration
4079:
1963:
6908:
6525:
6476:
6354:
6180:
5564:
4868:
4704:
3955:{\displaystyle H^{1}(P_{E},\mathbb {Z} /2)\to H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)}
3748:
3312:
3138:
1957:
1614:
1473:
1300:
348:
69:
65:
8:
6636:
Lazaroiu, C.; Shahbazi, C.S. (2019). "Real pinor bundles and real
Lipschitz structures".
6598:
6561:
5594:
4899:
4821:
4694:
1513:
1505:
1010:
343:
314:
A precise definition of spin structure on manifold was possible only after the notion of
39:
6529:
5924:) to be in the image of the arrow, which, by exactness, is classified by its image in H(
3309:
where the two left vertical maps are the double covering maps. Now, double coverings of
6849:
6762:
6744:
6715:
6687:
6663:
6645:
6541:
6515:
6398:
4864:
is a rational invariant, defined for any manifold, but it is not in general an integer.
4639:
4629:
4617:
4203:{\displaystyle \pi _{1}(\operatorname {SO} (n))\to \pi _{1}(P_{E})\to \pi _{1}(M)\to 1}
3833:
3339:
1896:
652:
380:
81:
6237:
5857:
5335:
respectively. This makes the Spin group both a bundle over the circle with fibre Spin(
1372:
319:
72:
where they are an essential ingredient in the definition of any theory with uncharged
6943:
6888:
6865:
6834:
6815:
6792:
6766:
6719:
6705:
6667:
6613:
6576:
6480:
6379:
Borel, A.; Hirzebruch, F. (1958). "Characteristic classes and homogeneous spaces I".
5563:
The above intuitive geometric picture may be made concrete as follows. Consider the
2167:
can be applied. From general theory of spectral sequences, there is an exact sequence
1807:
1540:
722:{\displaystyle \phi :P_{\operatorname {Spin} }\rightarrow P_{\operatorname {SO} }(E)}
6857:
6754:
6697:
6655:
6545:
6533:
6390:
6333:
6297:
6121:
5529:
5386:
4625:
3463:
1536:
268:
6856:. Lecture Notes in Mathematics. Vol. 676. Springer-Verlag. pp. 217â246.
6701:
6465:
6441:
6094:
4700:
4579:). More precisely, the space of the isomorphism classes of spin structures is an
1431:. Hence, a spin structure exists if and only if the second StiefelâWhitney class
3686:{\displaystyle 1\in H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)=\mathbb {Z} /2}
495:{\displaystyle \rho :\operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)}
6659:
6187:
bundles, and in particular no charged spinors can exist on a space that is not
6161:
5590:
4903:
4884:
4601:
a spin structure corresponds to a binary choice of whether a section of the SO(
1833:
1776:
236:
6758:
5845:. It reflects the fact that one may always locally lift an SO(n) bundle to a
6922:
6912:
6173:
6169:
5536:
5521:
1852:
89:
54:
1524:
may not admit any spinor bundle. In case it does, one says that the bundle
6192:
6129:
5319:
which is generated by the pair of covering transformations for the bundles
5046:
4872:
4580:
3965:
is the set of double coverings giving spin structures. Now, this subset of
1953:
1764:
is the mapping of groups presenting the spin group as a double-cover of SO(
1544:
1493:
734:
315:
326:(1968) extended this result to the non-orientable pseudo-Riemannian case.
108:, mathematicians ask whether or not a given oriented Riemannian manifold (
6537:
6317:
6259:
5893:
5471:
5385:, a spin structure can be equivalently thought of as a homotopy class of
4657:
1470:
1391:). The obstruction to having a spin structure is a certain element of H(
730:
398:
323:
6338:
6321:
6861:
6402:
6302:
6285:
5390:
5378:
4825:
4613:
1849:
1837:
1825:
1603:
340:
36:
5974:ÎČ. That is, the condition for the cancellation of the obstruction is
5539:. Therefore, the triple products of transition functions of the full
4880:
4861:
4854:
6749:
6520:
5427:
A spin structure exists when the bundle is orientable and the second
2104:
164:) = 0, then the set of the isomorphism classes of spin structures on
6394:
2232:
6692:
6650:
6165:
6091:
4633:
101:
85:
6240:(1956). "Sur l'extension du groupe structural d'un espace fibré".
3600:{\displaystyle \operatorname {Spin} (n)\to \operatorname {SO} (n)}
6133:
5520:
This has the following geometric interpretation, which is due to
4898:
A spin structure is analogous to a spin structure on an oriented
4534:{\displaystyle 1\in H^{1}(\operatorname {SO} (n),\mathbb {Z} /2)}
3466:
and change of coefficients, this is exactly the cohomology group
73:
6215:
638:{\displaystyle \pi _{P}:P_{\operatorname {Spin} }\rightarrow M}
117:
58:
5396:
Yet another definition is that a spin structure on a manifold
5045:, i.e., the scalar multiples of the identity. Thus there is a
5034:) consists of the diagonal elements coming from the inclusion
4656:
the second
Stiefel-Whitney class can be computed as the first
120:. One method for dealing with this problem is to require that
5806:
5586:
4541:
is in bijection with the kernel, we have the desired result.
1986:
is spin, the number of spin structures are in bijection with
6065:
5543:
bundle, which are the products of the triple product of the
586:{\displaystyle \pi :P_{\operatorname {SO} }(E)\rightarrow M}
6854:
Differential Geometrical Methods in Mathematical Physics II
5963:
and so the obstruction cannot be cancelled. By exactness,
5593:
by 2 and the third is reduction modulo 2. This induces a
4902:, but uses the Spin group, which is defined instead by the
183:
is assumed to be oriented, the first StiefelâWhitney class
6286:"Champs spinoriels et propagateurs en rélativité générale"
4605:) bundle switches sheets when one encircles the loop. If
2972:{\displaystyle H^{1}(P_{E},\mathbb {Z} /2)\to E_{3}^{0,1}}
329:
307:
is called the spinor bundle for a given spin structure on
4628:
this corresponds to a choice of periodic or antiperiodic
4612:
vanishes then these choices may be extended over the two-
4636:
going around each loop. Note that on a complex manifold
3455:{\displaystyle {\text{Hom}}(\pi _{1}(E),\mathbb {Z} /2)}
3399:, which is in bijection with the set of group morphisms
293:
and the spin representation of its structure group Spin(
1557:), which is a principal bundle under the action of the
1364:{\displaystyle P_{\operatorname {SO} }(E)\rightarrow M}
439:{\displaystyle P_{\operatorname {SO} }(E)\rightarrow M}
3173:
3135:
Now, a spin structure is exactly a double covering of
833:{\displaystyle \quad \phi (pq)=\phi (p)\rho (q)\quad }
6048:
is the Bockstein of the second StiefelâWhitney class
5987:
5608:
5162:
5057:
5030:
is a complex spinor representation. The center of U(
4914:
4747:
4665:
4642:
4477:
4425:
4269:
4219:
4111:
4082:
4030:
3971:
3859:
3836:
3778:
3751:
3699:
3613:
3563:
3531:
3472:
3405:
3362:
3342:
3315:
3171:
3141:
2991:
2902:
2840:
2694:
2632:
2318:
2176:
2115:
2076:
2044:
1992:
1966:
1922:
1899:
1860:
1663:
1617:
1335:
1303:
1273:
1246:
1204:
1171:
1119:
1072:
1023:
969:
923:
879:
846:
785:
745:
680:
655:
605:
551:
508:
452:
410:
383:
351:
235:
are defined to be the StiefelâWhitney classes of its
6783:
Lawson, H. Blaine; Michelsohn, Marie-Louise (1989).
6631:
6629:
6442:"Spin manifold and the second Stiefel-Whitney class"
5874:
bundle with a U(1) bundle with the same obstruction
2153:{\displaystyle \operatorname {SO} (n)\to P_{E}\to M}
1460:
76:. They are also of purely mathematical interest in
6887:. American Mathematical Society. pp. 174â189.
5497:. Thus, spin-structures correspond to elements of
1610:), by which we mean that there exists a bundle map
6232:
6230:
6115:
6022:
5792:
5266:
5138:
5004:
4759:
4676:
4648:
4533:
4463:
4408:
4252:
4202:
4094:
4068:
4016:
3954:
3842:
3822:
3764:
3737:
3685:
3599:
3549:
3517:
3454:
3391:
3348:
3328:
3298:
3154:
3124:
2971:
2885:
2826:
2680:
2615:
2298:
2152:
2095:
2062:
2030:
1978:
1944:
1905:
1885:
1708:
1623:
1535:This may be made rigorous through the language of
1402:) . For a spin structure the class is the second
1363:
1321:
1286:
1259:
1228:
1190:
1157:
1105:
1055:
1001:
955:
903:
865:
832:
771:
721:
661:
637:
585:
533:
494:
438:
389:
369:
6732:
6626:
4620:) they may automatically be extended over all of
1843:
1512:is a prescription for consistently associating a
534:{\displaystyle (P_{\operatorname {Spin} },\phi )}
6920:
6782:
6681:
6635:
6458:
6355:"Spin structures on pseudo-Riemannian manifolds"
1786:, if a spin structure exists then one says that
1106:{\displaystyle \phi _{2}\circ f=\phi _{1}\quad }
6378:
6227:
5938:To cancel the corresponding obstruction in the
1013:are called "equivalent" if there exists a Spin(
6848:Greub, Werner; Petry, Herbert-Rainer (2006) .
6596:Gompf, Robert E.; Stipsicz, Andras I. (1999).
6164:, but for the fermions on the worldvolumes of
5892:A legitimate U(1) bundle is classified by its
4253:{\displaystyle {\text{Hom}}(-,\mathbb {Z} /2)}
866:{\displaystyle p\in P_{\operatorname {Spin} }}
772:{\displaystyle \pi \circ \phi =\pi _{P}\quad }
6915:and spin structures for mathematics students.
6595:
6352:
5870:To cancel this obstruction, one tensors this
2681:{\displaystyle E_{\infty }^{0,1}=E_{3}^{0,1}}
1229:{\displaystyle q\in \operatorname {Spin} (n)}
904:{\displaystyle q\in \operatorname {Spin} (n)}
6911:by Sven-S. Porst is a short introduction to
6280:
5881:. Notice that this is an abuse of the word
4544:
3830:. If it vanishes, then the inverse image of
1329:as a spin structure on the principal bundle
6686:. Trends in Mathematics. pp. 229â235.
6183:charged spinors are sections of associated
6037:where we have used the fact that the third
4703:admits 2 inequivalent spin structures; see
4017:{\displaystyle H^{1}(P_{E},\mathbb {Z} /2)}
3772:of the second StiefelâWhitney class, hence
3518:{\displaystyle H^{1}(P_{E},\mathbb {Z} /2)}
2886:{\displaystyle H^{1}(P_{E},\mathbb {Z} /2)}
209:vanishes too. (The StiefelâWhitney classes
168:is acted upon freely and transitively by H(
6847:
5470:is spin. Intuitively, the lift gives the
4597:Intuitively, for each nontrivial cycle on
4260:, giving the sequence of cohomology groups
1709:{\displaystyle \phi (pg)=\phi (p)\rho (g)}
64:Spin structures have wide applications to
6805:
6748:
6691:
6649:
6558:
6519:
6502:Spinâstructures and homotopy equivalences
6337:
6301:
6258:
6236:
6066:Integral lifts of StiefelâWhitney classes
5260:
5232:
5182:
4923:
4516:
4446:
4391:
4341:
4296:
4235:
4051:
3999:
3937:
3887:
3720:
3671:
3652:
3500:
3437:
3107:
3069:
3019:
2930:
2868:
2806:
2746:
2594:
2553:
2455:
2402:
2013:
6499:
6362:Revista de la UniĂłn MatemĂĄtica Argentina
6262:(1963). "Spin structures on manifolds".
4775:, but for different reasons; see below.)
4471:is the kernel, and the inverse image of
1056:{\displaystyle f:P_{1}\rightarrow P_{2}}
6878:
6828:
6316:
5547:and U(1) component bundles, are either
4464:{\displaystyle H^{1}(M,\mathbb {Z} /2)}
4069:{\displaystyle H^{1}(M,\mathbb {Z} /2)}
3738:{\displaystyle H^{2}(M,\mathbb {Z} /2)}
2031:{\displaystyle H^{1}(M,\mathbb {Z} /2)}
1158:{\displaystyle \quad f(pq)=f(p)q\quad }
330:Spin structures on Riemannian manifolds
14:
6921:
6808:Dirac Operators in Riemannian Geometry
6736:Annals of Global Analysis and Geometry
6563:Dirac Operators in Riemannian Geometry
6353:Alagia, H. R.; SĂĄnchez, C. U. (1985),
6055:(this can be taken as a definition of
5819:The obstruction to the existence of a
5304:) is the quotient group obtained from
3550:{\displaystyle \operatorname {SO} (n)}
2063:{\displaystyle \operatorname {SO} (n)}
6418:"Elliptic complexes and index theory"
5532:. Instead it is sometimes −1.
4792:More generally, all even-dimensional
1821:quotient of a principal spin bundle.
1294:are two equivalent double coverings.
6850:"On the lifting of structure groups"
5515:
1496:. This means that at each point of
1297:The definition of spin structure on
446:with respect to the double covering
6322:"AlgÚbres de Clifford et K-théorie"
6023:{\displaystyle W_{3}=\beta w_{2}=0}
4893:
24:
6776:
6684:Geometric Methods in Physics XXXVI
5849:bundle, but one needs to choose a
5408:together with a spin structure on
5246:
5211:
5208:
5205:
5202:
5175:
5172:
5169:
5166:
5119:
5100:
5081:
5078:
5075:
5072:
4981:
3162:fitting into a commutative diagram
2700:
2638:
267:associated with the corresponding
25:
6960:
6902:
6430:from the original on 20 Aug 2018.
5477:
5281:. In other words, the group Spin(
3823:{\displaystyle w_{2}(1)=w_{2}(E)}
1461:Spin structures on vector bundles
1002:{\displaystyle (P_{2},\phi _{2})}
956:{\displaystyle (P_{1},\phi _{1})}
57:, giving rise to the notion of a
6415:
6097:of dimension 4 or less are spin.
5774:
5721:
5686:
5639:
5422:
4942:
3525:. Applying the same argument to
404:of the orthonormal frame bundle
53:allows one to define associated
6726:
6675:
6589:
6552:
6493:
6382:American Journal of Mathematics
6191:. An exception arises in some
6116:Application to particle physics
5942:bundle, this image needs to be
5512:although not in a natural way.
5153:). This is the twisted product
5015:To motivate this, suppose that
3392:{\displaystyle \pi _{1}(P_{E})}
1886:{\displaystyle \pi _{E}:E\to M}
1154:
1120:
1102:
829:
786:
768:
377:with an oriented vector bundle
88:. They form the foundation for
6833:. Springer. pp. 212â214.
6600:4-Manifolds and Kirby Calculus
6452:
6434:
6409:
6372:
6346:
6310:
6274:
6252:
5781:
5778:
5764:
5738:
5731:
5710:
5693:
5690:
5676:
5650:
5643:
5629:
5612:
5597:on cohomology, which contains
5558:
5257:
5251:
5222:
5216:
5194:
5188:
5130:
5124:
5114:
5111:
5105:
5092:
5086:
4996:
4993:
4987:
4975:
4969:
4960:
4957:
4951:
4933:
4918:
4867:This was originally proven by
4677:{\displaystyle {\text{mod }}2}
4528:
4509:
4503:
4494:
4458:
4436:
4403:
4384:
4378:
4369:
4356:
4353:
4324:
4311:
4308:
4286:
4273:
4247:
4225:
4194:
4191:
4185:
4172:
4169:
4156:
4143:
4140:
4137:
4131:
4122:
4086:
4063:
4041:
4011:
3982:
3949:
3930:
3924:
3915:
3902:
3899:
3870:
3817:
3811:
3795:
3789:
3732:
3710:
3664:
3645:
3639:
3630:
3594:
3588:
3579:
3576:
3570:
3544:
3538:
3512:
3483:
3449:
3430:
3424:
3411:
3386:
3373:
3284:
3267:
3262:
3256:
3243:
3237:
3231:
3219:
3205:
3193:
3188:
3182:
3119:
3097:
3084:
3081:
3062:
3056:
3047:
3034:
3031:
3002:
2945:
2942:
2913:
2880:
2851:
2821:
2818:
2789:
2776:
2758:
2729:
2606:
2584:
2568:
2565:
2546:
2540:
2531:
2512:
2467:
2448:
2442:
2433:
2417:
2414:
2395:
2389:
2380:
2361:
2290:
2266:
2204:
2180:
2144:
2131:
2128:
2122:
2087:
2057:
2051:
2025:
2003:
1970:
1952:vanishes. This is a result of
1939:
1933:
1877:
1844:Obstruction and classification
1703:
1697:
1691:
1685:
1676:
1667:
1539:. The collection of oriented
1378:
1355:
1352:
1346:
1316:
1304:
1223:
1217:
1148:
1142:
1133:
1124:
1040:
996:
970:
950:
924:
898:
892:
826:
820:
814:
808:
799:
790:
716:
710:
697:
629:
577:
574:
568:
541:is a spin structure on the SO(
528:
509:
489:
483:
474:
471:
465:
430:
427:
421:
364:
352:
13:
1:
6885:The wild world of 4-manifolds
6812:American Mathematical Society
6606:American Mathematical Society
6569:American Mathematical Society
6221:
6156:. In many physical theories
6084:
4847:
4562:universal coefficient theorem
1771:In the special case in which
334:
6909:Something on Spin Structures
6702:10.1007/978-3-030-01156-7_25
6638:Asian Journal of Mathematics
3336:are in bijection with index
7:
6879:Scorpan, Alexandru (2005).
6264:L'Enseignement Mathématique
6199:
5896:, which is an element of H(
5435:is in the image of the map
5312:with respect to the normal
4877:AtiyahâSinger index theorem
4875:, and can be proven by the
4687:
3557:, the non-trivial covering
1960:. Furthermore, in the case
1893:a spin structure exists on
297:) on the space of spinors Î
95:
10:
6965:
6806:Friedrich, Thomas (2000).
6789:Princeton University Press
6660:10.4310/AJM.2019.v23.n5.a3
6559:Friedrich, Thomas (2000).
6473:Princeton University Press
2096:{\displaystyle P_{E}\to M}
1913:if and only if the second
1602:) under the action of the
1543:of a vector bundle form a
1191:{\displaystyle p\in P_{1}}
153:vanishes. Furthermore, if
61:in differential geometry.
6326:Ann. Sci. Ăc. Norm. SupĂ©r
5400:is a complex line bundle
5300:Viewed another way, Spin(
4808:complex projective spaces
4794:complex projective spaces
4545:Remarks on classification
1810:of the tangent fibers of
1565:). A spin structure for
1287:{\displaystyle \phi _{2}}
1260:{\displaystyle \phi _{1}}
502:. In other words, a pair
6461:Michelsohn, Marie-Louise
6211:Orthonormal frame bundle
6138:associated vector bundle
6102:almost complex manifolds
5935:) under the next arrow.
5526:triple overlap condition
5339:), and a bundle over SO(
4780:complex projective plane
2982:giving an exact sequence
1945:{\displaystyle w_{2}(E)}
1591:) to a principal bundle
1559:special orthogonal group
6934:Structures on manifolds
6759:10.1023/A:1006713405277
6507:Geometry & Topology
6126:spinâstatistics theorem
5346:The fundamental group Ï
5343:) with fibre a circle.
4760:{\displaystyle n\neq 2}
4024:can be identified with
2834:for some filtration on
2165:Serre spectral sequence
1782:over the base manifold
339:A spin structure on an
245:The bundle of spinors Ï
6242:C. R. Acad. Sci. Paris
6041:StiefelâWhitney class
6024:
5972:Bockstein homomorphism
5811:Bockstein homomorphism
5794:
5377:If the manifold has a
5268:
5140:
5006:
4761:
4678:
4650:
4535:
4465:
4417:
4410:
4254:
4211:
4204:
4096:
4095:{\displaystyle E\to M}
4070:
4018:
3963:
3956:
3844:
3824:
3766:
3739:
3687:
3601:
3551:
3519:
3456:
3393:
3350:
3330:
3307:
3300:
3156:
3133:
3126:
2980:
2973:
2887:
2828:
2682:
2624:
2617:
2307:
2300:
2161:
2154:
2103:. Notice this gives a
2097:
2064:
2032:
1980:
1979:{\displaystyle E\to M}
1946:
1907:
1887:
1824:If the manifold has a
1806:) principal bundle of
1710:
1625:
1508:. A spinor bundle of
1365:
1323:
1288:
1261:
1230:
1192:
1159:
1107:
1057:
1003:
957:
913:
905:
867:
834:
773:
723:
663:
639:
587:
535:
496:
440:
391:
371:
18:Complex spin structure
6829:Karoubi, Max (2008).
6206:Metaplectic structure
6025:
5949:. In particular, if
5823:bundle is an element
5795:
5429:StiefelâWhitney class
5277:where U(1) = SO(2) =
5269:
5141:
5007:
4762:
4679:
4651:
4536:
4466:
4411:
4262:
4255:
4205:
4104:
4097:
4071:
4019:
3957:
3852:
3845:
3825:
3767:
3765:{\displaystyle w_{2}}
3740:
3688:
3602:
3552:
3520:
3457:
3394:
3351:
3331:
3329:{\displaystyle P_{E}}
3301:
3164:
3157:
3155:{\displaystyle P_{E}}
3127:
2984:
2974:
2895:
2888:
2829:
2683:
2618:
2311:
2301:
2169:
2155:
2108:
2098:
2065:
2033:
1981:
1947:
1915:StiefelâWhitney class
1908:
1888:
1711:
1626:
1624:{\displaystyle \phi }
1404:StiefelâWhitney class
1366:
1324:
1322:{\displaystyle (M,g)}
1289:
1262:
1231:
1193:
1160:
1108:
1058:
1009:on the same oriented
1004:
958:
906:
868:
835:
774:
738:
724:
664:
640:
588:
536:
497:
441:
392:
372:
370:{\displaystyle (M,g)}
318:had been introduced;
265:complex vector bundle
126:StiefelâWhitney class
78:differential geometry
29:differential geometry
6949:Mathematical physics
6929:Riemannian manifolds
6538:10.2140/gt.1997.1.41
6181:quantum field theory
6136:is a section of the
5985:
5606:
5565:short exact sequence
5354:)) is isomorphic to
5160:
5055:
4912:
4837:CalabiâYau manifolds
4822:orientable manifolds
4806:All odd-dimensional
4745:
4705:theta characteristic
4663:
4640:
4475:
4423:
4267:
4217:
4109:
4080:
4028:
3969:
3857:
3834:
3776:
3749:
3697:
3611:
3561:
3529:
3470:
3403:
3360:
3340:
3313:
3169:
3139:
2989:
2900:
2893:, hence we get a map
2838:
2692:
2630:
2316:
2174:
2113:
2074:
2042:
1990:
1964:
1958:Friedrich Hirzebruch
1920:
1897:
1858:
1661:
1615:
1474:topological manifold
1333:
1301:
1271:
1244:
1202:
1169:
1117:
1070:
1021:
967:
921:
917:Two spin structures
877:
844:
783:
743:
678:
653:
645:is a principal Spin(
603:
549:
506:
450:
408:
381:
349:
179:) . As the manifold
70:quantum field theory
66:mathematical physics
6530:1997math......5218G
6459:Lawson, H. Blaine;
6416:Pati, Vishwambhar.
6339:10.24033/asens.1163
6290:Bull. Soc. Math. Fr
6111:manifolds are spin.
5595:long exact sequence
5585:, where the second
4900:Riemannian manifold
4879:, by realizing the
4630:boundary conditions
2968:
2715:
2677:
2653:
2494:
2343:
2289:
2265:
2243:
2227:
2203:
1514:spin representation
1506:inner product space
1011:Riemannian manifold
545:)-principal bundle
344:Riemannian manifold
68:, in particular to
40:Riemannian manifold
6939:Algebraic topology
6862:10.1007/BFb0063673
6500:R. Gompf (1997). "
6446:Math.Stachexchange
6303:10.24033/bsmf.1604
6154:partition function
6020:
5790:
5379:cell decomposition
5264:
5136:
5002:
4883:as the index of a
4757:
4674:
4646:
4618:obstruction theory
4564:is isomorphic to H
4531:
4461:
4406:
4250:
4200:
4092:
4066:
4014:
3952:
3840:
3820:
3762:
3735:
3683:
3597:
3547:
3515:
3452:
3389:
3346:
3326:
3296:
3294:
3152:
3122:
2969:
2948:
2883:
2824:
2695:
2678:
2657:
2633:
2613:
2611:
2474:
2323:
2296:
2269:
2245:
2207:
2183:
2150:
2093:
2060:
2028:
1976:
1942:
1903:
1883:
1826:cell decomposition
1706:
1621:
1541:orthonormal frames
1516:to every point of
1361:
1319:
1284:
1257:
1226:
1188:
1155:
1103:
1053:
1017:)-equivariant map
999:
953:
901:
863:
830:
769:
719:
659:
635:
583:
531:
492:
436:
387:
367:
82:algebraic topology
6840:978-3-540-79889-7
6821:978-0-8218-2055-1
6798:978-0-691-08542-5
6711:978-3-030-01155-0
6582:978-0-8218-2055-1
6486:978-0-691-08542-5
6128:implies that the
5885:, as neither the
5805:where the second
5755:
5747:
5701:
5667:
5659:
5620:
5516:Geometric picture
5387:complex structure
5287:central extension
5039: : U(1) â U(
4669:
4649:{\displaystyle X}
4223:
3843:{\displaystyle 1}
3745:is precisely the
3693:, and the map to
3409:
3349:{\displaystyle 2}
3208:
2244:
1906:{\displaystyle E}
1808:orthonormal bases
1537:principal bundles
1484:vector bundle on
662:{\displaystyle M}
390:{\displaystyle E}
16:(Redirected from
6956:
6898:
6875:
6844:
6825:
6802:
6771:
6770:
6752:
6730:
6724:
6723:
6695:
6679:
6673:
6671:
6653:
6633:
6624:
6623:
6603:
6593:
6587:
6586:
6566:
6556:
6550:
6549:
6523:
6497:
6491:
6490:
6470:
6456:
6450:
6449:
6438:
6432:
6431:
6429:
6422:
6413:
6407:
6406:
6376:
6370:
6369:
6359:
6350:
6344:
6343:
6341:
6314:
6308:
6307:
6305:
6282:Lichnerowicz, A.
6278:
6272:
6271:
6256:
6250:
6249:
6234:
6132:of an uncharged
6122:particle physics
6095:smooth manifolds
6029:
6027:
6026:
6021:
6013:
6012:
5997:
5996:
5915:
5844:
5799:
5797:
5796:
5791:
5777:
5763:
5762:
5757:
5756:
5753:
5749:
5748:
5746:
5741:
5736:
5730:
5729:
5724:
5709:
5708:
5703:
5702:
5699:
5689:
5675:
5674:
5669:
5668:
5665:
5661:
5660:
5658:
5653:
5648:
5642:
5628:
5627:
5622:
5621:
5618:
5584:
5554:
5550:
5530:principal bundle
5511:
5496:
5465:
5418:
5334:
5330:
5311:
5273:
5271:
5270:
5265:
5250:
5249:
5243:
5242:
5241:
5240:
5235:
5215:
5214:
5187:
5186:
5185:
5179:
5178:
5145:
5143:
5142:
5137:
5123:
5122:
5104:
5103:
5085:
5084:
5044:
5029:
5011:
5009:
5008:
5003:
4947:
4946:
4945:
4932:
4931:
4926:
4766:
4764:
4763:
4758:
4683:
4681:
4680:
4675:
4670:
4667:
4655:
4653:
4652:
4647:
4626:particle physics
4560:), which by the
4540:
4538:
4537:
4532:
4524:
4519:
4493:
4492:
4470:
4468:
4467:
4462:
4454:
4449:
4435:
4434:
4415:
4413:
4412:
4407:
4399:
4394:
4368:
4367:
4349:
4344:
4336:
4335:
4323:
4322:
4304:
4299:
4285:
4284:
4259:
4257:
4256:
4251:
4243:
4238:
4224:
4221:
4209:
4207:
4206:
4201:
4184:
4183:
4168:
4167:
4155:
4154:
4121:
4120:
4101:
4099:
4098:
4093:
4075:
4073:
4072:
4067:
4059:
4054:
4040:
4039:
4023:
4021:
4020:
4015:
4007:
4002:
3994:
3993:
3981:
3980:
3961:
3959:
3958:
3953:
3945:
3940:
3914:
3913:
3895:
3890:
3882:
3881:
3869:
3868:
3849:
3847:
3846:
3841:
3829:
3827:
3826:
3821:
3810:
3809:
3788:
3787:
3771:
3769:
3768:
3763:
3761:
3760:
3744:
3742:
3741:
3736:
3728:
3723:
3709:
3708:
3692:
3690:
3689:
3684:
3679:
3674:
3660:
3655:
3629:
3628:
3606:
3604:
3603:
3598:
3556:
3554:
3553:
3548:
3524:
3522:
3521:
3516:
3508:
3503:
3495:
3494:
3482:
3481:
3464:Hurewicz theorem
3461:
3459:
3458:
3453:
3445:
3440:
3423:
3422:
3410:
3407:
3398:
3396:
3395:
3390:
3385:
3384:
3372:
3371:
3355:
3353:
3352:
3347:
3335:
3333:
3332:
3327:
3325:
3324:
3305:
3303:
3302:
3297:
3295:
3281:
3280:
3241:
3235:
3216:
3215:
3210:
3209:
3201:
3161:
3159:
3158:
3153:
3151:
3150:
3131:
3129:
3128:
3123:
3115:
3110:
3096:
3095:
3077:
3072:
3046:
3045:
3027:
3022:
3014:
3013:
3001:
3000:
2978:
2976:
2975:
2970:
2967:
2956:
2938:
2933:
2925:
2924:
2912:
2911:
2892:
2890:
2889:
2884:
2876:
2871:
2863:
2862:
2850:
2849:
2833:
2831:
2830:
2825:
2814:
2809:
2801:
2800:
2788:
2787:
2775:
2774:
2765:
2754:
2749:
2741:
2740:
2728:
2727:
2714:
2703:
2687:
2685:
2684:
2679:
2676:
2665:
2652:
2641:
2622:
2620:
2619:
2614:
2612:
2602:
2597:
2583:
2582:
2561:
2556:
2530:
2529:
2511:
2510:
2493:
2482:
2463:
2458:
2432:
2431:
2410:
2405:
2379:
2378:
2360:
2359:
2342:
2331:
2305:
2303:
2302:
2297:
2288:
2277:
2264:
2253:
2242:
2241:
2228:
2226:
2215:
2202:
2191:
2159:
2157:
2156:
2151:
2143:
2142:
2102:
2100:
2099:
2094:
2086:
2085:
2069:
2067:
2066:
2061:
2037:
2035:
2034:
2029:
2021:
2016:
2002:
2001:
1985:
1983:
1982:
1977:
1951:
1949:
1948:
1943:
1932:
1931:
1912:
1910:
1909:
1904:
1892:
1890:
1889:
1884:
1870:
1869:
1763:
1744:
1733:
1715:
1713:
1712:
1707:
1630:
1628:
1627:
1622:
1492:equipped with a
1370:
1368:
1367:
1362:
1345:
1344:
1328:
1326:
1325:
1320:
1293:
1291:
1290:
1285:
1283:
1282:
1266:
1264:
1263:
1258:
1256:
1255:
1235:
1233:
1232:
1227:
1197:
1195:
1194:
1189:
1187:
1186:
1164:
1162:
1161:
1156:
1112:
1110:
1109:
1104:
1101:
1100:
1082:
1081:
1062:
1060:
1059:
1054:
1052:
1051:
1039:
1038:
1008:
1006:
1005:
1000:
995:
994:
982:
981:
962:
960:
959:
954:
949:
948:
936:
935:
910:
908:
907:
902:
872:
870:
869:
864:
862:
861:
839:
837:
836:
831:
778:
776:
775:
770:
767:
766:
728:
726:
725:
720:
709:
708:
696:
695:
668:
666:
665:
660:
644:
642:
641:
636:
628:
627:
615:
614:
592:
590:
589:
584:
567:
566:
540:
538:
537:
532:
521:
520:
501:
499:
498:
493:
445:
443:
442:
437:
420:
419:
396:
394:
393:
388:
376:
374:
373:
368:
269:principal bundle
231:) of a manifold
52:
21:
6964:
6963:
6959:
6958:
6957:
6955:
6954:
6953:
6919:
6918:
6905:
6895:
6872:
6841:
6822:
6799:
6779:
6777:Further reading
6774:
6731:
6727:
6712:
6680:
6676:
6634:
6627:
6620:
6594:
6590:
6583:
6557:
6553:
6498:
6494:
6487:
6457:
6453:
6440:
6439:
6435:
6427:
6420:
6414:
6410:
6395:10.2307/2372795
6377:
6373:
6357:
6351:
6347:
6315:
6311:
6279:
6275:
6257:
6253:
6235:
6228:
6224:
6202:
6118:
6087:
6076:
6068:
6061:
6054:
6047:
6008:
6004:
5992:
5988:
5986:
5983:
5982:
5969:
5962:
5955:
5948:
5934:
5905:
5880:
5866:
5858:Äech cohomology
5855:
5842:
5831:
5829:
5773:
5758:
5752:
5751:
5750:
5742:
5737:
5735:
5734:
5725:
5720:
5719:
5704:
5698:
5697:
5696:
5685:
5670:
5664:
5663:
5662:
5654:
5649:
5647:
5646:
5638:
5623:
5617:
5616:
5615:
5607:
5604:
5603:
5582:
5567:
5561:
5552:
5548:
5518:
5498:
5483:
5480:
5436:
5425:
5409:
5349:
5333:Spin(2) â SO(2)
5332:
5320:
5318:
5305:
5245:
5244:
5236:
5231:
5230:
5229:
5225:
5201:
5200:
5181:
5180:
5165:
5164:
5163:
5161:
5158:
5157:
5118:
5117:
5099:
5098:
5071:
5070:
5056:
5053:
5052:
5035:
5016:
4941:
4940:
4936:
4927:
4922:
4921:
4913:
4910:
4909:
4896:
4894:Spin structures
4860:In general the
4850:
4746:
4743:
4742:
4733:. For example,
4724:
4701:Riemann surface
4690:
4666:
4664:
4661:
4660:
4641:
4638:
4637:
4611:
4593:
4578:
4567:
4559:
4547:
4520:
4515:
4488:
4484:
4476:
4473:
4472:
4450:
4445:
4430:
4426:
4424:
4421:
4420:
4395:
4390:
4363:
4359:
4345:
4340:
4331:
4327:
4318:
4314:
4300:
4295:
4280:
4276:
4268:
4265:
4264:
4239:
4234:
4220:
4218:
4215:
4214:
4179:
4175:
4163:
4159:
4150:
4146:
4116:
4112:
4110:
4107:
4106:
4081:
4078:
4077:
4055:
4050:
4035:
4031:
4029:
4026:
4025:
4003:
3998:
3989:
3985:
3976:
3972:
3970:
3967:
3966:
3941:
3936:
3909:
3905:
3891:
3886:
3877:
3873:
3864:
3860:
3858:
3855:
3854:
3835:
3832:
3831:
3805:
3801:
3783:
3779:
3777:
3774:
3773:
3756:
3752:
3750:
3747:
3746:
3724:
3719:
3704:
3700:
3698:
3695:
3694:
3675:
3670:
3656:
3651:
3624:
3620:
3612:
3609:
3608:
3607:corresponds to
3562:
3559:
3558:
3530:
3527:
3526:
3504:
3499:
3490:
3486:
3477:
3473:
3471:
3468:
3467:
3441:
3436:
3418:
3414:
3406:
3404:
3401:
3400:
3380:
3376:
3367:
3363:
3361:
3358:
3357:
3341:
3338:
3337:
3320:
3316:
3314:
3311:
3310:
3293:
3292:
3287:
3282:
3276:
3272:
3270:
3265:
3247:
3246:
3240:
3234:
3228:
3227:
3222:
3217:
3211:
3200:
3199:
3198:
3196:
3191:
3172:
3170:
3167:
3166:
3146:
3142:
3140:
3137:
3136:
3111:
3106:
3091:
3087:
3073:
3068:
3041:
3037:
3023:
3018:
3009:
3005:
2996:
2992:
2990:
2987:
2986:
2957:
2952:
2934:
2929:
2920:
2916:
2907:
2903:
2901:
2898:
2897:
2872:
2867:
2858:
2854:
2845:
2841:
2839:
2836:
2835:
2810:
2805:
2796:
2792:
2783:
2779:
2770:
2766:
2761:
2750:
2745:
2736:
2732:
2723:
2719:
2704:
2699:
2693:
2690:
2689:
2666:
2661:
2642:
2637:
2631:
2628:
2627:
2610:
2609:
2598:
2593:
2578:
2574:
2557:
2552:
2525:
2521:
2506:
2502:
2495:
2483:
2478:
2471:
2470:
2459:
2454:
2427:
2423:
2406:
2401:
2374:
2370:
2355:
2351:
2344:
2332:
2327:
2319:
2317:
2314:
2313:
2278:
2273:
2254:
2249:
2237:
2233:
2216:
2211:
2192:
2187:
2175:
2172:
2171:
2138:
2134:
2114:
2111:
2110:
2081:
2077:
2075:
2072:
2071:
2043:
2040:
2039:
2017:
2012:
1997:
1993:
1991:
1988:
1987:
1965:
1962:
1961:
1927:
1923:
1921:
1918:
1917:
1898:
1895:
1894:
1865:
1861:
1859:
1856:
1855:
1846:
1820:
1794:. Equivalently
1750:
1735:
1727:
1717:
1662:
1659:
1658:
1649:
1638:
1616:
1613:
1612:
1597:
1586:
1571:
1552:
1500:, the fibre of
1463:
1452:
1437:
1426:
1411:
1401:
1381:
1373:André Haefliger
1340:
1336:
1334:
1331:
1330:
1302:
1299:
1298:
1278:
1274:
1272:
1269:
1268:
1251:
1247:
1245:
1242:
1241:
1203:
1200:
1199:
1182:
1178:
1170:
1167:
1166:
1118:
1115:
1114:
1096:
1092:
1077:
1073:
1071:
1068:
1067:
1047:
1043:
1034:
1030:
1022:
1019:
1018:
990:
986:
977:
973:
968:
965:
964:
944:
940:
931:
927:
922:
919:
918:
878:
875:
874:
857:
853:
845:
842:
841:
784:
781:
780:
762:
758:
744:
741:
740:
704:
700:
691:
687:
679:
676:
675:
654:
651:
650:
623:
619:
610:
606:
604:
601:
600:
562:
558:
550:
547:
546:
516:
512:
507:
504:
503:
451:
448:
447:
415:
411:
409:
406:
405:
382:
379:
378:
350:
347:
346:
337:
332:
320:André Haefliger
302:
276:
250:
230:
214:
204:
189:
178:
159:
148:
133:
98:
42:
23:
22:
15:
12:
11:
5:
6962:
6952:
6951:
6946:
6941:
6936:
6931:
6917:
6916:
6904:
6903:External links
6901:
6900:
6899:
6893:
6876:
6870:
6845:
6839:
6826:
6820:
6803:
6797:
6778:
6775:
6773:
6772:
6743:(3): 221â240.
6725:
6710:
6674:
6644:(5): 749â836.
6625:
6618:
6612:â58, 186â187.
6588:
6581:
6551:
6492:
6485:
6451:
6433:
6408:
6371:
6345:
6332:(2): 161â270.
6309:
6273:
6251:
6225:
6223:
6220:
6219:
6218:
6213:
6208:
6201:
6198:
6162:tangent bundle
6144:lift of an SO(
6117:
6114:
6113:
6112:
6105:
6098:
6086:
6083:
6074:
6067:
6064:
6059:
6052:
6045:
6035:
6034:
6033:
6032:
6031:
6030:
6019:
6016:
6011:
6007:
6003:
6000:
5995:
5991:
5967:
5960:
5953:
5946:
5932:
5878:
5864:
5853:
5840:
5827:
5803:
5802:
5801:
5800:
5789:
5786:
5783:
5780:
5776:
5772:
5769:
5766:
5761:
5745:
5740:
5733:
5728:
5723:
5718:
5715:
5712:
5707:
5695:
5692:
5688:
5684:
5681:
5678:
5673:
5657:
5652:
5645:
5641:
5637:
5634:
5631:
5626:
5614:
5611:
5591:multiplication
5580:
5560:
5557:
5517:
5514:
5479:
5478:Classification
5476:
5431:of the bundle
5424:
5421:
5347:
5316:
5275:
5274:
5263:
5259:
5256:
5253:
5248:
5239:
5234:
5228:
5224:
5221:
5218:
5213:
5210:
5207:
5204:
5199:
5196:
5193:
5190:
5184:
5177:
5174:
5171:
5168:
5147:
5146:
5135:
5132:
5129:
5126:
5121:
5116:
5113:
5110:
5107:
5102:
5097:
5094:
5091:
5088:
5083:
5080:
5077:
5074:
5069:
5066:
5063:
5060:
5013:
5012:
5001:
4998:
4995:
4992:
4989:
4986:
4983:
4980:
4977:
4974:
4971:
4968:
4965:
4962:
4959:
4956:
4953:
4950:
4944:
4939:
4935:
4930:
4925:
4920:
4917:
4904:exact sequence
4895:
4892:
4891:
4890:
4889:
4888:
4885:Dirac operator
4865:
4849:
4846:
4845:
4844:
4833:
4818:
4804:
4790:
4776:
4756:
4753:
4750:
4722:
4708:
4689:
4686:
4673:
4645:
4609:
4591:
4576:
4565:
4557:
4546:
4543:
4530:
4527:
4523:
4518:
4514:
4511:
4508:
4505:
4502:
4499:
4496:
4491:
4487:
4483:
4480:
4460:
4457:
4453:
4448:
4444:
4441:
4438:
4433:
4429:
4405:
4402:
4398:
4393:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4366:
4362:
4358:
4355:
4352:
4348:
4343:
4339:
4334:
4330:
4326:
4321:
4317:
4313:
4310:
4307:
4303:
4298:
4294:
4291:
4288:
4283:
4279:
4275:
4272:
4249:
4246:
4242:
4237:
4233:
4230:
4227:
4199:
4196:
4193:
4190:
4187:
4182:
4178:
4174:
4171:
4166:
4162:
4158:
4153:
4149:
4145:
4142:
4139:
4136:
4133:
4130:
4127:
4124:
4119:
4115:
4091:
4088:
4085:
4065:
4062:
4058:
4053:
4049:
4046:
4043:
4038:
4034:
4013:
4010:
4006:
4001:
3997:
3992:
3988:
3984:
3979:
3975:
3951:
3948:
3944:
3939:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3912:
3908:
3904:
3901:
3898:
3894:
3889:
3885:
3880:
3876:
3872:
3867:
3863:
3839:
3819:
3816:
3813:
3808:
3804:
3800:
3797:
3794:
3791:
3786:
3782:
3759:
3755:
3734:
3731:
3727:
3722:
3718:
3715:
3712:
3707:
3703:
3682:
3678:
3673:
3669:
3666:
3663:
3659:
3654:
3650:
3647:
3644:
3641:
3638:
3635:
3632:
3627:
3623:
3619:
3616:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3546:
3543:
3540:
3537:
3534:
3514:
3511:
3507:
3502:
3498:
3493:
3489:
3485:
3480:
3476:
3451:
3448:
3444:
3439:
3435:
3432:
3429:
3426:
3421:
3417:
3413:
3388:
3383:
3379:
3375:
3370:
3366:
3345:
3323:
3319:
3291:
3288:
3286:
3283:
3279:
3275:
3271:
3269:
3266:
3264:
3261:
3258:
3255:
3252:
3249:
3248:
3245:
3242:
3239:
3236:
3233:
3230:
3229:
3226:
3223:
3221:
3218:
3214:
3207:
3204:
3197:
3195:
3192:
3190:
3187:
3184:
3181:
3178:
3175:
3174:
3149:
3145:
3121:
3118:
3114:
3109:
3105:
3102:
3099:
3094:
3090:
3086:
3083:
3080:
3076:
3071:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3044:
3040:
3036:
3033:
3030:
3026:
3021:
3017:
3012:
3008:
3004:
2999:
2995:
2966:
2963:
2960:
2955:
2951:
2947:
2944:
2941:
2937:
2932:
2928:
2923:
2919:
2915:
2910:
2906:
2882:
2879:
2875:
2870:
2866:
2861:
2857:
2853:
2848:
2844:
2823:
2820:
2817:
2813:
2808:
2804:
2799:
2795:
2791:
2786:
2782:
2778:
2773:
2769:
2764:
2760:
2757:
2753:
2748:
2744:
2739:
2735:
2731:
2726:
2722:
2718:
2713:
2710:
2707:
2702:
2698:
2675:
2672:
2669:
2664:
2660:
2656:
2651:
2648:
2645:
2640:
2636:
2608:
2605:
2601:
2596:
2592:
2589:
2586:
2581:
2577:
2573:
2570:
2567:
2564:
2560:
2555:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2528:
2524:
2520:
2517:
2514:
2509:
2505:
2501:
2498:
2496:
2492:
2489:
2486:
2481:
2477:
2473:
2472:
2469:
2466:
2462:
2457:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2430:
2426:
2422:
2419:
2416:
2413:
2409:
2404:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2377:
2373:
2369:
2366:
2363:
2358:
2354:
2350:
2347:
2345:
2341:
2338:
2335:
2330:
2326:
2322:
2321:
2295:
2292:
2287:
2284:
2281:
2276:
2272:
2268:
2263:
2260:
2257:
2252:
2248:
2240:
2236:
2231:
2225:
2222:
2219:
2214:
2210:
2206:
2201:
2198:
2195:
2190:
2186:
2182:
2179:
2149:
2146:
2141:
2137:
2133:
2130:
2127:
2124:
2121:
2118:
2092:
2089:
2084:
2080:
2059:
2056:
2053:
2050:
2047:
2027:
2024:
2020:
2015:
2011:
2008:
2005:
2000:
1996:
1975:
1972:
1969:
1941:
1938:
1935:
1930:
1926:
1902:
1882:
1879:
1876:
1873:
1868:
1864:
1845:
1842:
1834:tangent bundle
1818:
1777:tangent bundle
1747:
1746:
1725:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1647:
1636:
1620:
1595:
1584:
1569:
1550:
1462:
1459:
1450:
1435:
1424:
1409:
1399:
1380:
1377:
1360:
1357:
1354:
1351:
1348:
1343:
1339:
1318:
1315:
1312:
1309:
1306:
1281:
1277:
1254:
1250:
1238:
1237:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1185:
1181:
1177:
1174:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1099:
1095:
1091:
1088:
1085:
1080:
1076:
1050:
1046:
1042:
1037:
1033:
1029:
1026:
998:
993:
989:
985:
980:
976:
972:
952:
947:
943:
939:
934:
930:
926:
915:
914:
900:
897:
894:
891:
888:
885:
882:
860:
856:
852:
849:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
798:
795:
792:
789:
765:
761:
757:
754:
751:
748:
718:
715:
712:
707:
703:
699:
694:
690:
686:
683:
670:
658:
649:)-bundle over
634:
631:
626:
622:
618:
613:
609:
582:
579:
576:
573:
570:
565:
561:
557:
554:
530:
527:
524:
519:
515:
511:
491:
488:
485:
482:
479:
476:
473:
470:
467:
464:
461:
458:
455:
435:
432:
429:
426:
423:
418:
414:
386:
366:
363:
360:
357:
354:
336:
333:
331:
328:
298:
272:
246:
237:tangent bundle
228:
212:
202:
187:
176:
157:
146:
131:
97:
94:
55:spinor bundles
33:spin structure
9:
6:
4:
3:
2:
6961:
6950:
6947:
6945:
6942:
6940:
6937:
6935:
6932:
6930:
6927:
6926:
6924:
6914:
6910:
6907:
6906:
6896:
6894:9780821837498
6890:
6886:
6882:
6877:
6873:
6871:9783540357216
6867:
6863:
6859:
6855:
6851:
6846:
6842:
6836:
6832:
6827:
6823:
6817:
6813:
6809:
6804:
6800:
6794:
6790:
6786:
6785:Spin Geometry
6781:
6780:
6768:
6764:
6760:
6756:
6751:
6746:
6742:
6738:
6737:
6729:
6721:
6717:
6713:
6707:
6703:
6699:
6694:
6689:
6685:
6678:
6669:
6665:
6661:
6657:
6652:
6647:
6643:
6639:
6632:
6630:
6621:
6619:0-8218-0994-6
6615:
6611:
6607:
6602:
6601:
6592:
6584:
6578:
6574:
6570:
6565:
6564:
6555:
6547:
6543:
6539:
6535:
6531:
6527:
6522:
6517:
6513:
6509:
6508:
6503:
6496:
6488:
6482:
6478:
6474:
6469:
6468:
6467:Spin Geometry
6462:
6455:
6447:
6443:
6437:
6426:
6419:
6412:
6404:
6400:
6396:
6392:
6389:(2): 97â136.
6388:
6384:
6383:
6375:
6367:
6363:
6356:
6349:
6340:
6335:
6331:
6327:
6323:
6319:
6313:
6304:
6299:
6295:
6291:
6287:
6283:
6277:
6269:
6265:
6261:
6255:
6247:
6243:
6239:
6238:Haefliger, A.
6233:
6231:
6226:
6217:
6214:
6212:
6209:
6207:
6204:
6203:
6197:
6194:
6190:
6186:
6182:
6177:
6175:
6174:normal bundle
6171:
6170:string theory
6167:
6163:
6159:
6155:
6151:
6147:
6143:
6139:
6135:
6131:
6127:
6123:
6110:
6106:
6103:
6099:
6096:
6093:
6089:
6088:
6082:
6080:
6073:
6063:
6058:
6051:
6044:
6040:
6017:
6014:
6009:
6005:
6001:
5998:
5993:
5989:
5981:
5980:
5979:
5978:
5977:
5976:
5975:
5973:
5966:
5959:
5952:
5945:
5941:
5936:
5931:
5927:
5923:
5919:
5913:
5909:
5903:
5899:
5895:
5890:
5888:
5884:
5877:
5873:
5868:
5863:
5859:
5852:
5848:
5839:
5835:
5826:
5822:
5817:
5815:
5812:
5808:
5787:
5784:
5770:
5767:
5759:
5743:
5726:
5716:
5713:
5705:
5682:
5679:
5671:
5655:
5635:
5632:
5624:
5609:
5602:
5601:
5600:
5599:
5598:
5596:
5592:
5588:
5579:
5575:
5571:
5566:
5556:
5546:
5542:
5538:
5533:
5531:
5527:
5523:
5522:Edward Witten
5513:
5509:
5505:
5501:
5494:
5490:
5486:
5475:
5473:
5469:
5463:
5459:
5455:
5451:
5447:
5443:
5439:
5434:
5430:
5420:
5417:
5413:
5407:
5403:
5399:
5394:
5392:
5388:
5384:
5383:triangulation
5380:
5375:
5373:
5369:
5365:
5361:
5357:
5353:
5344:
5342:
5338:
5328:
5324:
5315:
5309:
5303:
5298:
5296:
5292:
5288:
5284:
5280:
5261:
5254:
5237:
5226:
5219:
5197:
5191:
5156:
5155:
5154:
5152:
5133:
5127:
5108:
5095:
5089:
5067:
5064:
5061:
5058:
5051:
5050:
5049:
5048:
5042:
5038:
5033:
5027:
5023:
5020: : Spin(
5019:
4999:
4990:
4984:
4978:
4972:
4966:
4963:
4954:
4948:
4937:
4928:
4915:
4908:
4907:
4906:
4905:
4901:
4886:
4882:
4878:
4874:
4870:
4866:
4863:
4859:
4858:
4856:
4852:
4851:
4842:
4838:
4834:
4831:
4827:
4823:
4820:All compact,
4819:
4816:
4812:
4809:
4805:
4802:
4798:
4795:
4791:
4788:
4784:
4781:
4777:
4774:
4770:
4767:. (Note that
4754:
4751:
4748:
4740:
4736:
4732:
4728:
4721:
4717:
4713:
4709:
4706:
4702:
4699:
4696:
4692:
4691:
4685:
4671:
4659:
4643:
4635:
4631:
4627:
4623:
4619:
4615:
4608:
4604:
4600:
4595:
4590:
4586:
4582:
4575:
4571:
4563:
4556:
4552:
4542:
4525:
4521:
4512:
4506:
4500:
4497:
4489:
4485:
4481:
4478:
4455:
4451:
4442:
4439:
4431:
4427:
4416:
4400:
4396:
4387:
4381:
4375:
4372:
4364:
4360:
4350:
4346:
4337:
4332:
4328:
4319:
4315:
4305:
4301:
4292:
4289:
4281:
4277:
4270:
4261:
4244:
4240:
4231:
4228:
4213:and applying
4210:
4197:
4188:
4180:
4176:
4164:
4160:
4151:
4147:
4134:
4128:
4125:
4117:
4113:
4103:
4089:
4083:
4060:
4056:
4047:
4044:
4036:
4032:
4008:
4004:
3995:
3990:
3986:
3977:
3973:
3962:
3946:
3942:
3933:
3927:
3921:
3918:
3910:
3906:
3896:
3892:
3883:
3878:
3874:
3865:
3861:
3851:
3850:under the map
3837:
3814:
3806:
3802:
3798:
3792:
3784:
3780:
3757:
3753:
3729:
3725:
3716:
3713:
3705:
3701:
3680:
3676:
3667:
3661:
3657:
3648:
3642:
3636:
3633:
3625:
3621:
3617:
3614:
3591:
3585:
3582:
3573:
3567:
3564:
3541:
3535:
3532:
3509:
3505:
3496:
3491:
3487:
3478:
3474:
3465:
3446:
3442:
3433:
3427:
3419:
3415:
3381:
3377:
3368:
3364:
3356:subgroups of
3343:
3321:
3317:
3306:
3289:
3277:
3273:
3259:
3253:
3250:
3224:
3212:
3202:
3185:
3179:
3176:
3163:
3147:
3143:
3132:
3116:
3112:
3103:
3100:
3092:
3088:
3078:
3074:
3065:
3059:
3053:
3050:
3042:
3038:
3028:
3024:
3015:
3010:
3006:
2997:
2993:
2983:
2979:
2964:
2961:
2958:
2953:
2949:
2939:
2935:
2926:
2921:
2917:
2908:
2904:
2894:
2877:
2873:
2864:
2859:
2855:
2846:
2842:
2815:
2811:
2802:
2797:
2793:
2784:
2780:
2771:
2767:
2762:
2755:
2751:
2742:
2737:
2733:
2724:
2720:
2716:
2711:
2708:
2705:
2696:
2673:
2670:
2667:
2662:
2658:
2654:
2649:
2646:
2643:
2634:
2626:In addition,
2623:
2603:
2599:
2590:
2587:
2579:
2575:
2571:
2562:
2558:
2549:
2543:
2537:
2534:
2526:
2522:
2518:
2515:
2507:
2503:
2499:
2497:
2490:
2487:
2484:
2479:
2475:
2464:
2460:
2451:
2445:
2439:
2436:
2428:
2424:
2420:
2411:
2407:
2398:
2392:
2386:
2383:
2375:
2371:
2367:
2364:
2356:
2352:
2348:
2346:
2339:
2336:
2333:
2328:
2324:
2310:
2306:
2293:
2285:
2282:
2279:
2274:
2270:
2261:
2258:
2255:
2250:
2246:
2238:
2234:
2229:
2223:
2220:
2217:
2212:
2208:
2199:
2196:
2193:
2188:
2184:
2177:
2168:
2166:
2160:
2147:
2139:
2135:
2125:
2119:
2116:
2107:
2106:
2090:
2082:
2078:
2054:
2048:
2045:
2022:
2018:
2009:
2006:
1998:
1994:
1973:
1967:
1959:
1955:
1936:
1928:
1924:
1916:
1900:
1880:
1874:
1871:
1866:
1862:
1854:
1853:vector bundle
1851:
1841:
1839:
1835:
1831:
1830:triangulation
1827:
1822:
1817:
1813:
1809:
1805:
1801:
1797:
1793:
1792:spin manifold
1789:
1785:
1781:
1778:
1774:
1769:
1767:
1761:
1757:
1754: : Spin(
1753:
1742:
1739:∈ Spin(
1738:
1731:
1724:
1720:
1700:
1694:
1688:
1682:
1679:
1673:
1670:
1664:
1657:
1656:
1655:
1653:
1646:
1642:
1635:
1631:
1618:
1609:
1605:
1601:
1594:
1590:
1583:
1579:
1575:
1568:
1564:
1560:
1556:
1549:
1546:
1542:
1538:
1533:
1531:
1527:
1523:
1519:
1515:
1511:
1507:
1503:
1499:
1495:
1491:
1488:of dimension
1487:
1483:
1479:
1475:
1472:
1468:
1458:
1456:
1449:
1445:
1441:
1434:
1430:
1423:
1419:
1415:
1408:
1405:
1398:
1394:
1390:
1386:
1376:
1374:
1358:
1349:
1341:
1337:
1313:
1310:
1307:
1295:
1279:
1275:
1252:
1248:
1240:In this case
1220:
1214:
1211:
1208:
1205:
1183:
1179:
1175:
1172:
1151:
1145:
1139:
1136:
1130:
1127:
1121:
1097:
1093:
1089:
1086:
1083:
1078:
1074:
1066:
1065:
1064:
1048:
1044:
1035:
1031:
1027:
1024:
1016:
1012:
991:
987:
983:
978:
974:
945:
941:
937:
932:
928:
912:
895:
889:
886:
883:
880:
858:
854:
850:
847:
823:
817:
811:
805:
802:
796:
793:
787:
763:
759:
755:
752:
749:
746:
736:
732:
713:
705:
701:
692:
688:
684:
681:
674:
671:
656:
648:
632:
624:
620:
616:
611:
607:
599:
596:
595:
594:
580:
571:
563:
559:
555:
552:
544:
525:
522:
517:
513:
486:
480:
477:
468:
462:
459:
456:
453:
433:
424:
416:
412:
403:
400:
384:
361:
358:
355:
345:
342:
327:
325:
321:
317:
312:
310:
306:
303:. The bundle
301:
296:
292:
288:
284:
280:
275:
270:
266:
262:
258:
254:
249:
243:
241:
238:
234:
227:
223:
219:
215:
208:
201:
197:
193:
186:
182:
175:
171:
167:
163:
156:
152:
145:
141:
137:
130:
127:
123:
119:
115:
111:
107:
103:
93:
91:
90:spin geometry
87:
83:
79:
75:
71:
67:
62:
60:
56:
50:
46:
41:
38:
34:
30:
19:
6884:
6853:
6830:
6807:
6784:
6750:math/9901137
6740:
6734:
6728:
6683:
6677:
6641:
6637:
6599:
6591:
6562:
6554:
6521:math/9705218
6511:
6505:
6501:
6495:
6466:
6454:
6445:
6436:
6411:
6386:
6380:
6374:
6365:
6361:
6348:
6329:
6325:
6312:
6293:
6289:
6276:
6267:
6263:
6254:
6245:
6241:
6193:supergravity
6188:
6184:
6178:
6157:
6149:
6145:
6141:
6130:wavefunction
6119:
6108:
6078:
6071:
6069:
6056:
6049:
6042:
6038:
6036:
5964:
5957:
5950:
5943:
5939:
5937:
5929:
5925:
5921:
5917:
5911:
5907:
5901:
5897:
5891:
5886:
5882:
5875:
5871:
5869:
5861:
5850:
5846:
5837:
5833:
5824:
5820:
5818:
5813:
5804:
5577:
5573:
5569:
5562:
5544:
5540:
5534:
5519:
5507:
5503:
5499:
5492:
5488:
5484:
5481:
5467:
5461:
5457:
5453:
5449:
5445:
5441:
5437:
5432:
5426:
5415:
5411:
5405:
5401:
5397:
5395:
5376:
5371:
5367:
5363:
5362:â 2, and to
5359:
5355:
5351:
5345:
5340:
5336:
5326:
5322:
5313:
5307:
5301:
5299:
5294:
5290:
5282:
5278:
5276:
5150:
5148:
5047:homomorphism
5040:
5036:
5031:
5025:
5021:
5017:
5014:
4897:
4840:
4829:
4828:or less are
4814:
4810:
4800:
4796:
4786:
4782:
4772:
4768:
4738:
4734:
4730:
4726:
4725:) vanishes,
4719:
4715:
4711:
4697:
4621:
4606:
4602:
4598:
4596:
4588:
4584:
4581:affine space
4573:
4569:
4554:
4550:
4548:
4418:
4263:
4212:
4105:
3964:
3853:
3462:. But, from
3308:
3165:
3134:
2985:
2981:
2896:
2625:
2312:
2308:
2170:
2162:
2109:
1954:Armand Borel
1847:
1823:
1815:
1811:
1803:
1799:
1795:
1791:
1787:
1783:
1779:
1772:
1770:
1765:
1759:
1755:
1751:
1748:
1740:
1736:
1729:
1722:
1718:
1654:) such that
1651:
1644:
1640:
1633:
1611:
1607:
1599:
1592:
1588:
1581:
1577:
1573:
1566:
1562:
1554:
1547:
1545:frame bundle
1534:
1529:
1525:
1521:
1517:
1509:
1501:
1497:
1494:fibre metric
1489:
1485:
1477:
1466:
1464:
1454:
1447:
1443:
1439:
1432:
1428:
1421:
1417:
1413:
1406:
1396:
1392:
1388:
1384:
1382:
1296:
1239:
1014:
916:
739:
735:covering map
672:
646:
597:
542:
401:
338:
316:fiber bundle
313:
308:
304:
299:
294:
290:
286:
282:
278:
273:
263:is then the
260:
256:
252:
247:
244:
239:
232:
225:
221:
217:
210:
206:
199:
195:
191:
184:
180:
173:
169:
165:
161:
154:
150:
143:
139:
135:
128:
121:
113:
109:
106:field theory
99:
63:
48:
44:
32:
26:
6913:orientation
6608:. pp.
6318:Karoubi, M.
5894:Chern class
5860:picture of
5559:The details
5537:spin bundle
5472:Chern class
5423:Obstruction
5389:over the 2-
5310:) Ă Spin(2)
4826:dimension 3
4658:chern class
4616:, then (by
1836:over the 1-
1471:paracompact
1379:Obstruction
731:equivariant
399:equivariant
324:Max Karoubi
287:spin frames
6923:Categories
6693:1607.02103
6651:1606.07894
6571:. p.
6475:. p.
6296:: 11â100.
6270:: 198â203.
6248:: 558â560.
6222:References
4869:Hirzebruch
4848:Properties
2163:hence the
1850:orientable
1802:if the SO(
1716:, for all
1604:spin group
1457:vanishes.
1371:is due to
1063:such that
341:orientable
335:Definition
37:orientable
6767:118698159
6720:104292702
6668:119598006
6514:: 41â50.
6260:J. Milnor
6148:) bundle
6104:are spin.
6002:β
5785:…
5782:⟶
5744:β
5739:⟶
5694:⟶
5651:⟶
5613:⟶
5610:⋯
5227:×
5115:→
5096:×
5068::
5062:×
5059:κ
4997:→
4985:
4979:×
4967:
4961:→
4949:
4934:→
4919:→
4752:≠
4668:mod
4501:
4482:∈
4376:
4357:→
4312:→
4274:→
4229:−
4195:→
4177:π
4173:→
4148:π
4144:→
4129:
4114:π
4087:→
3922:
3903:→
3637:
3618:∈
3586:
3580:→
3568:
3536:
3416:π
3365:π
3285:→
3268:→
3254:
3244:↓
3238:↓
3232:↓
3220:→
3206:~
3194:→
3180:
3085:→
3054:
3035:→
2946:→
2701:∞
2639:∞
2538:
2440:
2387:
2291:→
2267:→
2205:→
2181:→
2145:→
2132:→
2120:
2105:fibration
2088:→
2049:
1971:→
1878:→
1863:π
1695:ρ
1683:ϕ
1665:ϕ
1619:ϕ
1356:→
1276:ϕ
1249:ϕ
1215:
1209:∈
1176:∈
1094:ϕ
1084:∘
1075:ϕ
1041:→
988:ϕ
942:ϕ
890:
884:∈
851:∈
818:ρ
806:ϕ
788:ϕ
760:π
753:ϕ
750:∘
747:π
737:such that
698:→
682:ϕ
630:→
608:π
578:→
553:π
526:ϕ
481:
475:→
463:
454:ρ
431:→
116:) admits
6944:K-theory
6831:K-Theory
6463:(1989).
6425:Archived
6320:(1968).
6284:(1964).
6200:See also
6172:it is a
6166:D-branes
6092:oriented
6085:Examples
6039:integral
5553:(â1) = 1
5391:skeleton
4799:are not
4771:is also
4741:for all
4688:Examples
4634:fermions
4614:skeleton
4419:Because
2230:→
2070:-bundle
1838:skeleton
1721:∈
1632: :
1482:oriented
1375:(1956).
1165:for all
840:for all
102:geometry
96:Overview
86:K theory
74:fermions
6546:6906852
6526:Bibcode
6403:2372795
6368:: 64â78
6160:is the
6140:to the
6134:fermion
5325:) â SO(
5285:) is a
4881:Ă genus
4862:Ă genus
4855:Ă genus
4785:is not
4583:over H(
1848:For an
1775:is the
1758:) â SO(
1576:) is a
733:2-fold
118:spinors
104:and in
6891:
6868:
6837:
6818:
6795:
6765:
6718:
6708:
6666:
6616:
6579:
6544:
6483:
6401:
6216:Spinor
5883:bundle
5350:(Spin(
5289:of SO(
5024:) â U(
4624:. In
1749:where
1504:is an
1442:) â H(
1416:) â H(
729:is an
397:is an
220:) â H(
194:) â H(
138:) â H(
84:, and
59:spinor
35:on an
6763:S2CID
6745:arXiv
6716:S2CID
6688:arXiv
6664:S2CID
6646:arXiv
6542:S2CID
6516:arXiv
6428:(PDF)
6421:(PDF)
6399:JSTOR
6358:(PDF)
5807:arrow
5587:arrow
5549:1 = 1
5404:over
5381:or a
5374:= 2.
5321:Spin(
5306:Spin(
5293:) by
4873:Borel
4695:genus
2309:where
1828:or a
1814:is a
1790:is a
1606:Spin(
1469:be a
1453:) of
1427:) of
669:, and
593:when
289:over
259:over
205:) of
149:) of
6889:ISBN
6866:ISBN
6835:ISBN
6816:ISBN
6793:ISBN
6706:ISBN
6614:ISBN
6577:ISBN
6481:ISBN
6189:spin
6185:spin
6142:spin
6124:the
6109:spin
6107:All
6100:All
6090:All
5940:spin
5887:spin
5872:spin
5847:spin
5821:spin
5568:0 â
5545:spin
5541:spin
5448:) â
5331:and
4938:Spin
4871:and
4853:The
4841:spin
4839:are
4835:All
4830:spin
4815:spin
4813:are
4801:spin
4787:spin
4778:The
4773:spin
4739:spin
4731:spin
4632:for
3565:Spin
3177:Spin
2688:and
1956:and
1800:spin
1734:and
1726:Spin
1643:) â
1637:Spin
1596:Spin
1578:lift
1530:spin
1476:and
1465:Let
1267:and
1212:Spin
1198:and
1113:and
963:and
887:Spin
873:and
859:Spin
693:Spin
625:Spin
518:Spin
460:Spin
402:lift
31:, a
6858:doi
6755:doi
6698:doi
6656:doi
6534:doi
6504:".
6477:391
6391:doi
6334:doi
6298:doi
6246:243
6179:In
6168:in
6120:In
6062:).
5830:of
5589:is
5583:â 0
5551:or
5370:if
5358:if
4824:of
4737:is
4729:is
4710:If
4594:).
4222:Hom
3408:Hom
1798:is
1768:).
1580:of
1561:SO(
1528:is
1480:an
779:and
285:of
242:.)
100:In
27:In
6925::
6883:.
6864:.
6852:.
6814:.
6810:.
6791:.
6787:.
6761:.
6753:.
6741:18
6739:.
6714:.
6704:.
6696:.
6662:.
6654:.
6642:23
6640:.
6628:^
6610:55
6604:.
6575:.
6573:26
6567:.
6540:.
6532:.
6524:.
6510:.
6479:.
6471:.
6444:.
6423:.
6397:.
6387:80
6385:.
6366:32
6364:,
6360:,
6328:.
6324:.
6294:92
6292:.
6288:.
6266:.
6244:.
6229:^
6176:.
5910:,
5906:H(
5867:.
5832:H(
5816:.
5576:â
5572:â
5506:,
5491:,
5460:/2
5456:,
5444:,
5419:.
5414:â
5366:â
5297:.
5000:1.
4964:SO
4811:CP
4797:CP
4783:CP
4693:A
4684:.
4498:SO
4373:SO
4126:SO
3919:SO
3634:SO
3583:SO
3533:SO
3251:SO
3051:SO
2535:SO
2437:SO
2384:SO
2117:SO
2046:SO
1780:TM
1648:SO
1585:SO
1570:SO
1551:SO
1532:.
1446:,
1420:,
1395:,
1342:SO
706:SO
673:b)
598:a)
564:SO
478:SO
417:SO
311:.
281:â
277::
255:â
251::
240:TM
224:,
198:,
172:,
142:,
92:.
80:,
47:,
6897:.
6874:.
6860::
6843:.
6824:.
6801:.
6769:.
6757::
6747::
6722:.
6700::
6690::
6672:.
6670:.
6658::
6648::
6622:.
6585:.
6548:.
6536::
6528::
6518::
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6489:.
6448:.
6405:.
6393::
6342:.
6336::
6330:1
6306:.
6300::
6268:9
6158:E
6150:E
6146:N
6079:W
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6060:3
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6053:2
6050:w
6046:3
6043:W
6018:0
6015:=
6010:2
6006:w
5999:=
5994:3
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5944:w
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5255:1
5252:(
5247:U
5238:2
5233:Z
5223:)
5220:n
5217:(
5212:n
5209:i
5206:p
5203:S
5198:=
5195:)
5192:n
5189:(
5183:C
5176:n
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5170:p
5167:S
5151:n
5134:.
5131:)
5128:N
5125:(
5120:U
5112:)
5109:1
5106:(
5101:U
5093:)
5090:n
5087:(
5082:n
5079:i
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5028:)
5026:N
5022:n
5018:Îș
4994:)
4991:1
4988:(
4982:U
4976:)
4973:n
4970:(
4958:)
4955:n
4952:(
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4843:.
4832:.
4817:.
4803:.
4789:.
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4755:2
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4714:(
4712:H
4707:.
4698:g
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4607:w
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4568:(
4566:1
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4526:2
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4513:,
4510:)
4507:n
4504:(
4495:(
4490:1
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4479:1
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4382:n
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4329:P
4325:(
4320:1
4316:H
4309:)
4306:2
4302:/
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4287:(
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4271:0
4248:)
4245:2
4241:/
4236:Z
4232:,
4226:(
4198:1
4192:)
4189:M
4186:(
4181:1
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4157:(
4152:1
4141:)
4138:)
4135:n
4132:(
4123:(
4118:1
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4064:)
4061:2
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4033:H
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4009:2
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4000:Z
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3931:)
3928:n
3925:(
3916:(
3911:1
3907:H
3900:)
3897:2
3893:/
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3875:P
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3838:1
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3812:(
3807:2
3803:w
3799:=
3796:)
3793:1
3790:(
3785:2
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3758:2
3754:w
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3730:2
3726:/
3721:Z
3717:,
3714:M
3711:(
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3677:/
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3668:=
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3658:/
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3649:,
3646:)
3643:n
3640:(
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3506:/
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2717:=
2712:1
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