6144:
960:
1917:
6200:
gave a workable definition of a category of spectra and of maps (not just homotopy classes) between them, as useful in stable homotopy theory as the category of CW complexes is in the unstable case. (This is essentially the category described above, and it is still used for many purposes: for other
3825:
which is injective. Unfortunately, these two structures, with the addition of the smash product, lead to significant complexity in the theory of spectra because there cannot exist a single category of spectra which satisfies a list of five axioms relating these structures. The above adjunction is
5922:
5808:
6034:
4830:; in other words it behaves like the (derived) tensor product of abelian groups. A major problem with the smash product is that obvious ways of defining it make it associative and commutative only up to homotopy. Some more recent definitions of spectra, such as
3702:
748:
1711:
685:
One of the most important invariants of spectra are the homotopy groups of the spectrum. These groups mirror the definition of the stable homotopy groups of spaces since the structure of the suspension maps is integral in its definition. Given a spectrum
4721:
The stable homotopy category is additive: maps can be added by using a variant of the track addition used to define homotopy groups. Thus homotopy classes from one spectrum to another form an abelian group. Furthermore the stable homotopy category is
5439:
3590:
4585:
is defined to be the category whose objects are spectra and whose morphisms are homotopy classes of maps between spectra. Many other definitions of spectrum, some appearing very different, lead to equivalent stable homotopy categories.
4811:
5222:
One of the canonical complexities while working with spectra and defining a category of spectra comes from the fact each of these categories cannot satisfy five seemingly obvious axioms concerning the infinite loop space of a spectrum
4167:, a cofinal subspectrum is a subspectrum for which each cell of the parent spectrum is eventually contained in the subspectrum after a finite number of suspensions. Spectra can then be turned into a category by defining a
5366:
3025:
1062:
3080:
109:
5980:
5813:
5699:
3198:
2425:
6479:
6027:
3820:
6139:{\displaystyle {\begin{matrix}X&\xrightarrow {\eta } &\Omega ^{\infty }\Sigma ^{\infty }X\\{\mathord {=}}\downarrow &&\downarrow \theta \\X&\xrightarrow {i} &QX\end{matrix}}}
5293:
5692:
1716:
753:
545:
279:
3324:
2579:
955:{\displaystyle {\begin{aligned}\pi _{n}(E)&=\lim _{\to k}\pi _{n+k}(E_{k})\\&=\lim _{\to }\left(\cdots \to \pi _{n+k}(E_{k})\to \pi _{n+k+1}(E_{k+1})\to \cdots \right)\end{aligned}}}
3624:
4432:
3751:
1912:{\displaystyle {\begin{aligned}\pi _{i}(H(R/I)\wedge _{R}H(R/J))&\cong H_{i}\left(R/I\otimes ^{\mathbf {L} }R/J\right)\\&\cong \operatorname {Tor} _{i}^{R}(R/I,R/J)\end{aligned}}}
393:
5117:
4711:
2266:
4918:
2517:
3885:
2199:
1230:
598:
5189:
4642:
2470:
6205:(1970).) Important further theoretical advances have however been made since 1990, improving vastly the formal properties of spectra. Consequently, much recent literature uses
2301:
2063:
5549:
5517:
4837:
The smash product is compatible with the triangulated category structure. In particular the smash product of a distinguished triangle with a spectrum is a distinguished triangle.
4481:
2919:
2873:
5625:
5598:
4263:, where two such functions represent the same map if they coincide on some cofinal subspectrum. Intuitively such a map of spectra does not need to be everywhere defined, just
3492:
3422:
5488:
5374:
4944:
4552:
4327:
2630:
2604:
2364:
631:
2034:
is the group corresponding to vector bundles on the suspension of X. Topological K-theory is a generalized cohomology theory, so it gives a spectrum. The zeroth space is
1574:
740:
6566:
3944:
1417:
1329:
4201:
2032:
1980:
1261:
484:
5647:
5569:
5459:
4165:
3520:
2764:
1181:
664:
413:
1655:
1472:
6166:
4508:
4128:
4097:
4070:
2726:
334:
207:
140:
4736:
1161:
6421:
2811:
2694:
2671:
2130:
1600:
5241:
5212:
5016:
4996:
4572:
4374:
4354:
4289:
4261:
4241:
4221:
3771:
3617:
3512:
3465:
3445:
2947:
2843:
2788:
2321:
2103:
2083:
1699:
1679:
1620:
1496:
1437:
1375:
1352:
704:
461:
437:
180:
160:
4976:
5304:
6387:
3971:
There are three natural categories whose objects are spectra, whose morphisms are the functions, or maps, or homotopy classes defined below.
3826:
valid only in the homotopy categories of spaces and spectra, but not always with a specific category of spectra (not the homotopy category).
2955:
968:
5917:{\displaystyle \gamma :\left(\Sigma ^{\infty }E\right)\wedge \left(\Sigma ^{\infty }E'\right)\to \Sigma ^{\infty }\left(E\wedge E'\right)}
3036:
6172:
Because of this, the study of spectra is fractured based upon the model being used. For an overview, check out the article cited above.
5803:{\displaystyle \phi :\left(\Omega ^{\infty }E\right)\wedge \left(\Omega ^{\infty }E'\right)\to \Omega ^{\infty }\left(E\wedge E'\right)}
57:
48:
5930:
3030:
The construction of the suspension spectrum implies every space can be considered as a cohomology theory. In fact, it defines a functor
4271:(and maps), which is a major tool. There is a natural embedding of the category of pointed CW complexes into this category: it takes
6335:
3091:
2372:
1922:
showing the category of spectra keeps track of the derived information of commutative rings, where the smash product acts as the
5985:
1927:
3779:
1281:
3204:
5924:
which commutes with the unit object in both categories, and the commutative and associative isomorphisms in both categories.
5248:
4834:, eliminate this problem, and give a symmetric monoidal structure at the level of maps, before passing to homotopy classes.
4846:
5652:
3697:{\displaystyle \Omega ^{\infty }\Sigma ^{\infty }X={\underset {\to }{\operatorname {colim} {}}}\Omega ^{n}\Sigma ^{n}X}
1475:
6632:
6327:
3960:
2323:. There is a corresponding construction using real vector bundles instead of complex vector bundles, which gives an 8-
496:
215:
3213:
2525:
297:. This is one of the key points for introducing spectra because they form a natural home for stable homotopy theory.
4379:
1702:
3710:
339:
5519:
denote a category of spectra, the following five axioms can never be satisfied by the specific model of spectra:
5028:
4651:
4267:
become defined, and two maps that coincide on a cofinal subspectrum are said to be equivalent. This gives the
6627:
6206:
2204:
4861:
2475:
3841:
2143:
1186:
554:
6247:
5128:
4727:
4592:
2433:
2275:
2037:
6430:
6365:
5525:
5493:
4441:
3085:
from the homotopy category of CW complexes to the homotopy category of spectra. The morphisms are given by
1923:
286:
5434:{\displaystyle \Sigma ^{\infty }:{\text{Top}}_{*}\leftrightarrows {\text{Spectra}}_{*}:\Omega ^{\infty }}
2878:
6242:
4852:
4723:
2848:
601:
40:
5603:
5576:
3470:
3329:
2324:
6601:- contains excellent introduction to spectra and applications for constructing Adams spectral sequence
5464:
1939:
6561:
6435:
6370:
6193:
4826:
of spectra extends the smash product of CW complexes. It makes the stable homotopy category into a
3596:
1287:
1233:
674:
4927:
4513:
4302:
2613:
2587:
2347:
606:
6496:
6456:
6227:
6202:
4831:
2607:
2366:. This is a spectrum whose homotopy groups are given by the stable homotopy groups of spheres, so
1983:
1504:
709:
670:
440:
28:
6196:
in their work on generalized homology theories in the early 1960s. The 1964 doctoral thesis of
3916:
3585:{\displaystyle \Omega ^{\infty }E={\underset {\to n}{\operatorname {colim} {}}}\Omega ^{n}E_{n}}
1380:
1292:
1701:
into the category of spectra. This embedding forms the basis of spectral geometry, a model for
6383:
6331:
6295:
4827:
4174:
3888:
2133:
2001:
1949:
1239:
466:
290:
5632:
5554:
5444:
4806:{\displaystyle X\rightarrow Y\rightarrow Y\cup CX\rightarrow (Y\cup CX)\cup CY\cong \Sigma X}
4137:
2929:
discussed above. The homotopy groups of this spectrum are then the stable homotopy groups of
2731:
1166:
636:
398:
6575:
6488:
6440:
6375:
6285:
6237:
6197:
2137:
1625:
1442:
6554:
6523:
6452:
6397:
6180:
A version of the concept of a spectrum was introduced in the 1958 doctoral dissertation of
6151:
4486:
4434:(associativity of the smash product yields immediately that this is indeed a spectrum). A
4106:
4075:
4048:
2699:
312:
185:
118:
6550:
6519:
6448:
6393:
6181:
4726:(Vogt (1970)), the shift being given by suspension and the distinguished triangles by the
3952:
2926:
2767:
2342:
2336:
1067:
336:
of pointed topological spaces or pointed simplicial sets together with the structure maps
17:
6350:
2793:
2676:
2653:
2112:
1582:
6474:
6416:
6379:
6189:
5226:
5197:
5001:
4981:
4557:
4359:
4339:
4274:
4246:
4226:
4206:
3756:
3602:
3497:
3450:
3430:
2932:
2828:
2773:
2306:
2088:
2068:
1991:
1684:
1664:
1605:
1481:
1422:
1360:
1337:
689:
446:
422:
165:
145:
6580:
4949:
6621:
6598:
6510:
Lima, Elon Lages (1959), "The
Spanier–Whitehead duality in new homotopy categories",
6500:
6290:
6273:
6222:
6185:
4823:
4333:
3901:
2647:
2641:
2106:
1332:
416:
6604:
6594:
6412:
6346:
2921:(the structure maps are the identity.) For example, the suspension spectrum of the
289:
of spectra leading to many technical difficulties, but they all determine the same
6460:
6549:, Lecture Notes Series, No. 21, Matematisk Institut, Aarhus Universitet, Aarhus,
6544:
6321:
1930:
for commutative rings, a more refined theory than classical
Hochschild homology.
6317:
3910:
490:
32:
6492:
6444:
6357:
6232:
5490:
denote the category of based, compactly generated, weak
Hausdorff spaces, and
1355:
548:
44:
6299:
2272:, so all the spaces in the topological K-theory spectrum are given by either
1926:. Moreover, Eilenberg–Maclane spectra can be used to define theories such as
5361:{\displaystyle QX=\mathop {\text{colim}} _{\to n}\Omega ^{n}\Sigma ^{n}X}
2922:
6530:
Lima, Elon Lages (1960), "Stable
Postnikov invariants and their duals",
1705:. One of the important properties of this embedding are the isomorphisms
3946:
corresponds to the identity.) For example, the spectrum of topological
3838:
is a spectrum such that the adjoint of the structure map (i.e., the map
3020:{\displaystyle \pi _{n}(\Sigma ^{\infty }X)=\pi _{n}^{\mathbb {S} }(X)}
1057:{\displaystyle \Sigma :\pi _{n+k}(E_{n})\to \pi _{n+k+1}(\Sigma E_{n})}
5461:
in both the category of spaces and the category of spectra. If we let
2650:
representing various cobordism theories. This includes real cobordism
1987:
182:
is equivalent to computing the homotopy classes of maps to the space
6610:
6118:
6051:
5551:
is a symmetric monoidal category with respect to the smash product
3075:{\displaystyle \Sigma ^{\infty }:h{\text{CW}}\to h{\text{Spectra}}}
2519:. Note the smash product gives a product structure on this spectrum
104:{\displaystyle {\mathcal {E}}^{*}:{\text{CW}}^{op}\to {\text{Ab}}}
5975:{\displaystyle \theta :\Omega ^{\infty }\Sigma ^{\infty }X\to QX}
5214:
can be a spectrum or (by using its suspension spectrum) a space.
47:. Every such cohomology theory is representable, as follows from
4099:
is a sequence of subcomplexes that is also a spectrum. As each
1419:
can be identified with the set of homotopy classes of maps from
6188:
wrote further on the subject in 1959. Spectra were adopted by
3193:{\displaystyle ={\underset {\to n}{\operatorname {colim} {}}}}
2420:{\displaystyle \pi _{n}(\mathbb {S} )=\pi _{n}^{\mathbb {S} }}
5018:. We define the generalized homology theory of a spectrum
222:
64:
965:
where the maps are induced from the composition of the map
305:
There are many variations of the definition: in general, a
6022:{\displaystyle X\in \operatorname {Ob} ({\text{Top}}_{*})}
6345:
Elmendorf, Anthony D.; KĹ™ĂĹľ, Igor; Mandell, Michael A.;
6213:(2001) for a unified treatment of these new approaches.
3815:{\displaystyle X\to \Omega ^{\infty }\Sigma ^{\infty }X}
2341:
One of the quintessential examples of a spectrum is the
4589:
Finally, we can define the suspension of a spectrum by
1681:. Note this construction can be used to embed any ring
6039:
5288:{\displaystyle Q:{\text{Top}}_{*}\to {\text{Top}}_{*}}
6611:"Are spectra really the same as cohomology theories?"
6154:
6037:
5988:
5933:
5816:
5702:
5655:
5635:
5606:
5579:
5557:
5528:
5496:
5467:
5447:
5377:
5307:
5251:
5229:
5200:
5132:
5131:
5031:
5004:
4984:
4952:
4930:
4865:
4864:
4739:
4654:
4595:
4560:
4516:
4489:
4444:
4382:
4362:
4342:
4305:
4277:
4249:
4229:
4209:
4177:
4140:
4109:
4078:
4051:
3919:
3913:
in terms of smash products commute "up to homotopy" (
3844:
3782:
3759:
3713:
3627:
3605:
3523:
3500:
3473:
3453:
3433:
3332:
3216:
3094:
3039:
2958:
2935:
2881:
2851:
2831:
2796:
2776:
2734:
2702:
2679:
2656:
2616:
2590:
2528:
2478:
2436:
2375:
2350:
2309:
2278:
2207:
2146:
2115:
2091:
2071:
2040:
2004:
1952:
1714:
1687:
1667:
1628:
1608:
1585:
1507:
1484:
1445:
1425:
1383:
1363:
1340:
1295:
1242:
1189:
1169:
1070:
971:
751:
712:
692:
639:
609:
557:
499:
469:
449:
425:
401:
342:
315:
218:
188:
168:
148:
142:
such that evaluating the cohomology theory in degree
121:
60:
5687:{\displaystyle \Sigma ^{\infty }S^{0}=\mathbb {S} }
4648:is invertible, as we can desuspend too, by setting
2646:Another canonical example of spectra come from the
6480:Proceedings of the Cambridge Philosophical Society
6160:
6138:
6021:
5974:
5916:
5802:
5686:
5641:
5619:
5592:
5563:
5543:
5511:
5482:
5453:
5433:
5360:
5287:
5235:
5206:
5183:
5111:
5010:
4990:
4970:
4938:
4912:
4805:
4705:
4636:
4566:
4546:
4502:
4475:
4426:
4368:
4348:
4321:
4283:
4255:
4235:
4215:
4195:
4159:
4122:
4091:
4064:
3938:
3879:
3814:
3765:
3745:
3696:
3611:
3584:
3506:
3486:
3459:
3439:
3416:
3318:
3192:
3074:
3019:
2941:
2913:
2867:
2837:
2821:A spectrum may be constructed out of a space. The
2805:
2782:
2758:
2720:
2688:
2665:
2624:
2598:
2573:
2511:
2464:
2419:
2358:
2315:
2295:
2260:
2193:
2124:
2097:
2077:
2057:
2026:
1974:
1911:
1693:
1673:
1649:
1614:
1594:
1568:
1490:
1466:
1431:
1411:
1369:
1346:
1323:
1255:
1224:
1175:
1155:
1056:
954:
734:
698:
658:
625:
592:
539:
493:(1974): a spectrum (or CW-spectrum) is a sequence
478:
455:
431:
407:
387:
328:
273:
201:
174:
154:
134:
103:
6567:Transactions of the American Mathematical Society
6605:An untitled book project about symmetric spectra
5122:and define its generalized cohomology theory by
838:
783:
540:{\displaystyle E:=\{E_{n}\}_{n\in \mathbb {N} }}
274:{\displaystyle {\mathcal {E}}^{k}(X)\cong \left}
6419:(2001), "Model categories of diagram spectra",
6351:"Modern foundations for stable homotopy theory"
3319:{\displaystyle \left\simeq \left\simeq \cdots }
2574:{\displaystyle S^{n}\wedge S^{m}\simeq S^{n+m}}
6422:Proceedings of the London Mathematical Society
6364:, Amsterdam: North-Holland, pp. 213–253,
4841:Generalized homology and cohomology of spectra
3707:and this construction comes with an inclusion
2610:, this forms the initial object, analogous to
2430:We can write down this spectrum explicitly as
4717:The triangulated homotopy category of spectra
4438:of maps between spectra corresponds to a map
4427:{\displaystyle (E\wedge X)_{n}=E_{n}\wedge X}
8:
6274:"Is there a convenient category of spectra?"
4978:is the set of homotopy classes of maps from
4541:
4535:
4203:to be a function from a cofinal subspectrum
3746:{\displaystyle X\to \Omega ^{n}\Sigma ^{n}X}
2506:
2494:
520:
506:
388:{\displaystyle S^{1}\wedge X_{n}\to X_{n+1}}
51:. This means that, given a cohomology theory
5112:{\displaystyle E_{n}X=\pi _{n}(E\wedge X)=}
4706:{\displaystyle (\Sigma ^{-1}E)_{n}=E_{n-1}}
2606:. Moreover, if considering the category of
3967:Functions, maps, and homotopies of spectra
3891:of a ring is an example of an Ω-spectrum.
6579:
6564:(1962), "Generalized homology theories",
6434:
6369:
6289:
6153:
6087:
6086:
6073:
6063:
6038:
6036:
6010:
6005:
5987:
5954:
5944:
5932:
5884:
5858:
5832:
5815:
5770:
5744:
5718:
5701:
5696:Either there is a natural transformation
5680:
5679:
5670:
5660:
5654:
5634:
5611:
5605:
5584:
5578:
5556:
5535:
5530:
5527:
5503:
5498:
5495:
5474:
5469:
5466:
5446:
5425:
5412:
5407:
5397:
5392:
5382:
5376:
5349:
5339:
5323:
5318:
5306:
5279:
5274:
5264:
5259:
5250:
5228:
5199:
5156:
5137:
5130:
5090:
5089:
5083:
5052:
5036:
5030:
5003:
4983:
4951:
4932:
4931:
4929:
4896:
4895:
4889:
4870:
4863:
4738:
4691:
4678:
4662:
4653:
4622:
4609:
4594:
4559:
4515:
4494:
4488:
4458:
4443:
4412:
4399:
4381:
4361:
4341:
4310:
4304:
4276:
4248:
4228:
4208:
4176:
4145:
4139:
4114:
4108:
4083:
4077:
4056:
4050:
3924:
3918:
3865:
3849:
3843:
3803:
3793:
3781:
3758:
3734:
3724:
3712:
3685:
3675:
3663:
3654:
3642:
3632:
3626:
3604:
3576:
3566:
3549:
3540:
3528:
3522:
3499:
3478:
3472:
3452:
3432:
3400:
3384:
3358:
3342:
3331:
3290:
3268:
3242:
3226:
3215:
3178:
3162:
3142:
3133:
3118:
3102:
3093:
3067:
3056:
3044:
3038:
3002:
3001:
3000:
2995:
2976:
2963:
2957:
2934:
2899:
2886:
2880:
2856:
2850:
2830:
2795:
2775:
2733:
2701:
2678:
2655:
2618:
2617:
2615:
2592:
2591:
2589:
2559:
2546:
2533:
2527:
2485:
2481:
2480:
2477:
2456:
2443:
2439:
2438:
2435:
2411:
2410:
2409:
2404:
2390:
2389:
2380:
2374:
2352:
2351:
2349:
2308:
2280:
2279:
2277:
2261:{\displaystyle K^{2n+1}(X)\cong K^{1}(X)}
2243:
2212:
2206:
2176:
2151:
2145:
2114:
2090:
2070:
2042:
2041:
2039:
2009:
2003:
1957:
1951:
1894:
1880:
1865:
1860:
1833:
1823:
1822:
1810:
1796:
1771:
1756:
1741:
1723:
1715:
1713:
1686:
1666:
1627:
1607:
1584:
1545:
1506:
1483:
1444:
1424:
1388:
1382:
1362:
1339:
1300:
1294:
1247:
1241:
1210:
1197:
1188:
1168:
1144:
1116:
1097:
1078:
1069:
1045:
1017:
1001:
982:
970:
922:
897:
881:
862:
841:
818:
799:
786:
760:
752:
750:
717:
711:
691:
644:
638:
617:
608:
578:
565:
556:
531:
530:
523:
513:
498:
468:
448:
424:
400:
373:
360:
347:
341:
320:
314:
260:
227:
221:
220:
217:
193:
187:
167:
147:
126:
120:
96:
84:
79:
69:
63:
62:
59:
6323:Stable homotopy and generalised homology
6029:which that there is a commuting diagram:
4913:{\displaystyle \displaystyle \pi _{n}E=}
2512:{\displaystyle \mathbb {S} _{0}=\{0,1\}}
1938:As a second important example, consider
6259:
3880:{\displaystyle X_{n}\to \Omega X_{n+1}}
2194:{\displaystyle K^{2n}(X)\cong K^{0}(X)}
1225:{\displaystyle \Sigma E_{n}\to E_{n+1}}
593:{\displaystyle \Sigma E_{n}\to E_{n+1}}
419:. The smash product of a pointed space
5184:{\displaystyle \displaystyle E^{n}X=.}
4637:{\displaystyle (\Sigma E)_{n}=E_{n+1}}
3207:eventually stabilizes. By this we mean
2632:in the category of commutative rings.
2465:{\displaystyle \mathbb {S} _{i}=S^{i}}
6405:Modern articles developing the theory
3909:such that the diagrams that describe
2770:. In fact, for any topological group
2296:{\displaystyle \mathbb {Z} \times BU}
2058:{\displaystyle \mathbb {Z} \times BU}
1478:with homotopy concentrated in degree
439:with a circle is homeomorphic to the
7:
6267:
6265:
6263:
5927:There is a natural weak equivalence
5544:{\displaystyle {\text{Spectra}}_{*}}
5512:{\displaystyle {\text{Spectra}}_{*}}
4855:of a spectrum to be those given by
4476:{\displaystyle (E\wedge I^{+})\to F}
6546:Boardman's stable homotopy category
6278:Journal of Pure and Applied Algebra
5218:Technical complexities with spectra
2914:{\displaystyle X_{n}=S^{n}\wedge X}
2696:, framed cobordism, spin cobordism
6168:is the unit map in the adjunction.
6074:
6070:
6064:
6060:
5955:
5951:
5945:
5941:
5885:
5881:
5859:
5855:
5833:
5829:
5771:
5767:
5745:
5741:
5719:
5715:
5661:
5657:
5612:
5608:
5585:
5581:
5426:
5422:
5383:
5379:
5346:
5336:
5153:
5080:
4886:
4797:
4659:
4599:
4307:
3858:
3804:
3800:
3794:
3790:
3731:
3721:
3682:
3672:
3643:
3639:
3633:
3629:
3599:of the spectrum. For a CW complex
3563:
3529:
3525:
3479:
3475:
3397:
3381:
3359:
3355:
3343:
3339:
3287:
3265:
3239:
3223:
3175:
3159:
3119:
3115:
3103:
3099:
3045:
3041:
2977:
2973:
2868:{\displaystyle \Sigma ^{\infty }X}
2857:
2853:
1190:
1170:
1137:
1038:
972:
610:
558:
470:
25:
6581:10.1090/S0002-9947-1962-0137117-6
6477:(1961). "Bordism and cobordism".
5620:{\displaystyle \Omega ^{\infty }}
5593:{\displaystyle \Sigma ^{\infty }}
4583:homotopy category of (CW) spectra
3487:{\displaystyle \Omega ^{\infty }}
3467:there is an inverse construction
3417:{\displaystyle \left\simeq \left}
285:Note there are several different
6380:10.1016/B978-044481779-2/50007-9
6207:modified definitions of spectrum
5483:{\displaystyle {\text{Top}}_{*}}
3959:For many more examples, see the
1824:
1579:Then the corresponding spectrum
49:Brown's representability theorem
6272:Lewis, L. Gaunce (1991-08-30).
5629:The unit for the smash product
1928:topological Hochschild homology
6468:Historically relevant articles
6362:Handbook of algebraic topology
6201:accounts, see Adams (1974) or
6099:
6093:
6016:
6001:
5963:
5877:
5763:
5403:
5324:
5270:
5174:
5149:
5106:
5076:
5070:
5058:
4965:
4953:
4906:
4882:
4782:
4767:
4764:
4749:
4743:
4675:
4655:
4606:
4596:
4529:
4517:
4467:
4464:
4445:
4396:
4383:
4187:
3950:-theory is a ring spectrum. A
3930:
3855:
3786:
3717:
3666:
3553:
3205:Freudenthal suspension theorem
3187:
3155:
3146:
3127:
3095:
3061:
3014:
3008:
2985:
2969:
2394:
2386:
2255:
2249:
2233:
2227:
2188:
2182:
2166:
2160:
2021:
2015:
1969:
1963:
1902:
1874:
1782:
1779:
1765:
1749:
1735:
1729:
1644:
1632:
1563:
1551:
1535:
1532:
1520:
1508:
1461:
1449:
1406:
1394:
1318:
1306:
1203:
1150:
1109:
1106:
1103:
1071:
1051:
1035:
1010:
1007:
994:
937:
934:
915:
890:
887:
874:
855:
842:
824:
811:
787:
772:
766:
729:
723:
571:
366:
239:
233:
93:
1:
4847:Generalized cohomology theory
4004:that commute with the maps ÎŁ
3887:) is a weak equivalence. The
681:Homotopy groups of a spectrum
45:generalized cohomology theory
6291:10.1016/0022-4049(91)90030-6
5810:or a natural transformation
5441:, the and the smash product
4939:{\displaystyle \mathbb {S} }
4574:taken to be the basepoint.
4547:{\displaystyle \sqcup \{*\}}
4322:{\displaystyle \Sigma ^{n}Y}
3956:may be defined analogously.
2625:{\displaystyle \mathbb {Z} }
2599:{\displaystyle \mathbb {S} }
2584:induces a ring structure on
2359:{\displaystyle \mathbb {S} }
1934:Topological complex K-theory
626:{\displaystyle \Sigma E_{n}}
301:The definition of a spectrum
6328:University of Chicago Press
5371:a pair of adjoint functors
4946:is the sphere spectrum and
3986:is a sequence of maps from
3961:list of cohomology theories
1569:{\displaystyle =H^{n}(X;A)}
1232:. A spectrum is said to be
735:{\displaystyle \pi _{n}(E)}
669:For other definitions, see
6649:
4844:
3939:{\displaystyle S^{0}\to X}
2639:
2334:
1703:derived algebraic geometry
1659:Eilenberg–MacLane spectrum
1412:{\displaystyle H^{n}(X;A)}
1324:{\displaystyle H^{n}(X;A)}
1282:Eilenberg–Maclane spectrum
1279:
1276:Eilenberg–Maclane spectrum
1163:given by functoriality of
706:define the homotopy group
18:Eilenberg–MacLane spectrum
6493:10.1017/s0305004100035064
6445:10.1112/S0024611501012692
4818:Smash products of spectra
2790:there is a Thom spectrum
2065:while the first space is
551:together with inclusions
4853:(stable) homotopy groups
4579:stable homotopy category
4196:{\displaystyle f:E\to F}
4134: + 1)-cell in
3427:for some finite integer
2027:{\displaystyle K^{1}(X)}
1975:{\displaystyle K^{0}(X)}
1331:with coefficients in an
1256:{\displaystyle \pi _{k}}
1183:) and the structure map
489:The following is due to
479:{\displaystyle \Sigma X}
295:stable homotopy category
6248:Adams spectral sequence
5649:is the sphere spectrum
5642:{\displaystyle \wedge }
5564:{\displaystyle \wedge }
5454:{\displaystyle \wedge }
4376:is a spectrum given by
4160:{\displaystyle E_{j+1}}
3494:which takes a spectrum
2759:{\displaystyle MString}
1476:Eilenberg–MacLane space
1176:{\displaystyle \Sigma }
659:{\displaystyle E_{n+1}}
408:{\displaystyle \wedge }
6209:: see Michael Mandell
6162:
6147:
6140:
6023:
5976:
5918:
5804:
5688:
5643:
5621:
5594:
5565:
5545:
5513:
5484:
5455:
5435:
5369:
5362:
5296:
5289:
5237:
5208:
5185:
5113:
5012:
4992:
4972:
4940:
4914:
4807:
4730:sequences of spectra
4707:
4646:translation suspension
4638:
4568:
4548:
4510:is the disjoint union
4504:
4477:
4428:
4370:
4356:and a pointed complex
4350:
4323:
4285:
4257:
4237:
4217:
4197:
4161:
4124:
4093:
4066:
3940:
3881:
3823:
3816:
3767:
3747:
3705:
3698:
3613:
3593:
3586:
3508:
3488:
3461:
3441:
3425:
3418:
3320:
3201:
3194:
3083:
3076:
3028:
3021:
2943:
2915:
2869:
2839:
2807:
2784:
2760:
2722:
2690:
2667:
2626:
2600:
2582:
2575:
2513:
2466:
2428:
2421:
2360:
2317:
2297:
2262:
2195:
2126:
2099:
2079:
2059:
2028:
1976:
1924:derived tensor product
1920:
1913:
1695:
1675:
1651:
1650:{\displaystyle K(A,n)}
1616:
1596:
1577:
1570:
1492:
1468:
1467:{\displaystyle K(A,n)}
1433:
1413:
1371:
1348:
1325:
1263:are zero for negative
1257:
1226:
1177:
1157:
1058:
963:
956:
736:
700:
660:
627:
594:
541:
480:
457:
433:
409:
389:
330:
283:
275:
203:
176:
156:
136:
113:
105:
6543:Vogt, Rainer (1970),
6411:Mandell, Michael A.;
6243:Suspension (topology)
6163:
6161:{\displaystyle \eta }
6141:
6030:
6024:
5977:
5919:
5805:
5689:
5644:
5622:
5595:
5566:
5546:
5514:
5485:
5456:
5436:
5363:
5300:
5290:
5244:
5238:
5209:
5186:
5114:
5013:
4993:
4973:
4941:
4915:
4808:
4708:
4639:
4569:
4549:
4505:
4503:{\displaystyle I^{+}}
4478:
4429:
4371:
4351:
4324:
4286:
4258:
4238:
4218:
4198:
4162:
4125:
4123:{\displaystyle E_{j}}
4094:
4092:{\displaystyle F_{n}}
4067:
4065:{\displaystyle E_{n}}
3941:
3882:
3817:
3775:
3768:
3748:
3699:
3620:
3614:
3587:
3516:
3509:
3489:
3462:
3442:
3419:
3321:
3209:
3195:
3087:
3077:
3032:
3022:
2951:
2944:
2916:
2870:
2840:
2808:
2785:
2761:
2723:
2721:{\displaystyle MSpin}
2691:
2668:
2627:
2601:
2576:
2521:
2514:
2467:
2422:
2368:
2361:
2318:
2298:
2263:
2196:
2127:
2100:
2080:
2060:
2029:
1982:is defined to be the
1977:
1914:
1707:
1696:
1676:
1652:
1617:
1597:
1571:
1500:
1493:
1469:
1434:
1414:
1372:
1349:
1326:
1258:
1227:
1178:
1158:
1059:
957:
744:
737:
701:
661:
628:
595:
542:
481:
458:
434:
410:
390:
331:
329:{\displaystyle X_{n}}
276:
211:
204:
202:{\displaystyle E^{k}}
177:
157:
137:
135:{\displaystyle E^{k}}
106:
53:
6562:Whitehead, George W.
6152:
6035:
5986:
5931:
5814:
5700:
5653:
5633:
5604:
5577:
5555:
5526:
5494:
5465:
5445:
5375:
5305:
5249:
5227:
5198:
5129:
5029:
5002:
4982:
4950:
4928:
4862:
4737:
4652:
4593:
4558:
4514:
4487:
4442:
4380:
4360:
4340:
4303:
4275:
4247:
4227:
4207:
4175:
4138:
4107:
4076:
4049:
3978:between two spectra
3917:
3842:
3780:
3757:
3711:
3625:
3603:
3521:
3498:
3471:
3451:
3431:
3330:
3214:
3092:
3037:
2956:
2933:
2879:
2849:
2829:
2794:
2774:
2732:
2700:
2677:
2673:, complex cobordism
2654:
2614:
2588:
2526:
2476:
2434:
2373:
2348:
2307:
2276:
2205:
2144:
2113:
2089:
2069:
2038:
2002:
1950:
1940:topological K-theory
1712:
1685:
1665:
1626:
1606:
1583:
1505:
1482:
1443:
1423:
1381:
1361:
1338:
1293:
1240:
1187:
1167:
1156:{\displaystyle \to }
1068:
969:
749:
710:
690:
637:
607:
555:
497:
467:
447:
423:
399:
340:
313:
216:
186:
166:
146:
119:
58:
6532:Summa Brasil. Math.
6512:Summa Brasil. Math.
6415:; Schwede, Stefan;
6198:J. Michael Boardman
6194:George W. Whitehead
6122:
6055:
5600:is left-adjoint to
4293:suspension spectrum
4269:category of spectra
3773:, hence gives a map
3597:infinite loop space
3447:. For a CW complex
3007:
2823:suspension spectrum
2817:Suspension spectrum
2728:, string cobordism
2416:
1870:
1657:; it is called the
1288:singular cohomology
675:simplicial spectrum
633:as a subcomplex of
115:there exist spaces
6633:Spectra (topology)
6595:Spectral Sequences
6475:Atiyah, Michael F.
6228:Symmetric spectrum
6158:
6136:
6134:
6019:
5972:
5914:
5800:
5684:
5639:
5617:
5590:
5561:
5541:
5509:
5480:
5451:
5431:
5358:
5331:
5285:
5233:
5204:
5181:
5180:
5109:
5008:
4988:
4968:
4936:
4910:
4909:
4851:We can define the
4803:
4703:
4634:
4564:
4544:
4500:
4473:
4424:
4366:
4346:
4319:
4281:
4253:
4233:
4213:
4193:
4157:
4120:
4089:
4062:
3936:
3877:
3812:
3763:
3743:
3694:
3669:
3609:
3582:
3560:
3504:
3484:
3457:
3437:
3414:
3316:
3190:
3153:
3072:
3017:
2991:
2939:
2911:
2865:
2835:
2806:{\displaystyle MG}
2803:
2780:
2756:
2718:
2689:{\displaystyle MU}
2686:
2666:{\displaystyle MO}
2663:
2622:
2596:
2571:
2509:
2462:
2417:
2400:
2356:
2313:
2293:
2258:
2191:
2125:{\displaystyle BU}
2122:
2095:
2075:
2055:
2024:
1984:Grothendieck group
1972:
1909:
1907:
1856:
1691:
1671:
1647:
1612:
1595:{\displaystyle HA}
1592:
1566:
1498:. We write this as
1488:
1464:
1429:
1409:
1367:
1344:
1321:
1253:
1222:
1173:
1153:
1054:
952:
950:
846:
794:
732:
696:
671:symmetric spectrum
656:
623:
590:
537:
476:
453:
441:reduced suspension
429:
405:
385:
326:
271:
199:
172:
152:
132:
101:
29:algebraic topology
6389:978-0-444-81779-2
6123:
6056:
6008:
5533:
5501:
5472:
5410:
5395:
5321:
5317:
5277:
5262:
5236:{\displaystyle Q}
5207:{\displaystyle X}
5011:{\displaystyle Y}
4991:{\displaystyle X}
4832:symmetric spectra
4828:monoidal category
4567:{\displaystyle *}
4369:{\displaystyle X}
4349:{\displaystyle E}
4284:{\displaystyle Y}
4256:{\displaystyle F}
4236:{\displaystyle E}
4216:{\displaystyle G}
4045:Given a spectrum
3889:K-theory spectrum
3766:{\displaystyle n}
3655:
3612:{\displaystyle X}
3541:
3514:and forms a space
3507:{\displaystyle E}
3460:{\displaystyle X}
3440:{\displaystyle N}
3134:
3070:
3059:
2942:{\displaystyle X}
2838:{\displaystyle X}
2783:{\displaystyle G}
2608:symmetric spectra
2325:periodic spectrum
2316:{\displaystyle U}
2134:classifying space
2098:{\displaystyle U}
2078:{\displaystyle U}
1694:{\displaystyle R}
1674:{\displaystyle A}
1615:{\displaystyle n}
1491:{\displaystyle n}
1432:{\displaystyle X}
1370:{\displaystyle X}
1347:{\displaystyle A}
837:
782:
699:{\displaystyle E}
456:{\displaystyle X}
432:{\displaystyle X}
291:homotopy category
175:{\displaystyle X}
155:{\displaystyle k}
99:
82:
16:(Redirected from
6640:
6614:
6584:
6583:
6557:
6539:
6526:
6504:
6463:
6438:
6400:
6373:
6355:
6341:
6304:
6303:
6293:
6269:
6238:Mapping spectrum
6167:
6165:
6164:
6159:
6145:
6143:
6142:
6137:
6135:
6114:
6097:
6092:
6091:
6078:
6077:
6068:
6067:
6047:
6028:
6026:
6025:
6020:
6015:
6014:
6009:
6006:
5981:
5979:
5978:
5973:
5959:
5958:
5949:
5948:
5923:
5921:
5920:
5915:
5913:
5909:
5908:
5889:
5888:
5876:
5872:
5871:
5863:
5862:
5845:
5841:
5837:
5836:
5809:
5807:
5806:
5801:
5799:
5795:
5794:
5775:
5774:
5762:
5758:
5757:
5749:
5748:
5731:
5727:
5723:
5722:
5693:
5691:
5690:
5685:
5683:
5675:
5674:
5665:
5664:
5648:
5646:
5645:
5640:
5626:
5624:
5623:
5618:
5616:
5615:
5599:
5597:
5596:
5591:
5589:
5588:
5570:
5568:
5567:
5562:
5550:
5548:
5547:
5542:
5540:
5539:
5534:
5531:
5518:
5516:
5515:
5510:
5508:
5507:
5502:
5499:
5489:
5487:
5486:
5481:
5479:
5478:
5473:
5470:
5460:
5458:
5457:
5452:
5440:
5438:
5437:
5432:
5430:
5429:
5417:
5416:
5411:
5408:
5402:
5401:
5396:
5393:
5387:
5386:
5367:
5365:
5364:
5359:
5354:
5353:
5344:
5343:
5330:
5322:
5319:
5294:
5292:
5291:
5286:
5284:
5283:
5278:
5275:
5269:
5268:
5263:
5260:
5242:
5240:
5239:
5234:
5213:
5211:
5210:
5205:
5190:
5188:
5187:
5182:
5164:
5163:
5142:
5141:
5118:
5116:
5115:
5110:
5093:
5088:
5087:
5057:
5056:
5041:
5040:
5017:
5015:
5014:
5009:
4997:
4995:
4994:
4989:
4977:
4975:
4974:
4971:{\displaystyle }
4969:
4945:
4943:
4942:
4937:
4935:
4919:
4917:
4916:
4911:
4899:
4894:
4893:
4875:
4874:
4812:
4810:
4809:
4804:
4712:
4710:
4709:
4704:
4702:
4701:
4683:
4682:
4670:
4669:
4643:
4641:
4640:
4635:
4633:
4632:
4614:
4613:
4573:
4571:
4570:
4565:
4553:
4551:
4550:
4545:
4509:
4507:
4506:
4501:
4499:
4498:
4482:
4480:
4479:
4474:
4463:
4462:
4433:
4431:
4430:
4425:
4417:
4416:
4404:
4403:
4375:
4373:
4372:
4367:
4355:
4353:
4352:
4347:
4328:
4326:
4325:
4320:
4315:
4314:
4290:
4288:
4287:
4282:
4262:
4260:
4259:
4254:
4242:
4240:
4239:
4234:
4222:
4220:
4219:
4214:
4202:
4200:
4199:
4194:
4166:
4164:
4163:
4158:
4156:
4155:
4130:suspends to an (
4129:
4127:
4126:
4121:
4119:
4118:
4098:
4096:
4095:
4090:
4088:
4087:
4072:, a subspectrum
4071:
4069:
4068:
4063:
4061:
4060:
3945:
3943:
3942:
3937:
3929:
3928:
3886:
3884:
3883:
3878:
3876:
3875:
3854:
3853:
3821:
3819:
3818:
3813:
3808:
3807:
3798:
3797:
3772:
3770:
3769:
3764:
3752:
3750:
3749:
3744:
3739:
3738:
3729:
3728:
3703:
3701:
3700:
3695:
3690:
3689:
3680:
3679:
3670:
3665:
3664:
3647:
3646:
3637:
3636:
3618:
3616:
3615:
3610:
3591:
3589:
3588:
3583:
3581:
3580:
3571:
3570:
3561:
3559:
3551:
3550:
3533:
3532:
3513:
3511:
3510:
3505:
3493:
3491:
3490:
3485:
3483:
3482:
3466:
3464:
3463:
3458:
3446:
3444:
3443:
3438:
3423:
3421:
3420:
3415:
3413:
3409:
3405:
3404:
3389:
3388:
3371:
3367:
3363:
3362:
3347:
3346:
3325:
3323:
3322:
3317:
3309:
3305:
3301:
3300:
3279:
3278:
3255:
3251:
3247:
3246:
3231:
3230:
3199:
3197:
3196:
3191:
3183:
3182:
3167:
3166:
3154:
3152:
3144:
3143:
3123:
3122:
3107:
3106:
3081:
3079:
3078:
3073:
3071:
3068:
3060:
3057:
3049:
3048:
3026:
3024:
3023:
3018:
3006:
3005:
2999:
2981:
2980:
2968:
2967:
2948:
2946:
2945:
2940:
2920:
2918:
2917:
2912:
2904:
2903:
2891:
2890:
2874:
2872:
2871:
2866:
2861:
2860:
2844:
2842:
2841:
2836:
2812:
2810:
2809:
2804:
2789:
2787:
2786:
2781:
2765:
2763:
2762:
2757:
2727:
2725:
2724:
2719:
2695:
2693:
2692:
2687:
2672:
2670:
2669:
2664:
2631:
2629:
2628:
2623:
2621:
2605:
2603:
2602:
2597:
2595:
2580:
2578:
2577:
2572:
2570:
2569:
2551:
2550:
2538:
2537:
2518:
2516:
2515:
2510:
2490:
2489:
2484:
2471:
2469:
2468:
2463:
2461:
2460:
2448:
2447:
2442:
2426:
2424:
2423:
2418:
2415:
2414:
2408:
2393:
2385:
2384:
2365:
2363:
2362:
2357:
2355:
2322:
2320:
2319:
2314:
2302:
2300:
2299:
2294:
2283:
2267:
2265:
2264:
2259:
2248:
2247:
2226:
2225:
2200:
2198:
2197:
2192:
2181:
2180:
2159:
2158:
2138:Bott periodicity
2131:
2129:
2128:
2123:
2105:is the infinite
2104:
2102:
2101:
2096:
2084:
2082:
2081:
2076:
2064:
2062:
2061:
2056:
2045:
2033:
2031:
2030:
2025:
2014:
2013:
1981:
1979:
1978:
1973:
1962:
1961:
1918:
1916:
1915:
1910:
1908:
1898:
1884:
1869:
1864:
1849:
1845:
1841:
1837:
1829:
1828:
1827:
1814:
1801:
1800:
1775:
1761:
1760:
1745:
1728:
1727:
1700:
1698:
1697:
1692:
1680:
1678:
1677:
1672:
1656:
1654:
1653:
1648:
1621:
1619:
1618:
1613:
1601:
1599:
1598:
1593:
1575:
1573:
1572:
1567:
1550:
1549:
1497:
1495:
1494:
1489:
1473:
1471:
1470:
1465:
1438:
1436:
1435:
1430:
1418:
1416:
1415:
1410:
1393:
1392:
1376:
1374:
1373:
1368:
1353:
1351:
1350:
1345:
1330:
1328:
1327:
1322:
1305:
1304:
1262:
1260:
1259:
1254:
1252:
1251:
1231:
1229:
1228:
1223:
1221:
1220:
1202:
1201:
1182:
1180:
1179:
1174:
1162:
1160:
1159:
1154:
1149:
1148:
1133:
1132:
1102:
1101:
1089:
1088:
1063:
1061:
1060:
1055:
1050:
1049:
1034:
1033:
1006:
1005:
993:
992:
961:
959:
958:
953:
951:
947:
943:
933:
932:
914:
913:
886:
885:
873:
872:
845:
830:
823:
822:
810:
809:
793:
765:
764:
741:
739:
738:
733:
722:
721:
705:
703:
702:
697:
665:
663:
662:
657:
655:
654:
632:
630:
629:
624:
622:
621:
599:
597:
596:
591:
589:
588:
570:
569:
546:
544:
543:
538:
536:
535:
534:
518:
517:
485:
483:
482:
477:
462:
460:
459:
454:
438:
436:
435:
430:
414:
412:
411:
406:
394:
392:
391:
386:
384:
383:
365:
364:
352:
351:
335:
333:
332:
327:
325:
324:
309:is any sequence
280:
278:
277:
272:
270:
266:
265:
264:
232:
231:
226:
225:
208:
206:
205:
200:
198:
197:
181:
179:
178:
173:
161:
159:
158:
153:
141:
139:
138:
133:
131:
130:
110:
108:
107:
102:
100:
97:
92:
91:
83:
80:
74:
73:
68:
67:
21:
6648:
6647:
6643:
6642:
6641:
6639:
6638:
6637:
6628:Homotopy theory
6618:
6617:
6609:
6591:
6560:
6542:
6529:
6509:
6473:
6470:
6417:Shipley, Brooke
6410:
6407:
6390:
6358:James., Ioan M.
6353:
6344:
6338:
6318:Adams, J. Frank
6316:
6313:
6308:
6307:
6271:
6270:
6261:
6256:
6219:
6182:Elon Lages Lima
6178:
6150:
6149:
6133:
6132:
6124:
6112:
6106:
6105:
6096:
6083:
6082:
6069:
6059:
6057:
6045:
6033:
6032:
6004:
5984:
5983:
5950:
5940:
5929:
5928:
5901:
5894:
5890:
5880:
5864:
5854:
5853:
5849:
5828:
5827:
5823:
5812:
5811:
5787:
5780:
5776:
5766:
5750:
5740:
5739:
5735:
5714:
5713:
5709:
5698:
5697:
5666:
5656:
5651:
5650:
5631:
5630:
5607:
5602:
5601:
5580:
5575:
5574:
5553:
5552:
5529:
5524:
5523:
5497:
5492:
5491:
5468:
5463:
5462:
5443:
5442:
5421:
5406:
5391:
5378:
5373:
5372:
5345:
5335:
5303:
5302:
5273:
5258:
5247:
5246:
5225:
5224:
5220:
5196:
5195:
5152:
5133:
5127:
5126:
5079:
5048:
5032:
5027:
5026:
5000:
4999:
4980:
4979:
4948:
4947:
4926:
4925:
4885:
4866:
4860:
4859:
4849:
4843:
4820:
4735:
4734:
4719:
4687:
4674:
4658:
4650:
4649:
4618:
4605:
4591:
4590:
4556:
4555:
4512:
4511:
4490:
4485:
4484:
4454:
4440:
4439:
4408:
4395:
4378:
4377:
4358:
4357:
4338:
4337:
4306:
4301:
4300:
4273:
4272:
4245:
4244:
4225:
4224:
4205:
4204:
4173:
4172:
4141:
4136:
4135:
4110:
4105:
4104:
4079:
4074:
4073:
4052:
4047:
4046:
4041:
4031:
4022:
4012:
4003:
3994:
3969:
3953:module spectrum
3920:
3915:
3914:
3897:
3861:
3845:
3840:
3839:
3832:
3799:
3789:
3778:
3777:
3755:
3754:
3730:
3720:
3709:
3708:
3681:
3671:
3656:
3638:
3628:
3623:
3622:
3601:
3600:
3572:
3562:
3552:
3542:
3524:
3519:
3518:
3496:
3495:
3474:
3469:
3468:
3449:
3448:
3429:
3428:
3396:
3380:
3379:
3375:
3354:
3338:
3337:
3333:
3328:
3327:
3286:
3264:
3263:
3259:
3238:
3222:
3221:
3217:
3212:
3211:
3174:
3158:
3145:
3135:
3114:
3098:
3090:
3089:
3040:
3035:
3034:
2972:
2959:
2954:
2953:
2931:
2930:
2927:sphere spectrum
2895:
2882:
2877:
2876:
2852:
2847:
2846:
2827:
2826:
2819:
2792:
2791:
2772:
2771:
2730:
2729:
2698:
2697:
2675:
2674:
2652:
2651:
2644:
2638:
2612:
2611:
2586:
2585:
2555:
2542:
2529:
2524:
2523:
2479:
2474:
2473:
2452:
2437:
2432:
2431:
2376:
2371:
2370:
2346:
2345:
2343:sphere spectrum
2339:
2337:Sphere spectrum
2333:
2331:Sphere spectrum
2305:
2304:
2274:
2273:
2239:
2208:
2203:
2202:
2172:
2147:
2142:
2141:
2111:
2110:
2087:
2086:
2067:
2066:
2036:
2035:
2005:
2000:
1999:
1953:
1948:
1947:
1942:. At least for
1936:
1906:
1905:
1847:
1846:
1818:
1806:
1802:
1792:
1785:
1752:
1719:
1710:
1709:
1683:
1682:
1663:
1662:
1624:
1623:
1604:
1603:
1581:
1580:
1541:
1503:
1502:
1480:
1479:
1441:
1440:
1421:
1420:
1384:
1379:
1378:
1359:
1358:
1336:
1335:
1296:
1291:
1290:
1284:
1278:
1273:
1243:
1238:
1237:
1206:
1193:
1185:
1184:
1165:
1164:
1140:
1112:
1093:
1074:
1066:
1065:
1041:
1013:
997:
978:
967:
966:
949:
948:
918:
893:
877:
858:
851:
847:
828:
827:
814:
795:
775:
756:
747:
746:
713:
708:
707:
688:
687:
683:
640:
635:
634:
613:
605:
604:
574:
561:
553:
552:
519:
509:
495:
494:
465:
464:
445:
444:
421:
420:
397:
396:
369:
356:
343:
338:
337:
316:
311:
310:
303:
293:, known as the
256:
249:
245:
219:
214:
213:
189:
184:
183:
164:
163:
144:
143:
122:
117:
116:
78:
61:
56:
55:
23:
22:
15:
12:
11:
5:
6646:
6644:
6636:
6635:
6630:
6620:
6619:
6616:
6615:
6607:
6602:
6590:
6589:External links
6587:
6586:
6585:
6574:(2): 227–283,
6558:
6540:
6527:
6506:
6505:
6469:
6466:
6465:
6464:
6436:10.1.1.22.3815
6429:(2): 441–512,
6406:
6403:
6402:
6401:
6388:
6371:10.1.1.55.8006
6342:
6336:
6312:
6309:
6306:
6305:
6284:(3): 233–246.
6258:
6257:
6255:
6252:
6251:
6250:
6245:
6240:
6235:
6230:
6225:
6218:
6215:
6190:Michael Atiyah
6184:. His advisor
6177:
6174:
6170:
6169:
6157:
6131:
6128:
6125:
6121:
6117:
6113:
6111:
6108:
6107:
6104:
6101:
6098:
6095:
6090:
6085:
6084:
6081:
6076:
6072:
6066:
6062:
6058:
6054:
6050:
6046:
6044:
6041:
6040:
6018:
6013:
6003:
6000:
5997:
5994:
5991:
5971:
5968:
5965:
5962:
5957:
5953:
5947:
5943:
5939:
5936:
5925:
5912:
5907:
5904:
5900:
5897:
5893:
5887:
5883:
5879:
5875:
5870:
5867:
5861:
5857:
5852:
5848:
5844:
5840:
5835:
5831:
5826:
5822:
5819:
5798:
5793:
5790:
5786:
5783:
5779:
5773:
5769:
5765:
5761:
5756:
5753:
5747:
5743:
5738:
5734:
5730:
5726:
5721:
5717:
5712:
5708:
5705:
5694:
5682:
5678:
5673:
5669:
5663:
5659:
5638:
5627:
5614:
5610:
5587:
5583:
5571:
5560:
5538:
5506:
5477:
5450:
5428:
5424:
5420:
5415:
5405:
5400:
5390:
5385:
5381:
5357:
5352:
5348:
5342:
5338:
5334:
5329:
5326:
5316:
5313:
5310:
5282:
5272:
5267:
5257:
5254:
5232:
5219:
5216:
5203:
5192:
5191:
5179:
5176:
5173:
5170:
5167:
5162:
5159:
5155:
5151:
5148:
5145:
5140:
5136:
5120:
5119:
5108:
5105:
5102:
5099:
5096:
5092:
5086:
5082:
5078:
5075:
5072:
5069:
5066:
5063:
5060:
5055:
5051:
5047:
5044:
5039:
5035:
5007:
4987:
4967:
4964:
4961:
4958:
4955:
4934:
4922:
4921:
4908:
4905:
4902:
4898:
4892:
4888:
4884:
4881:
4878:
4873:
4869:
4842:
4839:
4819:
4816:
4815:
4814:
4802:
4799:
4796:
4793:
4790:
4787:
4784:
4781:
4778:
4775:
4772:
4769:
4766:
4763:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4718:
4715:
4700:
4697:
4694:
4690:
4686:
4681:
4677:
4673:
4668:
4665:
4661:
4657:
4631:
4628:
4625:
4621:
4617:
4612:
4608:
4604:
4601:
4598:
4563:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4522:
4519:
4497:
4493:
4472:
4469:
4466:
4461:
4457:
4453:
4450:
4447:
4423:
4420:
4415:
4411:
4407:
4402:
4398:
4394:
4391:
4388:
4385:
4365:
4345:
4336:of a spectrum
4318:
4313:
4309:
4299:th complex is
4280:
4252:
4232:
4212:
4192:
4189:
4186:
4183:
4180:
4154:
4151:
4148:
4144:
4117:
4113:
4086:
4082:
4059:
4055:
4036:
4027:
4017:
4008:
3999:
3990:
3968:
3965:
3935:
3932:
3927:
3923:
3905:is a spectrum
3896:
3893:
3874:
3871:
3868:
3864:
3860:
3857:
3852:
3848:
3831:
3828:
3811:
3806:
3802:
3796:
3792:
3788:
3785:
3762:
3742:
3737:
3733:
3727:
3723:
3719:
3716:
3693:
3688:
3684:
3678:
3674:
3668:
3662:
3659:
3653:
3650:
3645:
3641:
3635:
3631:
3608:
3579:
3575:
3569:
3565:
3558:
3555:
3548:
3545:
3539:
3536:
3531:
3527:
3503:
3481:
3477:
3456:
3436:
3412:
3408:
3403:
3399:
3395:
3392:
3387:
3383:
3378:
3374:
3370:
3366:
3361:
3357:
3353:
3350:
3345:
3341:
3336:
3315:
3312:
3308:
3304:
3299:
3296:
3293:
3289:
3285:
3282:
3277:
3274:
3271:
3267:
3262:
3258:
3254:
3250:
3245:
3241:
3237:
3234:
3229:
3225:
3220:
3189:
3186:
3181:
3177:
3173:
3170:
3165:
3161:
3157:
3151:
3148:
3141:
3138:
3132:
3129:
3126:
3121:
3117:
3113:
3110:
3105:
3101:
3097:
3066:
3063:
3055:
3052:
3047:
3043:
3016:
3013:
3010:
3004:
2998:
2994:
2990:
2987:
2984:
2979:
2975:
2971:
2966:
2962:
2938:
2910:
2907:
2902:
2898:
2894:
2889:
2885:
2875:is a spectrum
2864:
2859:
2855:
2834:
2818:
2815:
2802:
2799:
2779:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2717:
2714:
2711:
2708:
2705:
2685:
2682:
2662:
2659:
2640:Main article:
2637:
2634:
2620:
2594:
2568:
2565:
2562:
2558:
2554:
2549:
2545:
2541:
2536:
2532:
2508:
2505:
2502:
2499:
2496:
2493:
2488:
2483:
2459:
2455:
2451:
2446:
2441:
2413:
2407:
2403:
2399:
2396:
2392:
2388:
2383:
2379:
2354:
2335:Main article:
2332:
2329:
2312:
2292:
2289:
2286:
2282:
2257:
2254:
2251:
2246:
2242:
2238:
2235:
2232:
2229:
2224:
2221:
2218:
2215:
2211:
2190:
2187:
2184:
2179:
2175:
2171:
2168:
2165:
2162:
2157:
2154:
2150:
2121:
2118:
2094:
2074:
2054:
2051:
2048:
2044:
2023:
2020:
2017:
2012:
2008:
1992:vector bundles
1971:
1968:
1965:
1960:
1956:
1935:
1932:
1904:
1901:
1897:
1893:
1890:
1887:
1883:
1879:
1876:
1873:
1868:
1863:
1859:
1855:
1852:
1850:
1848:
1844:
1840:
1836:
1832:
1826:
1821:
1817:
1813:
1809:
1805:
1799:
1795:
1791:
1788:
1786:
1784:
1781:
1778:
1774:
1770:
1767:
1764:
1759:
1755:
1751:
1748:
1744:
1740:
1737:
1734:
1731:
1726:
1722:
1718:
1717:
1690:
1670:
1646:
1643:
1640:
1637:
1634:
1631:
1611:
1591:
1588:
1565:
1562:
1559:
1556:
1553:
1548:
1544:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1487:
1463:
1460:
1457:
1454:
1451:
1448:
1428:
1408:
1405:
1402:
1399:
1396:
1391:
1387:
1366:
1343:
1320:
1317:
1314:
1311:
1308:
1303:
1299:
1280:Main article:
1277:
1274:
1272:
1269:
1250:
1246:
1219:
1216:
1213:
1209:
1205:
1200:
1196:
1192:
1172:
1152:
1147:
1143:
1139:
1136:
1131:
1128:
1125:
1122:
1119:
1115:
1111:
1108:
1105:
1100:
1096:
1092:
1087:
1084:
1081:
1077:
1073:
1053:
1048:
1044:
1040:
1037:
1032:
1029:
1026:
1023:
1020:
1016:
1012:
1009:
1004:
1000:
996:
991:
988:
985:
981:
977:
974:
946:
942:
939:
936:
931:
928:
925:
921:
917:
912:
909:
906:
903:
900:
896:
892:
889:
884:
880:
876:
871:
868:
865:
861:
857:
854:
850:
844:
840:
836:
833:
831:
829:
826:
821:
817:
813:
808:
805:
802:
798:
792:
789:
785:
781:
778:
776:
774:
771:
768:
763:
759:
755:
754:
742:as the colimit
731:
728:
725:
720:
716:
695:
682:
679:
653:
650:
647:
643:
620:
616:
612:
587:
584:
581:
577:
573:
568:
564:
560:
533:
529:
526:
522:
516:
512:
508:
505:
502:
475:
472:
452:
428:
404:
382:
379:
376:
372:
368:
363:
359:
355:
350:
346:
323:
319:
302:
299:
269:
263:
259:
255:
252:
248:
244:
241:
238:
235:
230:
224:
196:
192:
171:
151:
129:
125:
95:
90:
87:
77:
72:
66:
31:, a branch of
24:
14:
13:
10:
9:
6:
4:
3:
2:
6645:
6634:
6631:
6629:
6626:
6625:
6623:
6612:
6608:
6606:
6603:
6600:
6599:Allen Hatcher
6596:
6593:
6592:
6588:
6582:
6577:
6573:
6569:
6568:
6563:
6559:
6556:
6552:
6548:
6547:
6541:
6537:
6533:
6528:
6525:
6521:
6517:
6513:
6508:
6507:
6502:
6498:
6494:
6490:
6486:
6482:
6481:
6476:
6472:
6471:
6467:
6462:
6458:
6454:
6450:
6446:
6442:
6437:
6432:
6428:
6424:
6423:
6418:
6414:
6413:May, J. Peter
6409:
6408:
6404:
6399:
6395:
6391:
6385:
6381:
6377:
6372:
6367:
6363:
6359:
6352:
6348:
6347:May, J. Peter
6343:
6339:
6337:9780226005249
6333:
6329:
6325:
6324:
6319:
6315:
6314:
6310:
6301:
6297:
6292:
6287:
6283:
6279:
6275:
6268:
6266:
6264:
6260:
6253:
6249:
6246:
6244:
6241:
6239:
6236:
6234:
6231:
6229:
6226:
6224:
6223:Ring spectrum
6221:
6220:
6216:
6214:
6212:
6208:
6204:
6199:
6195:
6191:
6187:
6186:Edwin Spanier
6183:
6175:
6173:
6155:
6146:
6129:
6126:
6119:
6115:
6109:
6102:
6088:
6079:
6052:
6048:
6042:
6011:
5998:
5995:
5992:
5989:
5969:
5966:
5960:
5937:
5934:
5926:
5910:
5905:
5902:
5898:
5895:
5891:
5873:
5868:
5865:
5850:
5846:
5842:
5838:
5824:
5820:
5817:
5796:
5791:
5788:
5784:
5781:
5777:
5759:
5754:
5751:
5736:
5732:
5728:
5724:
5710:
5706:
5703:
5695:
5676:
5671:
5667:
5636:
5628:
5572:
5558:
5536:
5522:
5521:
5520:
5504:
5475:
5448:
5418:
5413:
5398:
5388:
5368:
5355:
5350:
5340:
5332:
5327:
5314:
5311:
5308:
5299:
5295:
5280:
5265:
5255:
5252:
5243:
5230:
5217:
5215:
5201:
5177:
5171:
5168:
5165:
5160:
5157:
5146:
5143:
5138:
5134:
5125:
5124:
5123:
5103:
5100:
5097:
5094:
5084:
5073:
5067:
5064:
5061:
5053:
5049:
5045:
5042:
5037:
5033:
5025:
5024:
5023:
5021:
5005:
4985:
4962:
4959:
4956:
4903:
4900:
4890:
4879:
4876:
4871:
4867:
4858:
4857:
4856:
4854:
4848:
4840:
4838:
4835:
4833:
4829:
4825:
4824:smash product
4817:
4800:
4794:
4791:
4788:
4785:
4779:
4776:
4773:
4770:
4761:
4758:
4755:
4752:
4746:
4740:
4733:
4732:
4731:
4729:
4725:
4716:
4714:
4698:
4695:
4692:
4688:
4684:
4679:
4671:
4666:
4663:
4647:
4629:
4626:
4623:
4619:
4615:
4610:
4602:
4587:
4584:
4580:
4575:
4561:
4538:
4532:
4526:
4523:
4520:
4495:
4491:
4470:
4459:
4455:
4451:
4448:
4437:
4421:
4418:
4413:
4409:
4405:
4400:
4392:
4389:
4386:
4363:
4343:
4335:
4334:smash product
4330:
4316:
4311:
4298:
4295:in which the
4294:
4278:
4270:
4266:
4250:
4230:
4210:
4190:
4184:
4181:
4178:
4170:
4152:
4149:
4146:
4142:
4133:
4115:
4111:
4102:
4084:
4080:
4057:
4053:
4043:
4039:
4035:
4032: →
4030:
4026:
4020:
4016:
4013: →
4011:
4007:
4002:
3998:
3993:
3989:
3985:
3981:
3977:
3972:
3966:
3964:
3962:
3957:
3955:
3954:
3949:
3933:
3925:
3921:
3912:
3908:
3904:
3903:
3902:ring spectrum
3895:Ring spectrum
3894:
3892:
3890:
3872:
3869:
3866:
3862:
3850:
3846:
3837:
3829:
3827:
3822:
3809:
3783:
3774:
3760:
3740:
3735:
3725:
3714:
3704:
3691:
3686:
3676:
3660:
3657:
3651:
3648:
3619:
3606:
3598:
3592:
3577:
3573:
3567:
3556:
3546:
3543:
3537:
3534:
3515:
3501:
3454:
3434:
3424:
3410:
3406:
3401:
3393:
3390:
3385:
3376:
3372:
3368:
3364:
3351:
3348:
3334:
3313:
3310:
3306:
3302:
3297:
3294:
3291:
3283:
3280:
3275:
3272:
3269:
3260:
3256:
3252:
3248:
3243:
3235:
3232:
3227:
3218:
3208:
3206:
3203:which by the
3200:
3184:
3179:
3171:
3168:
3163:
3149:
3139:
3136:
3130:
3124:
3111:
3108:
3086:
3082:
3064:
3053:
3050:
3031:
3027:
3011:
2996:
2992:
2988:
2982:
2964:
2960:
2950:
2936:
2928:
2924:
2908:
2905:
2900:
2896:
2892:
2887:
2883:
2862:
2832:
2824:
2816:
2814:
2800:
2797:
2777:
2769:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2715:
2712:
2709:
2706:
2703:
2683:
2680:
2660:
2657:
2649:
2643:
2642:Thom spectrum
2635:
2633:
2609:
2581:
2566:
2563:
2560:
2556:
2552:
2547:
2543:
2539:
2534:
2530:
2520:
2503:
2500:
2497:
2491:
2486:
2457:
2453:
2449:
2444:
2427:
2405:
2401:
2397:
2381:
2377:
2367:
2344:
2338:
2330:
2328:
2326:
2310:
2290:
2287:
2284:
2271:
2252:
2244:
2240:
2236:
2230:
2222:
2219:
2216:
2213:
2209:
2185:
2177:
2173:
2169:
2163:
2155:
2152:
2148:
2139:
2135:
2119:
2116:
2108:
2107:unitary group
2092:
2072:
2052:
2049:
2046:
2018:
2010:
2006:
1997:
1993:
1989:
1985:
1966:
1958:
1954:
1945:
1941:
1933:
1931:
1929:
1925:
1919:
1899:
1895:
1891:
1888:
1885:
1881:
1877:
1871:
1866:
1861:
1857:
1853:
1851:
1842:
1838:
1834:
1830:
1819:
1815:
1811:
1807:
1803:
1797:
1793:
1789:
1787:
1776:
1772:
1768:
1762:
1757:
1753:
1746:
1742:
1738:
1732:
1724:
1720:
1706:
1704:
1688:
1668:
1660:
1641:
1638:
1635:
1629:
1609:
1589:
1586:
1576:
1560:
1557:
1554:
1546:
1542:
1538:
1529:
1526:
1523:
1517:
1514:
1511:
1499:
1485:
1477:
1458:
1455:
1452:
1446:
1426:
1403:
1400:
1397:
1389:
1385:
1364:
1357:
1341:
1334:
1333:abelian group
1315:
1312:
1309:
1301:
1297:
1289:
1283:
1275:
1270:
1268:
1266:
1248:
1244:
1235:
1217:
1214:
1211:
1207:
1198:
1194:
1145:
1141:
1134:
1129:
1126:
1123:
1120:
1117:
1113:
1098:
1094:
1090:
1085:
1082:
1079:
1075:
1046:
1042:
1030:
1027:
1024:
1021:
1018:
1014:
1002:
998:
989:
986:
983:
979:
975:
962:
944:
940:
929:
926:
923:
919:
910:
907:
904:
901:
898:
894:
882:
878:
869:
866:
863:
859:
852:
848:
834:
832:
819:
815:
806:
803:
800:
796:
790:
779:
777:
769:
761:
757:
743:
726:
718:
714:
693:
680:
678:
676:
672:
667:
651:
648:
645:
641:
618:
614:
603:
585:
582:
579:
575:
566:
562:
550:
527:
524:
514:
510:
503:
500:
492:
487:
473:
450:
442:
426:
418:
417:smash product
402:
380:
377:
374:
370:
361:
357:
353:
348:
344:
321:
317:
308:
300:
298:
296:
292:
288:
282:
267:
261:
257:
253:
250:
246:
242:
236:
228:
210:
194:
190:
169:
149:
127:
123:
112:
88:
85:
75:
70:
52:
50:
46:
42:
39:is an object
38:
34:
30:
19:
6571:
6565:
6545:
6535:
6531:
6515:
6511:
6487:(2): 200–8.
6484:
6478:
6426:
6425:, Series 3,
6420:
6361:
6322:
6311:Introductory
6281:
6277:
6210:
6179:
6171:
6031:
5573:The functor
5370:
5301:
5297:
5245:
5221:
5193:
5121:
5019:
4923:
4850:
4836:
4821:
4728:mapping cone
4724:triangulated
4720:
4645:
4588:
4582:
4578:
4576:
4435:
4331:
4296:
4292:
4268:
4264:
4168:
4131:
4100:
4044:
4037:
4033:
4028:
4024:
4018:
4014:
4009:
4005:
4000:
3996:
3991:
3987:
3983:
3979:
3975:
3973:
3970:
3958:
3951:
3947:
3906:
3900:
3898:
3835:
3833:
3824:
3776:
3706:
3621:
3594:
3517:
3426:
3210:
3202:
3088:
3084:
3033:
3029:
2952:
2822:
2820:
2648:Thom spectra
2645:
2636:Thom spectra
2583:
2522:
2429:
2369:
2340:
2269:
1995:
1943:
1937:
1921:
1708:
1658:
1578:
1501:
1377:, the group
1285:
1264:
964:
745:
684:
668:
549:CW complexes
488:
306:
304:
294:
284:
212:
114:
54:
41:representing
36:
26:
6203:Rainer Vogt
4171:of spectra
3911:ring axioms
3595:called the
2825:of a space
1990:of complex
491:Frank Adams
162:on a space
33:mathematics
6622:Categories
6518:: 91–148,
6254:References
6233:G-spectrum
4845:See also:
4265:eventually
3836:Ω-spectrum
3830:Ω-spectrum
3753:for every
2845:, denoted
1622:-th space
1356:CW complex
1234:connective
1064:(that is,
602:suspension
463:, denoted
287:categories
6538:: 193–251
6501:122937421
6431:CiteSeerX
6366:CiteSeerX
6300:0022-4049
6156:η
6103:θ
6100:↓
6094:↓
6075:∞
6071:Σ
6065:∞
6061:Ω
6053:η
6012:∗
5999:
5993:∈
5964:→
5956:∞
5952:Σ
5946:∞
5942:Ω
5935:θ
5899:∧
5886:∞
5882:Σ
5878:→
5860:∞
5856:Σ
5847:∧
5834:∞
5830:Σ
5818:γ
5785:∧
5772:∞
5768:Ω
5764:→
5746:∞
5742:Ω
5733:∧
5720:∞
5716:Ω
5704:ϕ
5662:∞
5658:Σ
5637:∧
5613:∞
5609:Ω
5586:∞
5582:Σ
5559:∧
5537:∗
5505:∗
5476:∗
5449:∧
5427:∞
5423:Ω
5414:∗
5404:⇆
5399:∗
5384:∞
5380:Σ
5347:Σ
5337:Ω
5333:
5325:→
5281:∗
5271:→
5266:∗
5158:−
5154:Σ
5101:∧
5081:Σ
5065:∧
5050:π
4887:Σ
4868:π
4798:Σ
4795:≅
4786:∪
4774:∪
4765:→
4756:∪
4750:→
4744:→
4696:−
4664:−
4660:Σ
4600:Σ
4562:∗
4539:∗
4533:⊔
4468:→
4452:∧
4419:∧
4390:∧
4308:Σ
4188:→
4103:-cell in
3931:→
3859:Ω
3856:→
3805:∞
3801:Σ
3795:∞
3791:Ω
3787:→
3732:Σ
3722:Ω
3718:→
3683:Σ
3673:Ω
3667:→
3661:
3644:∞
3640:Σ
3634:∞
3630:Ω
3564:Ω
3554:→
3547:
3530:∞
3526:Ω
3480:∞
3476:Ω
3398:Σ
3382:Σ
3373:≃
3360:∞
3356:Σ
3344:∞
3340:Σ
3314:⋯
3311:≃
3288:Σ
3266:Σ
3257:≃
3240:Σ
3224:Σ
3176:Σ
3160:Σ
3147:→
3140:
3120:∞
3116:Σ
3104:∞
3100:Σ
3062:→
3046:∞
3042:Σ
2993:π
2978:∞
2974:Σ
2961:π
2906:∧
2858:∞
2854:Σ
2553:≃
2540:∧
2402:π
2378:π
2285:×
2237:≅
2170:≅
2047:×
1946:compact,
1872:
1854:≅
1820:⊗
1790:≅
1754:∧
1721:π
1286:Consider
1245:π
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1191:Σ
1171:Σ
1138:Σ
1107:→
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980:π
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611:Σ
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528:∈
471:Σ
403:∧
367:→
354:∧
243:≅
209:, that is
94:→
71:∗
6349:(1995),
6320:(1974).
6217:See also
6116:→
6049:→
5906:′
5869:′
5792:′
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4483:, where
4436:homotopy
3976:function
2923:0-sphere
2268:for all
1998:. Also,
1354:. For a
1271:Examples
395:, where
307:spectrum
37:spectrum
6555:0275431
6524:0116332
6453:1806878
6398:1361891
6360:(ed.),
6176:History
5532:Spectra
5500:Spectra
5409:Spectra
5298:sending
4644:. This
4291:to the
3069:Spectra
2925:is the
2140:we get
2132:is its
2085:. Here
1986:of the
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600:of the
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1988:monoid
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