Homotopy groups of spheres - Knowledge (XXG)

36: 1264: 992: 1000: 1369: 93: 6214: 5284:-component). This exact sequence is similar to the ones coming from the Hopf fibration; the difference is that it works for all even-dimensional spheres, albeit at the expense of ignoring 2-torsion. Combining the results for odd and even dimensional spheres shows that much of the odd torsion of unstable homotopy groups is determined by the odd torsion of the stable homotopy groups. 1576: 140: 152: 6065:. In principle this gives an effective algorithm for computing all homotopy groups of any finite simply connected simplicial complex, but in practice it is too cumbersome to use for computing anything other than the first few nontrivial homotopy groups as the simplicial complex becomes much more complicated every time one kills a homotopy group. 1439: 8733:

are the products of cyclic groups of the infinite or prime power orders shown in the table. (For largely historical reasons, stable homotopy groups are usually given as products of cyclic groups of prime power order, while tables of unstable homotopy groups often give them as products of the smallest

6102:

to the calculation of homology groups of its repeated loop spaces. The Serre spectral sequence relates the homology of a space to that of its loop space, so can sometimes be used to calculate the homology of loop spaces. The Serre spectral sequence tends to have many non-zero differentials, which are

1446:

Any continuous mapping from a circle to an ordinary sphere can be continuously deformed to a one-point mapping, and so its homotopy class is trivial. One way to visualize this is to imagine a rubber-band wrapped around a frictionless ball: the band can always be slid off the ball. The homotopy group

6228:

The motivic Adams spectral sequence converges to the motivic stable homotopy groups of spheres. By comparing the motivic one over the complex numbers with the classical one, Isaksen gives rigorous proof of computations up to the 59-stem. In particular, Isaksen computes the Coker J of the 56-stem is

1235:

has exactly the same homotopy groups as a solitary point (as does a Euclidean space of any dimension), and the real plane with a point removed has the same groups as a circle, so groups alone are not enough to distinguish spaces. Although the loss of discrimination power is unfortunate, it can also

1512:

All the interesting cases of homotopy groups of spheres involve mappings from a higher-dimensional sphere onto one of lower dimension. Unfortunately, the only example which can easily be visualized is not interesting: there are no nontrivial mappings from the ordinary sphere to the circle. Hence,

1376:

Mappings from a 2-sphere to a 2-sphere can be visualized as wrapping a plastic bag around a ball and then sealing it. The sealed bag is topologically equivalent to a 2-sphere, as is the surface of the ball. The bag can be wrapped more than once by twisting it and wrapping it back over the ball.

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The low-dimensional examples of homotopy groups of spheres provide a sense of the subject, because these special cases can be visualized in ordinary 3-dimensional space. However, such visualizations are not mathematical proofs, and do not capture the possible complexity of maps between spheres.

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with a nonstandard trivialization of the normal 2-plane bundle. Until the advent of more sophisticated algebraic methods in the early 1950s (Serre) the Pontrjagin isomorphism was the main tool for computing the homotopy groups of spheres. In 1954 the Pontrjagin isomorphism was generalized by

6249:

page on positive stems. Wang and Xu develops a method using the Kahn–Priddy map to deduce Adams differentials for the sphere spectrum inductively. They give detailed argument for several Adams differentials and compute the 60 and 61-stem. A geometric corollary of their result is the sphere

1686:

which can be used to calculate some of the groups. An important method for calculating the various groups is the concept of stable algebraic topology, which finds properties that are independent of the dimensions. Typically these only hold for larger dimensions. The first such result was

791:, pronounced "modulo", means to take the topological space on the left (the disk) and in it join together as one all the points on the right (the circle). The region is 2-dimensional, which is why topology calls the resulting topological space a 2-sphere. Generalized, 381:

under addition. For example, every point on a circle can be mapped continuously onto a point of another circle; as the first point is moved around the first circle, the second point may cycle several times around the second circle, depending on the particular

3629: 1036:, where "closed curves" are continuous maps from the circle into the complex plane, and where two closed curves produce the same integral result if they are homotopic in the topological space consisting of the plane minus the points of singularity. 1027:

is a continuous path between continuous maps; two maps connected by a homotopy are said to be homotopic. The idea common to all these concepts is to discard variations that do not affect outcomes of interest. An important practical example is the

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A variation of this last approach uses a backwards version of the Adams–Novikov spectral sequence for Brown–Peterson cohomology: the limit is known, and the initial terms involve unknown stable homotopy groups of spheres that one is trying to

3428: 5267: 437:

turned out to be a central question in algebraic topology that has contributed to development of many of its fundamental techniques and has served as a stimulating focus of research. One of the main discoveries is that the homotopy groups

6045:. This is usually done by constructing suitable fibrations and taking the associated long exact sequences of homotopy groups; spectral sequences are a systematic way of organizing the complicated information that this process generates. 5683:-homomorphism is the subgroup of "well understood" or "easy" elements of the stable homotopy groups. These well understood elements account for most elements of the stable homotopy groups of spheres in small dimensions. The quotient of 5991:

used the composition product and Toda brackets to label many of the elements of homotopy groups. There are also higher Toda brackets of several elements, defined when suitable lower Toda brackets vanish. This parallels the theory of

4526: 1286:

around one's finger: it can be wrapped once, twice, three times and so on. The wrapping can be in either of two directions, and wrappings in opposite directions will cancel out after a deformation. The homotopy group

1128:" — two pointed spheres joined at their distinguished point. The two maps to be added map the upper and lower spheres separately, agreeing on the distinguished point, and composition with the pinch gives the sum map. 4364: 1381:

and so the bag is allowed to pass through itself.) The twist can be in one of two directions and opposite twists can cancel out by deformation. The total number of twists after cancellation is an integer, called the

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from purely algebraic data. One can then pullback these motivic Adams differentials to the motivic sphere, and then use the Betti realization functor to push forward them to the classical sphere. Using this method,

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hard to control, and too many ambiguities appear for higher homotopy groups. Consequently, it has been superseded by more powerful spectral sequences with fewer non-zero differentials, which give more information.

155:

This picture mimics part of the Hopf fibration, an interesting mapping from the three-dimensional sphere to the two-dimensional sphere. This mapping is the generator of the third homotopy group of the 2-sphere.

1994:, are all trivial. It therefore came as a great surprise historically that the corresponding homotopy groups are not trivial in general. This is the case that is of real importance: the higher homotopy groups 5821: 6808:

can be reduced to a question about stable homotopy groups of spheres. For example, knowledge of stable homotopy groups of degree up to 48 has been used to settle the Kervaire invariant problem in dimension

4588: 6888:. The stable homotopy groups are highlighted in blue, the unstable ones in red. Each homotopy group is the product of the cyclic groups of the orders given in the table, using the following conventions: 4103: 3281: 854:, and sweep each point on it to one point above (the North Pole), producing the northern hemisphere, and to one point below (the South Pole), producing the southern hemisphere. For each positive integer 4047: 3988: 6182:-component of the stable homotopy groups. The initial terms of the Adams spectral sequence are themselves quite hard to compute: this is sometimes done using an auxiliary spectral sequence called the 1583:

is a nontrivial mapping of the 3-sphere to the 2-sphere, and generates the third homotopy group of the 2-sphere. Each colored circle maps to the corresponding point on the 2-sphere shown bottom right.

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separately. The first few homotopy groups of spheres can be computed using ad hoc variations of the ideas above; beyond this point, most methods for computing homotopy groups of spheres are based on

6716: 1315:

under addition: a homotopy class is identified with an integer by counting the number of times a mapping in the homotopy class wraps around the circle. This integer can also be thought of as the

6000:. Every element of the stable homotopy groups of spheres can be expressed using composition products and higher Toda brackets in terms of certain well known elements, called Hopf elements. 1107:

fixed), so that two maps are in the same class if they are homotopic. Just as one point is distinguished, so one class is distinguished: all maps (or curves) homotopic to the constant map

5676: 3515: 979:. For some spaces the choice matters, but for a sphere all points are equivalent so the choice is a matter of convenience. For spheres constructed as a repeated suspension, the point 983:, which is on the equator of all the levels of suspension, works well; for the disk with collapsed rim, the point resulting from the collapse of the rim is another obvious choice. 6240:

The Kahn–Priddy map induces a map of Adams spectral sequences from the suspension spectrum of infinite real projective space to the sphere spectrum. It is surjective on the Adams

1124:

with the introduction of addition, defined via an "equator pinch". This pinch maps the equator of a pointed sphere (here a circle) to the distinguished point, producing a "

9508: 6535: 6511: 2949: 5987:

of these elements. The Toda bracket is not quite an element of a stable homotopy group, because it is only defined up to addition of products of certain other elements.

2026:

The following table gives an idea of the complexity of the higher homotopy groups even for spheres of dimension 8 or less. In this table, the entries are either a) the

3310: 5090: 10164: 1649:

introduced the notion of homotopy and used the notion of a homotopy group, without using the language of group theory. A more rigorous approach was adopted by

10724: 10124: 10075: 1466:

This result generalizes to higher dimensions. All mappings from a lower-dimensional sphere into a sphere of higher dimension are similarly trivial: if

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Most of the groups are finite. The only infinite groups are either on the main diagonal or immediately above the jagged line (highlighted in yellow).

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consisting only of the number zero. This group is often denoted by 0. Showing this rigorously requires more care, however, due to the existence of

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Gheorghe, Bogdan; Wang, Guozhen; Xu, Zhouli (2021), "The special fiber of the motivic deformation of the stable homotopy category is algebraic",

4951:. In the case of odd torsion there are more precise results; in this case there is a big difference between odd and even dimensional spheres. If 4259: 4146: 3688: 11298: 11203: 11167: 10980: 10880: 10849: 10645: 10606: 10490: 6789:

above, and therefore the stable homotopy groups of spheres, are used in the classification of possible smooth structures on a topological or

1388:

of the mapping. As in the case mappings from the circle to the circle, this degree identifies the homotopy group with the group of integers,

6110:

can be used to compute many homotopy groups of spheres; it is based on some fibrations used by Toda in his calculations of homotopy groups.

2018:, are surprisingly complex and difficult to compute, and the effort to compute them has generated a significant amount of new mathematics. 6190: 57: 1282:

The simplest case concerns the ways that a circle (1-sphere) can be wrapped around another circle. This can be visualized by wrapping a

5760: 268:

of mappings are summarized. An "addition" operation defined on these equivalence classes makes the set of equivalence classes into an

4140:

are zero. Thus the long exact sequences again break into families of split short exact sequences, implying two families of relations.

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and Heinz Hopf, on the grounds that the groups were commutative so could not be the right generalizations of the fundamental group.)

11015: 10756: 10409: 79: 4055: 3230: 6297:

is a map between motivic spheres. The Gheorghe–Wang–Xu theorem identifies the motivic Adams spectral sequence for the cofiber of

6017: 3999: 3940: 6568: 480: 5735:, and these are also considered to be "well understood".) Tables of homotopy groups of spheres sometimes omit the "easy" part 4998:. This is in some sense the best possible result, as these groups are known to have elements of this order for some values of 6641: 6431: 3655: 3084: 1692: 1491: 6202: 2968:

The groups below the jagged black line are constant along the diagonals (as indicated by the red, green and blue coloring).

10872: 10704: 10440: 1165:, and otherwise follows the same procedure. The null homotopic class acts as the identity of the group addition, and for 6487: 787:

is punctured and spread flat it produces a disk; this construction repairs the puncture, like pulling a drawstring. The

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The fact that the groups below the jagged line in the table above are constant along the diagonals is explained by the

11195: 10699: 10435: 6460: 6206: 10430: 6821: 1695:, published in 1937. Stable algebraic topology flourished between 1945 and 1966 with many important results. In 1953 50: 44: 10236:; Neisendorfer, Joseph A. (November 1979), "The double suspension and exponents of the homotopy groups of spheres", 4804:. In more recent work the argument is usually reversed, with cobordism groups computed in terms of homotopy groups. 11159: 10972: 10508:

Isaksen, Daniel C.; Wang, Guozhen; Xu, Zhouli (2023), "Stable homotopy groups of spheres: from dimension 0 to 90",

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can be lifted to a map into the real line and the nullhomotopy descends to the downstairs space (via composition).

1016: 552:

in three-dimensional space—the surface, not the solid ball—is just one example of what a sphere means in topology.

11357: 3624:{\displaystyle 0\rightarrow \pi _{i}(S^{3})\rightarrow \pi _{i}(S^{2})\rightarrow \pi _{i-1}(S^{1})\rightarrow 0.} 6790: 6628:-spheres, up to orientation-preserving diffeomorphism; the non-trivial elements of this group are represented by 1622: 1542: 6061:

to compute the first non-trivial homotopy group and then killing (eliminating) it with a fibration involving an

147:

is a nontrivial mapping of the 3-sphere to the 2-sphere, and generates the third homotopy group of the 2-sphere.

61: 11403: 6021: 5649: 5429: 4971: 2052: 1184: 484: 521:

The study of homotopy groups of spheres builds on a great deal of background material, here briefly reviewed.

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The results above about odd torsion only hold for odd-dimensional spheres: for even-dimensional spheres, the

817:, and the construction joins its ends to make a circle. An equivalent description is that the boundary of an 11032:

Wang, Guozhen; Xu, Zhouli (2017), "The triviality of the 61-stem in the stable homotopy groups of spheres",

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that measures how many times the sphere is wrapped around itself. This degree identifies the homotopy group

10694: 5833:, where multiplication is given by composition of representing maps, and any element of non-zero degree is 10084: 6837: 6334: 1660: 1655: 10743:, Graduate Studies in Mathematics, vol. 5, Providence, Rhode Island: American Mathematical Society, 11249: 11034: 10548: 10238: 6920: 6183: 6107: 5827: 4851: 3641: 1372:

Illustration of how a 2-sphere can be wrapped twice around another 2-sphere. Edges should be identified.

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Illustration of how a 2-sphere can be wrapped twice around another 2-sphere. Edges should be identified.

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was used by Serre to prove some of the results mentioned previously. He used the fact that taking the

9978: 6935: 6583: 6579: 5752: 3506: 1320: 324:

is homotopic (i.e., continuously deformable) to a constant mapping, i.e., a mapping that maps all of

10089: 1758: 10666: 6483: 5481: 1696: 1460: 1324: 1263: 1232: 1121: 991: 788: 747: 11378: 6518: 6494: 2932: 1762: 11266: 11112: 11092: 11069: 11043: 10915: 10640:, Contemporary Mathematics, vol. 181, Providence, R.I.: Amer. Math. Soc., pp. 299–316, 10565: 10517: 10354: 10292: 10255: 10233: 10229: 10115: 10025: 9470: 6801: 6475: 6468: 5838: 5485: 3423:{\displaystyle \cdots \to \pi _{i}(F)\to \pi _{i}(E)\to \pi _{i}(B)\to \pi _{i-1}(F)\to \cdots .} 1858: 1704: 1208: 1125: 522: 105: 6574:

The fact that the third stable homotopy group of spheres is cyclic of order 24, first proved by

5262:{\displaystyle \pi _{2m+k}(S^{2m})(p)=\pi _{2m+k-1}(S^{2m-1})(p)\oplus \pi _{2m+k}(S^{4m-1})(p)} 2964:) appear to be chaotic, but in fact there are many patterns, some obvious and some very subtle. 17: 3193:, and have been computed in numerous cases, although the general pattern is still elusive. For 11294: 11199: 11163: 11011: 10976: 10944: 10876: 10845: 10805: 10752: 10738: 10641: 10633: 10602: 10486: 10405: 10372: 10331: 10318: 10209: 10197: 10173: 10159: 9439: 6931: 6829: 6825: 6198: 6098:)-fold loop space by the Hurewicz theorem. This reduces the calculation of homotopy groups of 6042: 5842: 4900: 4813: 1700: 1664: 1650: 1368: 1335: 1331: 1100: 1040: 1008: 513:. Several important patterns have been established, yet much remains unknown and unexplained. 510: 506: 265: 244: 125: 92: 11244: 4862:. The 2-components are hardest to calculate, and in several ways behave differently from the 1682:

is also credited with the introduction of homotopy groups in his 1935 paper and also for the

11408: 11282: 11258: 11102: 11083: 11053: 11003: 10907: 10839: 10795: 10785: 10744: 10594: 10557: 10527: 10476: 10397: 10364: 10308: 10284: 10247: 10133: 10094: 10055: 10034: 8769: 6621: 6193:

is a more powerful version of the Adams spectral sequence replacing ordinary cohomology mod

6165: 6058: 5701:-homomorphism is considered to be the "hard" part of the stable homotopy groups of spheres ( 5540: 5465: 3088: 1883: 1875: 1688: 1683: 1675: 1118: 1033: 530: 121: 11213: 11177: 11134: 11065: 11025: 10990: 10956: 10927: 10890: 10859: 10817: 10766: 10686: 10655: 10616: 10577: 10500: 10460: 10419: 10384: 10304: 10267: 10221: 10185: 10147: 10106: 6938:) of the cyclic groups of those orders. Powers indicate repeated products. (Note that when 11338: 11209: 11173: 11130: 11061: 11021: 10986: 10952: 10923: 10886: 10855: 10835: 10813: 10762: 10682: 10651: 10612: 10590: 10573: 10539: 10496: 10456: 10415: 10380: 10300: 10263: 10217: 10181: 10143: 10102: 6833: 6402: 6256:

has a unique smooth structure, and it is the last odd dimensional one – the only ones are

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to an isomorphism expressing other groups of cobordism classes (e.g. of all manifolds) as

4720: 3651: 1679: 1532: 1384: 1029: 607: 360: 202: 10119: 2974:

The second and third rows of the table are the same starting in the third column (i.e.,

999: 11228: 10824: 10716: 8773: 6761: 6587: 6435: 6415: 5993: 5387: 5356: 4854:

with a finite abelian group. In particular the homotopy groups are determined by their

4797: 4738: 4610: 4594: 3991: 3921: 3301: 3217: 2926: 1879: 1671: 1646: 1618: 1580: 1316: 1056: 1020: 534: 396: 144: 133: 129: 10313: 10073:(1984), "Relations amongst Toda brackets and the Kervaire invariant in dimension 62", 844:

This construction, though simple, is of great theoretical importance. Take the circle

11397: 11334: 11270: 11153: 11149: 11116: 11073: 10662: 10629: 10070: 10060: 10039: 8312: 7022: 6947: 6824:

says that the stable homotopy groups of the spheres can be expressed in terms of the

6629: 6538:. The geometry near a critical point of such a map can be described by an element of 6342: 4664: 2027: 1826: 1754: 1448: 1216: 1201: 1180: 975: 337: 269: 10898:

Serre, Jean-Pierre (1951), "Homologie singulière des espaces fibrés. Applications",

10455:, Pure and Applied Mathematics, vol. 8, New York & London: Academic Press, 5468:, and it is reflected in the stable homotopy groups of spheres via the image of the 4597:

one problem, because such a fibration would imply that the failed relation is true.

1451:, with only one element, the identity element, and so it can be identified with the 260:. This summary does not distinguish between two mappings if one can be continuously 11315: 11278: 11185: 10964: 10773: 10426: 10396:, Undergraduate Texts in Mathematics, Springer-Verlag, New York, pp. 134–136, 10193: 10155: 6904: 6028: 5988: 5967: 4816:

showed that homotopy groups of spheres are all finite except for those of the form

3221: 2035: 1750: 1742: 814: 10392:

Fine, Benjamin; Rosenberger, Gerhard (1997), "8.1 Winding Number and Proof Five",

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which is contractible (it has the homotopy type of a point). In addition, because

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between the associated homotopy groups. In particular, if the map is a continuous

11107: 6012:

is any finite simplicial complex with finite fundamental group, in particular if

11057: 10998:

Walschap, Gerard (2004), "Chapter 3: Homotopy groups and bundles over spheres",

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Tables of homotopy groups of spheres are most conveniently organized by showing

6606: 6345: 6205:. The initial term is again quite hard to calculate; to do this one can use the 5830: 4521:{\displaystyle \pi _{30}(S^{16})\neq \pi _{30}(S^{31})\oplus \pi _{29}(S^{15}).} 1746: 1283: 101: 10776:(1973), "The nilpotency of elements of the stable homotopy groups of spheres", 10531: 10098: 6091:)-fold repeated loop space, which is equal to the first homology group of the ( 6049:"The method of killing homotopy groups", due to Cartan and Serre ( 11240: 11007: 10401: 10162:(1952a), "Espaces fibrés et groupes d'homotopie. I. Constructions générales", 6966: 6590: 6479: 6438: 6073: 5997: 4109: 1882:, which are generally easier to calculate; in particular, it shows that for a 1848: 1602: 1304: 1224: 822: 11319: 10948: 10831:

American Mathematical Society Translations, Ser. 2, Vol. 11, pp. 1–114 (1959)

10809: 10376: 10213: 10177: 5637:

The stable homotopy groups of spheres are the direct sum of the image of the

969:

Some theory requires selecting a fixed point on the sphere, calling the pair

11189: 11129:, The Univalent Foundations Program and Institute for Advanced Study, 2013, 10790: 6312:, which allows one to deduce motivic Adams differentials for the cofiber of 6213: 6127: 5834: 5558:

This last case accounts for the elements of unusually large finite order in

4606: 1861:, and any mapping to such a space can be deformed into a one-point mapping. 1766: 1699:

showed that there is a metastable range for the homotopy groups of spheres.

1575: 1378: 1212: 711: 617: 139: 113: 11002:, Graduate Texts in Mathematics, vol. 224, Springer-Verlag, New York, 10322: 3852:

The Hopf fibration may be constructed as follows: pairs of complex numbers

10679:

Proceedings of the International Congress of Mathematicians (Berlin, 1998)

411:

in a non-trivial fashion, and so is not equivalent to a one-point mapping.

151: 11290: 6076:

of a well behaved space shifts all the homotopy groups down by 1, so the

4359:{\displaystyle \pi _{i}(S^{8})=\pi _{i}(S^{15})\oplus \pi _{i-1}(S^{7}).} 4113: 1598: 1452: 1024: 1012: 665: 553: 526: 261: 164: 10481: 10368: 4246:{\displaystyle \pi _{i}(S^{4})=\pi _{i}(S^{7})\oplus \pi _{i-1}(S^{3}),} 3788:{\displaystyle \pi _{i}(S^{2})=\pi _{i}(S^{3})\oplus \pi _{i-1}(S^{1}).} 610:

found exactly one unit away from the origin. It is called the 2-sphere,

11262: 11245:"Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche" 11124: 10919: 10800: 10681:, Documenta Mathematica, Extra Volume, vol. II, pp. 465–472, 10598: 10569: 10296: 10259: 6896: 6035: 1312: 851: 784: 378: 10748: 6016:

is a sphere of dimension at least 2, then its homotopy groups are all

4936:. The case of 2-dimensional spheres is slightly different: the first 4719:, computed by the algebraic sum of their points, corresponding to the 10359: 8778: 6291:

method is so far the most efficient method at the prime 2. The class

685: 549: 479:

up to 90. The stable homotopy groups form the coefficient ring of an

188: 178: 10911: 10561: 10288: 10251: 10200:(1952b), "Espaces fibrés et groupes d'homotopie. II. Applications", 6734:

is the cyclic subgroup represented by homotopy spheres that bound a

2904:

The first row of this table is straightforward. The homotopy groups

395:. The first such surprise was the discovery of a mapping called the 11097: 11048: 10522: 1707:

to show that most of these groups are finite, the exceptions being

1438: 6376: 1574: 1367: 1096: 998: 990: 556:

defines a sphere rigidly, as a shape. Here are some alternatives.

150: 138: 91: 10347:

Proceedings of the Japan Academy, Series A, Mathematical Sciences

6596:

Stable homotopy groups of spheres are used to describe the group

5816:{\displaystyle \pi _{\ast }^{S}=\bigoplus _{k\geq 0}\pi _{k}^{S}} 5002:. Furthermore, the stable range can be extended in this case: if 719:

is the region contained by a circle, described by the inequality

5287:

For stable homotopy groups there are more precise results about

498:) are more erratic; nevertheless, they have been tabulated for 10935:

Serre, Jean-Pierre (1952), "Sur la suspension de Freudenthal",

10587:

Stable homotopy groups of spheres. A computer-assisted approach

10275:

Cohen, Joel M. (1968), "The decomposition of stable homotopy",

4713:

is the cobordism group of framed 0-dimensional submanifolds of

4583:{\displaystyle S^{15}\hookrightarrow S^{31}\rightarrow S^{16},} 3990:, where the bundle projection is a double covering), there are 3075:

These patterns follow from many different theoretical results.

1874:

has also been noted already, and is an easy consequence of the

1442:

A homotopy from a circle around a sphere down to a single point

201:-sphere may be defined geometrically as the set of points in a 11233:

Verhandlungen des Internationalen Mathematikerkongress, Zürich

6020:. To compute these groups, they are often factored into their 2951:, which has the same higher homotopy groups, is contractible. 29: 4775:

which corresponds to the framed 1-dimensional submanifold of

3487:

can be deformed to a point inside the higher-dimensional one

1219:), so that the two spaces have the same topology, then their 509:, a technique first applied to homotopy groups of spheres by 8768:

and 0 otherwise. The mod 8 behavior of the table comes from

6229:

0, and therefore by the work of Kervaire-Milnor, the sphere

4098:{\displaystyle S^{7}\hookrightarrow S^{15}\rightarrow S^{8}} 3276:{\displaystyle S^{1}\hookrightarrow S^{3}\rightarrow S^{2}.} 710:

This construction moves from geometry to pure topology. The

27:

How spheres of various dimensions can wrap around each other

9347: 9345: 9343: 6341:

identify these homotopy groups as certain quotients of the

4415:) and beyond. Although generalizations of the relations to 4042:{\displaystyle S^{3}\hookrightarrow S^{7}\rightarrow S^{4}} 3983:{\displaystyle S^{0}\hookrightarrow S^{1}\rightarrow S^{1}} 2086:). Extended tables of homotopy groups of spheres are given 10829:

Smooth manifolds and their applications in homotopy theory

1207:

A continuous map between two topological spaces induces a

1023:

is a function between spaces that preserves continuity. A

11126:

Homotopy type theory—univalent foundations of mathematics

9530: 9528: 6560:

sphere around the critical point maps into a topological

6354:. Under this correspondence, every nontrivial element in 1674:

in 1932. (His first paper was withdrawn on the advice of

616:, for reasons given below. The same idea applies for any 10589:, Lecture Notes in Mathematics, vol. 1423, Berlin: 7911: 5595:

have a cyclic subgroup of order 504, the denominator of

10841:

Complex cobordism and stable homotopy groups of spheres

6711:{\displaystyle \Theta _{n}/bP_{n+1}\to \pi _{n}^{S}/J,} 6474:

The stable homotopy groups of spheres are important in

3091:, which implies that the suspension homomorphism from 1063:

thus begins with continuous maps from a pointed circle

385:

The most interesting and surprising results occur when

11194:, Chicago lectures in mathematics (revised ed.), 9698: 9670: 9668: 9666: 9473: 4613:

established an isomorphism between the homotopy group

3005:). This isomorphism is induced by the Hopf fibration 1377:(There is no requirement for the continuous map to be 9594: 9582: 6644: 6521: 6497: 5763: 5652: 5093: 4540: 4430: 4262: 4149: 4058: 4002: 3943: 3691: 3518: 3313: 3233: 2935: 9953: 9464: 6930:

Where entry is a product, the homotopy group is the

6478:, which studies the structure of singular points of 6178:, and converges to something closely related to the 5498:

a cyclic group of order equal to the denominator of

4869:

In the same paper, Serre found the first place that

1911:, is isomorphic to the first nonzero homology group 1179:) — the homotopy groups of spheres — the groups are 1011:

is its continuity structure, formalized in terms of

1003:

Addition of two circle maps keeping base point fixed

995:

Homotopy of two circle maps keeping base point fixed

9781: 9779: 9617: 9615: 5705:). (Adams also introduced certain order 2 elements 821:-dimensional disk is glued to a point, producing a 136:are surprisingly complex and difficult to compute. 10937:Comptes Rendus de l'Académie des Sciences, Série I 10202:Comptes Rendus de l'Académie des Sciences, Série I 10165:Comptes Rendus de l'Académie des Sciences, Série I 9905: 9502: 6836:, leading to an identification of K-theory of the 6710: 6529: 6505: 6338: 5815: 5670: 5261: 4605:Homotopy groups of spheres are closely related to 4582: 4520: 4421:are often true, they sometimes fail; for example, 4358: 4245: 4097: 4041: 3982: 3787: 3623: 3433:For this specific bundle, each group homomorphism 3422: 3275: 2954:Beyond the first row, the higher homotopy groups ( 2943: 1829:. The reason is that a continuous mapping from an 128:, forgetting about their precise geometry. Unlike 10969:Composition methods in homotopy groups of spheres 1117:are called null homotopic. The classes become an 116:can wrap around each other. They are examples of 6919:Where the entry is ∞, the homotopy group is the 5645:-invariant, a homomorphism from these groups to 5314:-primary component of the stable homotopy group 1601:to the ordinary 2-sphere, and was discovered by 212:located at a unit distance from the origin. The 10971:, Annals of Mathematics Studies, vol. 49, 10621: 9917: 6303:as the algebraic Novikov spectral sequence for 678:)-dimensional space. For example, the 1-sphere 289:falls into three regimes, depending on whether 11231:(1932), "Höherdimensionale Homotopiegruppen", 9941: 9893: 6197:with a generalized cohomology theory, such as 5826:of the stable homotopy groups of spheres is a 1791:up to 90, and, as of 2023, unknown for larger 11321:This week's finds in mathematical physics 102 10717:"Differential topology forty-six years later" 10120:"Configurations, braids, and homotopy groups" 9998: 9351: 6870:The following table shows many of the groups 6817:for which the question was open at the time.) 6320: 5084:in terms of that of odd-dimensional spheres, 3937:Similarly (in addition to the Hopf fibration 2087: 1670:Higher homotopy groups were first defined by 1490:. This can be shown as a consequence of the 1334:) of the homotopy group with the integers is 525:provides the larger context, itself built on 415:The question of computing the homotopy group 132:, which are also topological invariants, the 8: 10778:Journal of the Mathematical Society of Japan 10725:Notices of the American Mathematical Society 10469:Memoirs of the American Mathematical Society 10125:Journal of the American Mathematical Society 6632:). More precisely, there is an injective map 6054: 6050: 5054:, and an epimorphism if equality holds. The 4663:which are "framed", i.e. have a trivialized 3481:to zero, since the lower-dimensional sphere 1191:all maps are null homotopic, then the group 1103:based on homotopy (keeping the "base point" 11390:in MacTutor History of Mathematics archive. 10467:Isaksen, Daniel C. (2019), "Stable Stems", 6775:, in which case the image has index 1 or 2. 5641:-homomorphism, and the kernel of the Adams 5495:is congruent to 2, 4, 5, or 6 modulo 8; and 3505:. Thus the long exact sequence breaks into 1931:-sphere, this immediately implies that for 1200:consists of one element, and is called the 606:This is the set of points in 3-dimensional 233:summarizes the different ways in which the 11377:O'Connor, J. J.; Robertson, E. F. (2001), 11356:O'Connor, J. J.; Robertson, E. F. (1996), 11000:Metric structures in differential geometry 10076:Journal of the London Mathematical Society 6553:, by considering the way in which a small 6327:The computation of the homotopy groups of 4873:-torsion occurs in the homotopy groups of 4681:is homotopic to a differentiable map with 1878:: this theorem links homotopy groups with 1851:. Consequently, its image is contained in 1737:. Others who worked in this area included 1087:, where maps from one pair to another map 124:terms, the structure of spheres viewed as 11106: 11096: 11047: 10799: 10789: 10546:(1963), "Groups of homotopy spheres: I", 10521: 10480: 10358: 10312: 10137: 10088: 10059: 10038: 9491: 9478: 9472: 6756:th stable homotopy group of spheres, and 6697: 6691: 6686: 6667: 6655: 6649: 6643: 6523: 6522: 6520: 6499: 6498: 6496: 5807: 5802: 5786: 5773: 5768: 5762: 5671:{\displaystyle \mathbb {Q} /\mathbb {Z} } 5664: 5663: 5658: 5654: 5653: 5651: 5232: 5210: 5176: 5148: 5120: 5098: 5092: 4571: 4558: 4545: 4539: 4506: 4493: 4477: 4464: 4448: 4435: 4429: 4344: 4325: 4309: 4296: 4280: 4267: 4261: 4231: 4212: 4196: 4183: 4167: 4154: 4148: 4089: 4076: 4063: 4057: 4033: 4020: 4007: 4001: 3974: 3961: 3948: 3942: 3773: 3754: 3738: 3725: 3709: 3696: 3690: 3606: 3587: 3571: 3558: 3542: 3529: 3517: 3390: 3368: 3346: 3324: 3312: 3264: 3251: 3238: 3232: 2937: 2936: 2934: 2055:of such groups (written, for example, as 1847:can always be deformed so that it is not 80:Learn how and when to remove this message 10069:Barratt, Michael G.; Jones, John D. S.; 9821: 9570: 8745:-component that is accounted for by the 6892:The entry "⋅" denotes the trivial group. 6212: 4850:), when the group is the product of the 2092: 1605:, who constructed a nontrivial map from 1437: 1262: 293:is less than, equal to, or greater than 43:This article includes a list of general 11371:MacTutor History of Mathematics archive 10002: 9965: 9929: 9869: 9857: 9845: 9833: 9809: 9770: 9734: 9710: 9674: 9657: 9645: 9633: 9546: 9534: 9519: 9451: 9423: 9411: 9399: 9387: 9375: 9363: 9339: 6434:, which states that every non-constant 6201:or, more usually, a piece of it called 5888:is nonzero and 12 times a generator of 5006:is odd then the double suspension from 4699:-dimensional submanifold. For example, 4116:instead of complex numbers. Here, too, 3493:. This corresponds to the vanishing of 11191:A Concise Course in Algebraic Topology 10638:The Čech centennial (Boston, MA, 1993) 9881: 9648:, Stable homotopy groups, pp. 385–393. 6800:, about the existence of manifolds of 5343:, in which case it is cyclic of order 1395:These two results generalize: for all 1095:. These maps (or equivalently, closed 336:. Therefore the homotopy group is the 11283:"Stable Algebraic Topology 1945–1966" 11143:General algebraic topology references 10740:Algebraic curves and Riemann surfaces 9797: 9758: 9722: 9686: 5702: 5679:. Roughly speaking, the image of the 4914:, and has a unique subgroup of order 4877:dimensional spheres, by showing that 4369:The three fibrations have base space 3817:at least 3, the first row shows that 3186:: they are finite abelian groups for 475:and have been computed for values of 7: 10634:"On the computation of stable stems" 10510:Publications mathématiques de l'IHÉS 9994: 9982: 9785: 9746: 9699:Cohen, Moore & Neisendorfer 1979 9621: 9606: 9558: 6084:is the first homotopy group of its ( 5464:. This period 8 pattern is known as 3286:The general theory of fiber bundles 487:. The unstable homotopy groups (for 171:-sphere for brevity, and denoted as 10667:"Toward a global understanding of π 9997:. The 2-components can be found in 9979:table of homotopy groups of spheres 6463:that every continuous map from the 6385:that is not Brunnian over the disk 5058:-torsion of the intermediate group 3206:unstable homotopy groups of spheres 1890:, the first nonzero homotopy group 1417: 10394:The Fundamental Theorem of Algebra 9954:Barratt, Jones & Mahowald 1984 9435: 6813:. (This was the smallest value of 6646: 4781:defined by the standard embedding 4593:the first non-trivial case of the 4402:) as mentioned above, but not for 3934:sends any such pair to its ratio. 3892:form a 3-sphere, and their ratios 3849:is at least 3, as observed above. 1587:The first nontrivial example with 1236:make certain computations easier. 49:it lacks sufficient corresponding 25: 11340:Stable homotopy groups of spheres 10431:"Spheres, homotopy groups of the" 10118:; Wong, Yan Loi; Wu, Jie (2006), 10023:(1966), "On the groups J(X) IV", 10001:, and the 3- and 5-components in 8749:-homomorphism is cyclic of order 6907:of that order (generally written 6375:may be represented by a Brunnian 6018:finitely generated abelian groups 5580:. For example, the stable groups 3170:stable homotopy groups of spheres 473:stable homotopy groups of spheres 6767:. This is an isomorphism unless 6418:(corresponding to an integer of 6057:) involves repeatedly using the 6034:, and calculating each of these 5080:gives the torsion at odd primes 2918:of the 1-sphere are trivial for 1857:with a point removed; this is a 1007:The distinguishing feature of a 112:describe how spheres of various 34: 18:Stable homotopy group of spheres 10332:"Remarks on zeta functions and 10046:Adams, J (1968), "Correction", 8715:Table of stable homotopy groups 6191:Adams–Novikov spectral sequence 4531:Thus there can be no fibration 505:. Most modern computations use 481:extraordinary cohomology theory 9977:These tables are based on the 9497: 9484: 8734:number of cyclic groups.) For 6679: 6486:. Such singularities arise as 6432:fundamental theorem of algebra 6235:has a unique smooth structure. 5256: 5250: 5247: 5225: 5200: 5194: 5191: 5169: 5138: 5132: 5129: 5113: 4609:classes of manifolds. In 1938 4564: 4551: 4512: 4499: 4483: 4470: 4454: 4441: 4350: 4337: 4315: 4302: 4286: 4273: 4237: 4224: 4202: 4189: 4173: 4160: 4082: 4069: 4026: 4013: 3967: 3954: 3779: 3766: 3744: 3731: 3715: 3702: 3615: 3612: 3599: 3580: 3577: 3564: 3551: 3548: 3535: 3522: 3411: 3408: 3402: 3383: 3380: 3374: 3361: 3358: 3352: 3339: 3336: 3330: 3317: 3257: 3244: 1659:where the related concepts of 1492:cellular approximation theorem 1: 11380:Marie Ennemond Camille Jordan 10873:American Mathematical Society 10869:The wild world of 4-manifolds 10844:(2nd ed.), AMS Chelsea, 10620:Also see the corrections in ( 10139:10.1090/S0894-0347-05-00507-2 9999:Isaksen, Wang & Xu (2023) 9595:O'Connor & Robertson 1996 9583:O'Connor & Robertson 2001 8776:, whose image is underlined. 6609:classes of oriented homotopy 6321:Isaksen, Wang & Xu (2023) 6080:th homotopy group of a space 5476:a cyclic group of order 2 if 4389:. A fibration does exist for 2030:0, the infinite cyclic group 1769:. The stable homotopy groups 1039:The first homotopy group, or 11108:10.4310/ACTA.2021.v226.n2.a2 10585:Kochman, Stanley O. (1990), 10061:10.1016/0040-9383(68)90010-4 10040:10.1016/0040-9383(66)90004-8 9894:Gheorghe, Wang & Xu 2021 9503:{\textstyle \pi _{1}(S^{1})} 6903:, the homotopy group is the 6840:with stable homotopy groups. 6530:{\displaystyle \mathbb {R} } 6506:{\displaystyle \mathbb {R} } 6471:to itself has a fixed point. 6391:. For example, the Hopf map 4866:-components for odd primes. 3204:, the groups are called the 2944:{\displaystyle \mathbb {R} } 1676:Pavel Sergeyevich Alexandrov 1541:is simply connected, by the 1099:) are grouped together into 756:, described by the equality 11196:University of Chicago Press 11058:10.4007/annals.2017.186.2.3 10867:Scorpan, Alexandru (2005), 10700:Encyclopedia of Mathematics 10622:Kochman & Mahowald 1995 10436:Encyclopedia of Mathematics 9918:Fine & Rosenberger 1997 9352:Isaksen, Wang & Xu 2023 8719:The stable homotopy groups 6461:Brouwer fixed point theorem 6323:computes up to the 90-stem. 6207:chromatic spectral sequence 5906:is zero because the group 5845:implies Nishida's theorem. 5291:-torsion. For example, if 4858:-components for all primes 4108:constructed using pairs of 3992:generalized Hopf fibrations 3924:, a 2-sphere. The Hopf map 3922:complex plane plus infinity 3457:, induced by the inclusion 1597:concerns mappings from the 399:, which wraps the 3-sphere 275:The problem of determining 177:— generalizes the familiar 11425: 11160:Cambridge University Press 10973:Princeton University Press 10532:10.1007/s10240-023-00139-1 9942:Kervaire & Milnor 1963 6798:Kervaire invariant problem 6335:combinatorial group theory 5354: 4750:represents a generator of 3079:Stable and unstable groups 1653:in his 1895 set of papers 671:as a geometric object in ( 110:homotopy groups of spheres 11008:10.1007/978-0-387-21826-7 10402:10.1007/978-1-4612-1928-6 9465:Homotopy type theory 2013 6791:piecewise linear manifold 6430:can be used to prove the 6203:Brown–Peterson cohomology 5870:is nonzero and generates 5361:An important subgroup of 2088:at the end of the article 1645:In the late 19th century 1531:has the real line as its 833:: written in topology as 696:: written in topology as 264:to the other; thus, only 11285:, in I. M. James (ed.), 10693:Mahowald, Mark (2001) , 10099:10.1112/jlms/s2-30.3.533 6845:Table of homotopy groups 5472:-homomorphism which is: 5430:special orthogonal group 5073:can be strictly larger. 4737:. The projection of the 4650:of cobordism classes of 3845:are isomorphic whenever 3049:do not vanish. However, 2925:, because the universal 1240:Low-dimensional examples 1223:-th homotopy groups are 1149:begins with the pointed 485:stable cohomotopy theory 405:around the usual sphere 10695:"EHP spectral sequence" 10330:Deitmar, Anton (2006), 6736:parallelizable manifold 6620:, this is the group of 6285:The motivic cofiber of 6189:At the odd primes, the 6115:Adams spectral sequence 6070:Serre spectral sequence 6063:Eilenberg–MacLane space 5443:, the homotopy groups 4966:, then elements of the 3656:suspension homomorphism 1803:As noted already, when 694:Disk with collapsed rim 471:. These are called the 64:more precise citations. 10737:Miranda, Rick (1995), 9504: 7016:, which is denoted by 6895:Where the entry is an 6838:field with one element 6822:Barratt–Priddy theorem 6712: 6531: 6507: 6333:has been reduced to a 6218: 5817: 5733:≡ 1 or 2 (mod 8) 5672: 5432:. In the stable range 5386:, is the image of the 5263: 4808:Finiteness and torsion 4584: 4522: 4360: 4247: 4099: 4043: 3984: 3789: 3682:, giving isomorphisms 3650:, these sequences are 3625: 3424: 3300:shows that there is a 3277: 3128:is an isomorphism for 2945: 1667:were also introduced. 1584: 1443: 1373: 1330:The identification (a 1279: 1004: 996: 247:continuously into the 156: 148: 118:topological invariants 97: 11359:A history of Topology 11250:Mathematische Annalen 11035:Annals of Mathematics 10900:Annals of Mathematics 10791:10.2969/jmsj/02540707 10628:Kochman, Stanley O.; 10549:Annals of Mathematics 10277:Annals of Mathematics 10239:Annals of Mathematics 9505: 6921:infinite cyclic group 6713: 6532: 6508: 6339:Berrick et al. (2006) 6216: 6184:May spectral sequence 6108:EHP spectral sequence 6004:Computational methods 5818: 5673: 5264: 5035:is an isomorphism of 4852:infinite cyclic group 4585: 4523: 4361: 4248: 4100: 4044: 3985: 3790: 3626: 3507:short exact sequences 3425: 3278: 2946: 1578: 1441: 1371: 1338:as an equality: thus 1319:of a loop around the 1301:infinite cyclic group 1266: 1075:to the pointed space 1002: 994: 952:, and the suspension 831:Suspension of equator 330:to a single point of 154: 142: 95: 11293:, pp. 665–723, 10943:, Paris: 1340–1342, 9471: 9378:, Example 0.3, p. 6. 6760:is the image of the 6642: 6586:of a compact smooth 6519: 6495: 6490:of smooth maps from 5852:is the generator of 5761: 5697:by the image of the 5650: 5304:− 1) − 2 5091: 4955:is an odd prime and 4940:-torsion occurs for 4538: 4428: 4260: 4147: 4056: 4000: 3941: 3689: 3516: 3311: 3231: 3035:the homotopy groups 2933: 1968:The homology groups 1461:space-filling curves 1231:. However, the real 1135:-th homotopy group, 1131:More generally, the 1059:) topological space 903:has as equator the ( 537:as a basic example. 237:-dimensional sphere 120:, which reflect, in 11287:History of Topology 10836:Ravenel, Douglas C. 10540:Kervaire, Michel A. 10369:10.3792/pjaa.82.141 10230:Cohen, Frederick R. 10116:Cohen, Frederick R. 9906:Berrick et al. 2006 9426:, pp. 123–125. 6696: 6593:is divisible by 16. 6484:algebraic varieties 6401:corresponds to the 5864:(of order 2), then 5812: 5778: 5576:for such values of 4992:have order at most 3304:of homotopy groups 3302:long exact sequence 1697:George W. Whitehead 1625:the homotopy group 1617:, now known as the 1101:equivalence classes 746:, and its rim (or " 456:are independent of 312:, any mapping from 266:equivalence classes 187:) and the ordinary 11263:10.1007/BF01457962 11155:Algebraic Topology 10599:10.1007/BFb0083795 10208:, Paris: 393–395, 10198:Serre, Jean-Pierre 10172:, Paris: 288–290, 10160:Serre, Jean-Pierre 9882:Wang & Xu 2017 9500: 8741:, the part of the 6802:Kervaire invariant 6708: 6682: 6567:sphere around the 6527: 6503: 6476:singularity theory 6219: 6126:term given by the 6043:spectral sequences 5839:nilpotence theorem 5813: 5798: 5797: 5764: 5668: 5351:The J-homomorphism 5259: 4580: 4518: 4356: 4243: 4095: 4039: 3980: 3785: 3621: 3420: 3273: 3172:, and are denoted 3085:suspension theorem 2941: 1859:contractible space 1705:spectral sequences 1693:suspension theorem 1585: 1525:. This is because 1444: 1374: 1280: 1209:group homomorphism 1185:finitely generated 1126:bouquet of spheres 1119:abstract algebraic 1005: 997: 523:Algebraic topology 507:spectral sequences 377:with the group of 163:-dimensional unit 157: 149: 126:topological spaces 106:algebraic topology 98: 11318:(21 April 1997), 11300:978-0-444-82375-5 11222:Historical papers 11205:978-0-226-51183-2 11169:978-0-521-79540-1 10982:978-0-691-09586-8 10902:, Second Series, 10882:978-0-8218-3749-8 10851:978-0-8218-2967-7 10647:978-0-8218-0296-0 10630:Mahowald, Mark E. 10608:978-3-540-52468-7 10492:978-1-4704-3788-6 10482:10.1090/memo/1269 10279:, Second Series, 10242:, Second Series, 10071:Mahowald, Mark E. 9836:, pp. 67–74. 9326: 9325: 8712: 8711: 8311: 8310: 6932:cartesian product 6830:classifying space 6826:plus construction 6622:smooth structures 6580:Rokhlin's theorem 6199:complex cobordism 5843:complex cobordism 5782: 4814:Jean-Pierre Serre 4690:(1, 0, ..., 0) ⊂ 4657:-submanifolds of 2902: 2901: 1701:Jean-Pierre Serre 1665:fundamental group 1543:lifting criterion 1332:group isomorphism 1041:fundamental group 1009:topological space 981:(1, 0, 0, ..., 0) 750:") is the circle 511:Jean-Pierre Serre 353:, every map from 90: 89: 82: 16:(Redirected from 11416: 11389: 11388: 11387: 11368: 11367: 11366: 11352: 11351: 11350: 11345: 11330: 11329: 11328: 11303: 11291:Elsevier Science 11273: 11235: 11216: 11180: 11137: 11119: 11110: 11100: 11084:Acta Mathematica 11076: 11051: 11028: 10993: 10959: 10930: 10893: 10862: 10820: 10803: 10793: 10769: 10733: 10721: 10707: 10689: 10658: 10619: 10580: 10534: 10525: 10503: 10484: 10463: 10443: 10422: 10387: 10362: 10325: 10316: 10270: 10224: 10188: 10150: 10141: 10114:Berrick, A. J.; 10109: 10092: 10064: 10063: 10043: 10042: 10006: 9992: 9986: 9975: 9969: 9963: 9957: 9951: 9945: 9939: 9933: 9927: 9921: 9915: 9909: 9903: 9897: 9891: 9885: 9879: 9873: 9867: 9861: 9855: 9849: 9843: 9837: 9831: 9825: 9819: 9813: 9807: 9801: 9795: 9789: 9783: 9774: 9768: 9762: 9756: 9750: 9744: 9738: 9732: 9726: 9720: 9714: 9708: 9702: 9696: 9690: 9684: 9678: 9672: 9661: 9655: 9649: 9643: 9637: 9631: 9625: 9619: 9610: 9604: 9598: 9592: 9586: 9580: 9574: 9568: 9562: 9556: 9550: 9544: 9538: 9532: 9523: 9517: 9511: 9509: 9507: 9506: 9501: 9496: 9495: 9483: 9482: 9467:, Section 8.1, " 9461: 9455: 9449: 9443: 9433: 9427: 9421: 9415: 9409: 9403: 9397: 9391: 9385: 9379: 9373: 9367: 9361: 9355: 9349: 8779: 8770:Bott periodicity 8767: 8760: 8752: 8748: 8744: 8740: 8732: 8731: 8730: 8313: 8228: 8171: 8004: 7769: 7023: 7019: 7015: 6977: 6964: 6945: 6941: 6926: 6915: 6902: 6887: 6866: 6816: 6812: 6811:2 − 2 = 62 6807: 6804:1 in dimensions 6788: 6774: 6770: 6764: 6759: 6755: 6751: 6750: 6749: 6733: 6717: 6715: 6714: 6709: 6701: 6695: 6690: 6678: 6677: 6659: 6654: 6653: 6627: 6619: 6612: 6604: 6576:Vladimir Rokhlin 6566: 6559: 6552: 6537: 6536: 6534: 6533: 6528: 6526: 6513: 6512: 6510: 6509: 6504: 6502: 6466: 6458: 6429: 6400: 6390: 6384: 6374: 6367: 6353: 6332: 6317: 6311: 6302: 6296: 6290: 6279: 6273: 6267: 6261: 6255: 6248: 6234: 6196: 6181: 6177: 6166:Steenrod algebra 6164: 6160: 6146: 6145: 6125: 6101: 6097: 6090: 6083: 6079: 6059:Hurewicz theorem 6038: 6033: 6024: 6015: 6011: 5986: 5985: 5965: 5954: 5944: 5943: 5942: 5933:are elements of 5932: 5928: 5924: 5917: 5916: 5915: 5905: 5899: 5898: 5897: 5887: 5881: 5880: 5879: 5869: 5863: 5862: 5861: 5851: 5828:supercommutative 5822: 5820: 5819: 5814: 5811: 5806: 5796: 5777: 5772: 5742: 5734: 5727: 5726: 5725: 5713: 5700: 5696: 5695: 5694: 5682: 5678: 5677: 5675: 5674: 5669: 5667: 5662: 5657: 5644: 5640: 5633: 5632: 5630: 5629: 5626: 5623: 5616: 5614: 5613: 5610: 5607: 5594: 5579: 5575: 5553: 5541:Bernoulli number 5538: 5526: 5525: 5523: 5522: 5516: 5513: 5494: 5479: 5471: 5466:Bott periodicity 5463: 5456: 5442: 5427: 5419: 5385: 5378: 5346: 5342: 5335:is divisible by 5334: 5328:vanishes unless 5327: 5326: 5325: 5313: 5309: 5305: 5290: 5283: 5279: 5268: 5266: 5265: 5260: 5246: 5245: 5224: 5223: 5190: 5189: 5168: 5167: 5128: 5127: 5112: 5111: 5083: 5072: 5057: 5053: 5038: 5034: 5019: 5005: 5001: 4997: 4991: 4969: 4965: 4954: 4950: 4939: 4935: 4924: 4917: 4913: 4898: 4894: 4876: 4872: 4865: 4861: 4857: 4849: 4845: 4829: 4790: 4780: 4774: 4768: 4767: 4749: 4736: 4718: 4712: 4698: 4694: 4680: 4662: 4656: 4649: 4643: 4642: 4630: 4601:Framed cobordism 4589: 4587: 4586: 4581: 4576: 4575: 4563: 4562: 4550: 4549: 4527: 4525: 4524: 4519: 4511: 4510: 4498: 4497: 4482: 4481: 4469: 4468: 4453: 4452: 4440: 4439: 4420: 4414: 4407: 4401: 4394: 4388: 4381: 4374: 4365: 4363: 4362: 4357: 4349: 4348: 4336: 4335: 4314: 4313: 4301: 4300: 4285: 4284: 4272: 4271: 4252: 4250: 4249: 4244: 4236: 4235: 4223: 4222: 4201: 4200: 4188: 4187: 4172: 4171: 4159: 4158: 4139: 4127: 4104: 4102: 4101: 4096: 4094: 4093: 4081: 4080: 4068: 4067: 4048: 4046: 4045: 4040: 4038: 4037: 4025: 4024: 4012: 4011: 3989: 3987: 3986: 3981: 3979: 3978: 3966: 3965: 3953: 3952: 3933: 3919: 3918: 3916: 3915: 3907: 3904: 3891: 3889: 3880: 3869: 3848: 3844: 3830: 3816: 3812: 3794: 3792: 3791: 3786: 3778: 3777: 3765: 3764: 3743: 3742: 3730: 3729: 3714: 3713: 3701: 3700: 3681: 3649: 3639: 3630: 3628: 3627: 3622: 3611: 3610: 3598: 3597: 3576: 3575: 3563: 3562: 3547: 3546: 3534: 3533: 3504: 3492: 3486: 3480: 3466: 3456: 3429: 3427: 3426: 3421: 3401: 3400: 3373: 3372: 3351: 3350: 3329: 3328: 3299: 3282: 3280: 3279: 3274: 3269: 3268: 3256: 3255: 3243: 3242: 3203: 3192: 3185: 3184: 3183: 3167: 3156: 3138: 3127: 3108: 3089:Hans Freudenthal 3070: 3063: 3048: 3034: 3024: 3014: 3004: 2997: 2963: 2950: 2948: 2947: 2942: 2940: 2924: 2917: 2831: 2830: 2756: 2755: 2664: 2663: 2583: 2582: 2568: 2567: 2551: 2550: 2518: 2517: 2507: 2506: 2459: 2458: 2448: 2447: 2423: 2422: 2348: 2347: 2337: 2336: 2312: 2311: 2093: 2085: 2076: 2075: 2065: 2050: 2041: 2034:, b) the finite 2033: 2017: 2007: 1993: 1983: 1964: 1937: 1930: 1926: 1910: 1903: 1884:simply-connected 1876:Hurewicz theorem 1873: 1856: 1846: 1836: 1832: 1824: 1810: 1806: 1794: 1790: 1786: 1736: 1720: 1689:Hans Freudenthal 1684:Hurewicz theorem 1636: 1616: 1610: 1596: 1571: 1556: 1550: 1540: 1530: 1524: 1508: 1489: 1475: 1458: 1434: 1415: 1401: 1391: 1364: 1349: 1310: 1299:is therefore an 1298: 1278: 1259: 1230: 1222: 1199: 1190: 1178: 1174: 1168: 1164: 1152: 1148: 1134: 1116: 1106: 1094: 1090: 1086: 1074: 1062: 1054: 1034:complex analysis 982: 972: 964: 958: 951: 949: 948: 934: 933: 922: 921: 909: 902: 900: 899: 886: 885: 874: 873: 861: 857: 849: 839: 820: 812: 806: 800: 782: 780: 779: 768: 767: 755: 745: 743: 742: 731: 730: 718: 705: 683: 677: 668: 663: 661: 660: 647: 646: 635: 634: 622: 615: 601: 599: 598: 587: 586: 575: 574: 561:Implicit surface 543: 531:abstract algebra 504: 497: 478: 470: 459: 455: 436: 432: 410: 404: 394: 376: 359:to itself has a 358: 352: 335: 329: 323: 317: 311: 296: 292: 288: 259: 253: 251: 242: 236: 232: 215: 211: 200: 196: 186: 176: 170: 162: 85: 78: 74: 71: 65: 60:this article by 51:inline citations 38: 37: 30: 21: 11424: 11423: 11419: 11418: 11417: 11415: 11414: 11413: 11404:Homotopy theory 11394: 11393: 11385: 11383: 11376: 11364: 11362: 11355: 11348: 11346: 11343: 11333: 11326: 11324: 11314: 11311: 11301: 11277: 11239: 11227: 11224: 11206: 11184: 11170: 11148: 11145: 11140: 11123: 11080: 11031: 11018: 10997: 10983: 10963: 10934: 10912:10.2307/1969485 10897: 10883: 10866: 10852: 10834: 10825:Pontrjagin, Lev 10772: 10759: 10749:10.1090/gsm/005 10736: 10719: 10713:Milnor, John W. 10711: 10692: 10670: 10661: 10648: 10627: 10609: 10591:Springer-Verlag 10584: 10562:10.2307/1970128 10544:Milnor, John W. 10538: 10507: 10493: 10466: 10453:Homotopy theory 10447: 10427:Fuks, Dmitry B. 10425: 10412: 10391: 10342: 10329: 10289:10.2307/1970586 10274: 10252:10.2307/1971238 10228: 10192: 10154: 10113: 10090:10.1.1.212.1163 10068: 10045: 10021:Adams, J. Frank 10019: 10015: 10010: 10009: 9993: 9989: 9976: 9972: 9964: 9960: 9952: 9948: 9940: 9936: 9928: 9924: 9916: 9912: 9904: 9900: 9892: 9888: 9880: 9876: 9868: 9864: 9856: 9852: 9844: 9840: 9832: 9828: 9820: 9816: 9808: 9804: 9796: 9792: 9784: 9777: 9769: 9765: 9757: 9753: 9745: 9741: 9733: 9729: 9721: 9717: 9709: 9705: 9697: 9693: 9685: 9681: 9673: 9664: 9656: 9652: 9644: 9640: 9632: 9628: 9620: 9613: 9605: 9601: 9593: 9589: 9581: 9577: 9569: 9565: 9557: 9553: 9545: 9541: 9533: 9526: 9518: 9514: 9487: 9474: 9469: 9468: 9462: 9458: 9450: 9446: 9434: 9430: 9422: 9418: 9410: 9406: 9398: 9394: 9386: 9382: 9374: 9370: 9362: 9358: 9350: 9341: 9336: 9331: 9280: 9268:27⋅5⋅7⋅13⋅19⋅37 9227: 9175: 9122: 9069: 9016: 8967: 8910: 8861: 8820: 8762: 8754: 8750: 8746: 8742: 8735: 8729: 8724: 8723: 8722: 8720: 8717: 8676: 8633: 8590: 8547: 8504: 8461: 8418: 8375: 8263: 8226: 8206: 8169: 8149: 8094: 8039: 8002: 7982: 7927: 7914: 7865: 7807: 7767: 7747: 7689: 7631: 7573: 7515: 7457: 7399: 7341: 7283: 7225: 7167: 7110: 7017: 7014: 7010: 7006: 6998: 6990: 6986: 6976: 6970: 6963: 6957: 6951: 6943: 6939: 6934:(equivalently, 6924: 6914: 6908: 6900: 6881: 6871: 6860: 6850: 6847: 6834:symmetric group 6828:applied to the 6814: 6810: 6805: 6787: 6781: 6772: 6771:is of the form 6768: 6762: 6757: 6753: 6748: 6743: 6742: 6741: 6739: 6732: 6723: 6663: 6645: 6640: 6639: 6625: 6614: 6610: 6603: 6597: 6561: 6554: 6546: 6539: 6517: 6516: 6515: 6493: 6492: 6491: 6488:critical points 6464: 6452: 6445: 6423: 6419: 6411: 6403:Borromean rings 6392: 6386: 6380: 6369: 6361: 6355: 6349: 6328: 6313: 6310: 6304: 6298: 6292: 6286: 6275: 6269: 6263: 6257: 6251: 6247: 6241: 6230: 6217:Borromean rings 6194: 6179: 6168: 6162: 6158: 6152: 6144: 6134: 6133: 6132: 6130: 6124: 6118: 6099: 6092: 6085: 6081: 6077: 6036: 6031: 6022: 6013: 6009: 6006: 5994:Massey products 5971: 5970: 5956: 5946: 5941: 5938: 5937: 5936: 5934: 5930: 5926: 5922: 5914: 5911: 5910: 5909: 5907: 5901: 5896: 5893: 5892: 5891: 5889: 5883: 5878: 5875: 5874: 5873: 5871: 5865: 5860: 5857: 5856: 5855: 5853: 5849: 5759: 5758: 5749: 5743:to save space. 5736: 5729: 5724: 5719: 5718: 5717: 5715: 5712: 5706: 5698: 5693: 5688: 5687: 5686: 5684: 5680: 5648: 5647: 5646: 5642: 5638: 5627: 5624: 5621: 5620: 5618: 5611: 5608: 5606: 5600: 5599: 5597: 5596: 5588: 5581: 5577: 5569: 5559: 5552:− 1 ≡ 3 (mod 4) 5544: 5537: 5528: 5517: 5514: 5512: 5503: 5502: 5500: 5499: 5492: 5477: 5469: 5458: 5457:only depend on 5450: 5444: 5433: 5421: 5413: 5399: 5390: 5380: 5372: 5362: 5359: 5353: 5344: 5336: 5329: 5324: 5319: 5318: 5317: 5315: 5311: 5307: 5292: 5288: 5281: 5280:means take the 5273: 5228: 5206: 5172: 5144: 5116: 5094: 5089: 5088: 5081: 5078:James fibration 5066: 5059: 5055: 5040: 5039:-components if 5036: 5028: 5021: 5013: 5007: 5003: 4999: 4993: 4985: 4975: 4967: 4956: 4952: 4941: 4937: 4926: 4919: 4915: 4904: 4896: 4888: 4878: 4874: 4870: 4863: 4859: 4855: 4847: 4839: 4831: 4823: 4817: 4810: 4798:homotopy groups 4782: 4776: 4766: 4763: 4762: 4761: 4755: 4751: 4741: 4724: 4714: 4706: 4700: 4696: 4682: 4668: 4658: 4654: 4641: 4636: 4635: 4634: 4632: 4624: 4614: 4603: 4567: 4554: 4541: 4536: 4535: 4502: 4489: 4473: 4460: 4444: 4431: 4426: 4425: 4416: 4409: 4403: 4396: 4390: 4383: 4376: 4370: 4340: 4321: 4305: 4292: 4276: 4263: 4258: 4257: 4227: 4208: 4192: 4179: 4163: 4150: 4145: 4144: 4133: 4129: 4121: 4117: 4085: 4072: 4059: 4054: 4053: 4029: 4016: 4003: 3998: 3997: 3970: 3957: 3944: 3939: 3938: 3925: 3914: 3908: 3905: 3903: 3897: 3896: 3894: 3893: 3888: 3882: 3881:| + | 3879: 3873: 3871: 3867: 3860: 3853: 3846: 3838: 3832: 3824: 3818: 3814: 3806: 3799: 3769: 3750: 3734: 3721: 3705: 3692: 3687: 3686: 3675: 3665: 3658: 3645: 3635: 3602: 3583: 3567: 3554: 3538: 3525: 3514: 3513: 3498: 3494: 3488: 3482: 3474: 3468: 3458: 3450: 3440: 3434: 3386: 3364: 3342: 3320: 3309: 3308: 3287: 3260: 3247: 3234: 3229: 3228: 3214: 3212:Hopf fibrations 3194: 3187: 3182: 3177: 3176: 3175: 3173: 3168:are called the 3158: 3150: 3140: 3129: 3121: 3110: 3102: 3092: 3081: 3065: 3057: 3050: 3042: 3036: 3026: 3019: 3006: 2999: 2991: 2981: 2975: 2955: 2931: 2930: 2919: 2911: 2905: 2898: 2892: 2880: 2874: 2868: 2829: 2826: 2825: 2824: 2820: 2814: 2802: 2796: 2790: 2754: 2751: 2750: 2749: 2745: 2741: 2735: 2729: 2717: 2711: 2705: 2674: 2670: 2662: 2659: 2658: 2657: 2653: 2647: 2641: 2635: 2629: 2623: 2617: 2611: 2581: 2578: 2577: 2576: 2574: 2566: 2565: 2561: 2556: 2555: 2554: 2549: 2546: 2545: 2544: 2540: 2534: 2528: 2524: 2516: 2513: 2512: 2511: 2505: 2502: 2501: 2500: 2496: 2490: 2484: 2457: 2454: 2453: 2452: 2446: 2443: 2442: 2441: 2439: 2433: 2429: 2421: 2418: 2417: 2416: 2412: 2406: 2400: 2394: 2388: 2382: 2376: 2370: 2346: 2343: 2342: 2341: 2335: 2332: 2331: 2330: 2328: 2322: 2318: 2310: 2307: 2306: 2305: 2301: 2295: 2289: 2283: 2277: 2271: 2265: 2259: 2185: 2179: 2173: 2167: 2161: 2155: 2149: 2143: 2137: 2131: 2125: 2119: 2113: 2107: 2101: 2084: 2080: 2074: 2071: 2070: 2069: 2067: 2064: 2060: 2056: 2053:direct products 2049: 2043: 2039: 2031: 2024: 2009: 2001: 1995: 1985: 1977: 1969: 1958: 1945: 1939: 1932: 1928: 1920: 1912: 1905: 1897: 1891: 1880:homology groups 1865: 1852: 1838: 1834: 1830: 1818: 1812: 1808: 1804: 1801: 1792: 1788: 1780: 1770: 1730: 1722: 1714: 1708: 1680:Witold Hurewicz 1643: 1630: 1626: 1612: 1606: 1588: 1573: 1565: 1561: 1552: 1546: 1545:, any map from 1536: 1533:universal cover 1526: 1518: 1514: 1510: 1502: 1498: 1483: 1477: 1467: 1456: 1447:is therefore a 1436: 1428: 1424: 1409: 1403: 1396: 1389: 1366: 1358: 1354: 1343: 1339: 1308: 1292: 1288: 1272: 1268: 1261: 1253: 1249: 1242: 1228: 1220: 1198: 1192: 1188: 1176: 1170: 1166: 1154: 1150: 1142: 1136: 1132: 1108: 1104: 1092: 1088: 1076: 1064: 1060: 1048: 1044: 1030:residue theorem 989: 980: 971:(sphere, point) 970: 960: 953: 947: 941: 940: 939: 932: 929: 928: 927: 920: 917: 916: 915: 911: 904: 898: 893: 892: 891: 884: 881: 880: 879: 872: 869: 868: 867: 863: 859: 855: 845: 834: 818: 808: 807:. For example, 802: 792: 778: 775: 774: 773: 766: 763: 762: 761: 757: 751: 741: 738: 737: 736: 729: 726: 725: 724: 720: 714: 697: 679: 672: 666: 659: 654: 653: 652: 645: 642: 641: 640: 633: 630: 629: 628: 624: 623:; the equation 620: 611: 608:Euclidean space 597: 594: 593: 592: 585: 582: 581: 580: 573: 570: 569: 568: 564: 546: 541: 535:homotopy groups 519: 499: 488: 476: 461: 457: 449: 439: 434: 426: 416: 406: 400: 386: 370: 364: 354: 344: 331: 325: 319: 313: 302: 294: 290: 282: 276: 255: 249: 248: 238: 234: 226: 220: 213: 206: 203:Euclidean space 198: 192: 182: 172: 168: 160: 134:homotopy groups 130:homology groups 86: 75: 69: 66: 56:Please help to 55: 39: 35: 28: 23: 22: 15: 12: 11: 5: 11422: 11420: 11412: 11411: 11406: 11396: 11395: 11392: 11391: 11374: 11353: 11335:Hatcher, Allen 11331: 11310: 11309:External links 11307: 11306: 11305: 11299: 11275: 11257:(1): 637–665, 11237: 11223: 11220: 11219: 11218: 11204: 11182: 11168: 11150:Hatcher, Allen 11144: 11141: 11139: 11138: 11121: 11091:(2): 319–407, 11078: 11042:(2): 501–580, 11029: 11016: 10995: 10981: 10961: 10932: 10906:(3): 425–505, 10895: 10881: 10864: 10850: 10832: 10822: 10784:(4): 707–732, 10770: 10757: 10734: 10709: 10690: 10668: 10663:Mahowald, Mark 10659: 10646: 10625: 10607: 10582: 10556:(3): 504–537, 10536: 10505: 10491: 10464: 10445: 10423: 10410: 10389: 10353:(8): 141–146, 10340: 10327: 10283:(2): 305–320, 10272: 10246:(3): 549–565, 10234:Moore, John C. 10226: 10190: 10152: 10132:(2): 265–326, 10111: 10083:(3): 533–550, 10066: 10016: 10014: 10011: 10008: 10007: 10003:Ravenel (2003) 9987: 9970: 9958: 9946: 9934: 9922: 9910: 9898: 9886: 9874: 9862: 9850: 9838: 9826: 9814: 9802: 9790: 9775: 9763: 9751: 9739: 9727: 9715: 9703: 9691: 9679: 9662: 9650: 9638: 9636:, p. 342. 9626: 9611: 9609:, p. 203. 9599: 9587: 9575: 9563: 9551: 9539: 9537:, p. 349. 9524: 9522:, p. 348. 9512: 9499: 9494: 9490: 9486: 9481: 9477: 9456: 9444: 9428: 9416: 9404: 9392: 9390:, p. 129. 9380: 9368: 9366:, p. xii. 9356: 9338: 9337: 9335: 9332: 9330: 9327: 9324: 9323: 9314: 9311: 9308: 9305: 9296: 9293: 9287: 9281: 9275: 9271: 9270: 9261: 9258: 9255: 9252: 9243: 9240: 9234: 9228: 9222: 9218: 9217: 9208: 9205: 9202: 9199: 9190: 9187: 9181: 9176: 9170: 9166: 9165: 9156: 9153: 9150: 9147: 9138: 9135: 9129: 9123: 9117: 9113: 9112: 9099: 9096: 9093: 9090: 9085: 9082: 9076: 9070: 9064: 9060: 9059: 9050: 9047: 9044: 9041: 9032: 9029: 9023: 9017: 9011: 9007: 9006: 8997: 8994: 8991: 8988: 8983: 8980: 8974: 8968: 8962: 8958: 8957: 8944: 8941: 8938: 8935: 8926: 8923: 8917: 8911: 8905: 8901: 8900: 8891: 8888: 8885: 8882: 8877: 8874: 8868: 8862: 8856: 8852: 8851: 8846: 8843: 8840: 8837: 8832: 8829: 8824: 8821: 8815: 8811: 8810: 8807: 8804: 8801: 8798: 8795: 8792: 8789: 8786: 8774:J-homomorphism 8725: 8716: 8713: 8710: 8709: 8706: 8703: 8700: 8697: 8694: 8691: 8688: 8685: 8682: 8671: 8667: 8666: 8663: 8660: 8657: 8654: 8651: 8648: 8645: 8642: 8639: 8628: 8624: 8623: 8620: 8617: 8614: 8611: 8608: 8605: 8602: 8599: 8596: 8585: 8581: 8580: 8577: 8574: 8571: 8568: 8565: 8562: 8559: 8556: 8553: 8542: 8538: 8537: 8534: 8531: 8528: 8525: 8522: 8519: 8516: 8513: 8510: 8499: 8495: 8494: 8491: 8488: 8485: 8482: 8479: 8476: 8473: 8470: 8467: 8456: 8452: 8451: 8448: 8445: 8442: 8439: 8436: 8433: 8430: 8427: 8424: 8413: 8409: 8408: 8405: 8402: 8399: 8396: 8393: 8390: 8387: 8384: 8381: 8370: 8366: 8365: 8360: 8355: 8350: 8345: 8340: 8335: 8330: 8325: 8320: 8309: 8308: 8305: 8302: 8299: 8296: 8293: 8290: 8287: 8284: 8281: 8278: 8275: 8272: 8269: 8258: 8254: 8253: 8250: 8247: 8244: 8241: 8238: 8235: 8232: 8229: 8224: 8221: 8218: 8215: 8212: 8201: 8197: 8196: 8193: 8190: 8187: 8184: 8181: 8178: 8175: 8172: 8167: 8164: 8161: 8158: 8155: 8144: 8140: 8139: 8136: 8133: 8130: 8127: 8124: 8121: 8118: 8115: 8112: 8109: 8106: 8103: 8100: 8089: 8085: 8084: 8081: 8078: 8075: 8072: 8069: 8066: 8063: 8060: 8057: 8054: 8051: 8048: 8045: 8034: 8030: 8029: 8026: 8023: 8020: 8017: 8014: 8011: 8008: 8005: 8000: 7997: 7994: 7991: 7988: 7977: 7973: 7972: 7969: 7966: 7963: 7960: 7957: 7954: 7951: 7948: 7945: 7942: 7939: 7936: 7933: 7922: 7918: 7917: 7910: 7907: 7904: 7901: 7898: 7895: 7892: 7889: 7886: 7883: 7880: 7877: 7874: 7871: 7860: 7856: 7855: 7852: 7849: 7846: 7843: 7840: 7837: 7834: 7831: 7828: 7825: 7822: 7819: 7816: 7813: 7802: 7798: 7797: 7794: 7791: 7788: 7785: 7782: 7779: 7776: 7773: 7770: 7765: 7762: 7759: 7756: 7753: 7742: 7738: 7737: 7734: 7731: 7728: 7725: 7722: 7719: 7716: 7713: 7710: 7707: 7704: 7701: 7698: 7695: 7684: 7680: 7679: 7676: 7673: 7670: 7667: 7664: 7661: 7658: 7655: 7652: 7649: 7646: 7643: 7640: 7637: 7626: 7622: 7621: 7618: 7615: 7612: 7609: 7606: 7603: 7600: 7597: 7594: 7591: 7588: 7585: 7582: 7579: 7568: 7564: 7563: 7560: 7557: 7554: 7551: 7548: 7545: 7542: 7539: 7536: 7533: 7530: 7527: 7524: 7521: 7510: 7506: 7505: 7502: 7499: 7496: 7493: 7490: 7487: 7484: 7481: 7478: 7475: 7472: 7469: 7466: 7463: 7452: 7448: 7447: 7444: 7441: 7438: 7435: 7432: 7429: 7426: 7423: 7420: 7417: 7414: 7411: 7408: 7405: 7394: 7390: 7389: 7386: 7383: 7380: 7377: 7374: 7371: 7368: 7365: 7362: 7359: 7356: 7353: 7350: 7347: 7336: 7332: 7331: 7328: 7325: 7322: 7319: 7316: 7313: 7310: 7307: 7304: 7301: 7298: 7295: 7292: 7289: 7278: 7274: 7273: 7270: 7267: 7264: 7261: 7258: 7255: 7252: 7249: 7246: 7243: 7240: 7237: 7234: 7231: 7220: 7216: 7215: 7212: 7209: 7206: 7203: 7200: 7197: 7194: 7191: 7188: 7185: 7182: 7179: 7176: 7173: 7162: 7158: 7157: 7154: 7151: 7148: 7145: 7142: 7139: 7136: 7133: 7130: 7127: 7124: 7121: 7118: 7116: 7105: 7101: 7100: 7095: 7090: 7085: 7080: 7075: 7070: 7065: 7060: 7055: 7050: 7045: 7040: 7035: 7030: 7020:in the table. 7012: 7008: 7004: 6996: 6988: 6980: 6979: 6972: 6959: 6953: 6928: 6917: 6910: 6893: 6873: 6852: 6846: 6843: 6842: 6841: 6818: 6794: 6783: 6777: 6776: 6744: 6727: 6720: 6719: 6718: 6707: 6704: 6700: 6694: 6689: 6685: 6681: 6676: 6673: 6670: 6666: 6662: 6658: 6652: 6648: 6634: 6633: 6630:exotic spheres 6613:-spheres (for 6599: 6594: 6572: 6569:critical value 6541: 6525: 6501: 6472: 6447: 6444:The fact that 6442: 6421: 6416:winding number 6410: 6407: 6357: 6325: 6324: 6308: 6282: 6281: 6245: 6237: 6236: 6225: 6224: 6211: 6210: 6187: 6154: 6148: 6135: 6122: 6113:The classical 6111: 6104: 6066: 6005: 6002: 5939: 5912: 5894: 5876: 5858: 5824: 5823: 5810: 5805: 5801: 5795: 5792: 5789: 5785: 5781: 5776: 5771: 5767: 5748: 5747:Ring structure 5745: 5720: 5708: 5689: 5666: 5661: 5656: 5604: 5583: 5561: 5556: 5555: 5532: 5507: 5496: 5489: 5446: 5405: 5395: 5388:J-homomorphism 5364: 5357:J-homomorphism 5355:Main article: 5352: 5349: 5320: 5270: 5269: 5258: 5255: 5252: 5249: 5244: 5241: 5238: 5235: 5231: 5227: 5222: 5219: 5216: 5213: 5209: 5205: 5202: 5199: 5196: 5193: 5188: 5185: 5182: 5179: 5175: 5171: 5166: 5163: 5160: 5157: 5154: 5151: 5147: 5143: 5140: 5137: 5134: 5131: 5126: 5123: 5119: 5115: 5110: 5107: 5104: 5101: 5097: 5061: 5023: 5009: 4977: 4880: 4846:(for positive 4833: 4819: 4809: 4806: 4800:of spaces and 4764: 4753: 4739:Hopf fibration 4702: 4652:differentiable 4637: 4631:and the group 4616: 4611:Lev Pontryagin 4602: 4599: 4595:Hopf invariant 4591: 4590: 4579: 4574: 4570: 4566: 4561: 4557: 4553: 4548: 4544: 4529: 4528: 4517: 4514: 4509: 4505: 4501: 4496: 4492: 4488: 4485: 4480: 4476: 4472: 4467: 4463: 4459: 4456: 4451: 4447: 4443: 4438: 4434: 4367: 4366: 4355: 4352: 4347: 4343: 4339: 4334: 4331: 4328: 4324: 4320: 4317: 4312: 4308: 4304: 4299: 4295: 4291: 4288: 4283: 4279: 4275: 4270: 4266: 4254: 4253: 4242: 4239: 4234: 4230: 4226: 4221: 4218: 4215: 4211: 4207: 4204: 4199: 4195: 4191: 4186: 4182: 4178: 4175: 4170: 4166: 4162: 4157: 4153: 4131: 4119: 4106: 4105: 4092: 4088: 4084: 4079: 4075: 4071: 4066: 4062: 4050: 4049: 4036: 4032: 4028: 4023: 4019: 4015: 4010: 4006: 3977: 3973: 3969: 3964: 3960: 3956: 3951: 3947: 3912: 3901: 3886: 3877: 3865: 3858: 3834: 3820: 3801: 3796: 3795: 3784: 3781: 3776: 3772: 3768: 3763: 3760: 3757: 3753: 3749: 3746: 3741: 3737: 3733: 3728: 3724: 3720: 3717: 3712: 3708: 3704: 3699: 3695: 3671: 3660: 3632: 3631: 3620: 3617: 3614: 3609: 3605: 3601: 3596: 3593: 3590: 3586: 3582: 3579: 3574: 3570: 3566: 3561: 3557: 3553: 3550: 3545: 3541: 3537: 3532: 3528: 3524: 3521: 3496: 3470: 3467:, maps all of 3446: 3436: 3431: 3430: 3419: 3416: 3413: 3410: 3407: 3404: 3399: 3396: 3393: 3389: 3385: 3382: 3379: 3376: 3371: 3367: 3363: 3360: 3357: 3354: 3349: 3345: 3341: 3338: 3335: 3332: 3327: 3323: 3319: 3316: 3284: 3283: 3272: 3267: 3263: 3259: 3254: 3250: 3246: 3241: 3237: 3218:Hopf fibration 3216:The classical 3213: 3210: 3178: 3142: 3139:. The groups 3112: 3094: 3080: 3077: 3073: 3072: 3052: 3038: 3016: 2987: 2977: 2972: 2969: 2939: 2927:covering space 2907: 2900: 2899: 2896: 2893: 2890: 2887: 2884: 2881: 2878: 2875: 2872: 2869: 2866: 2863: 2860: 2857: 2854: 2851: 2848: 2845: 2842: 2839: 2833: 2832: 2827: 2821: 2818: 2815: 2812: 2809: 2806: 2803: 2800: 2797: 2794: 2791: 2788: 2785: 2782: 2779: 2776: 2773: 2770: 2767: 2764: 2758: 2757: 2752: 2746: 2743: 2739: 2736: 2733: 2730: 2727: 2724: 2721: 2718: 2715: 2712: 2709: 2706: 2703: 2700: 2697: 2694: 2691: 2688: 2685: 2682: 2676: 2675: 2672: 2668: 2665: 2660: 2654: 2651: 2648: 2645: 2642: 2639: 2636: 2633: 2630: 2627: 2624: 2621: 2618: 2615: 2612: 2609: 2606: 2603: 2600: 2597: 2594: 2591: 2585: 2584: 2579: 2572: 2569: 2563: 2559: 2557: 2552: 2547: 2541: 2538: 2535: 2532: 2529: 2526: 2522: 2519: 2514: 2508: 2503: 2497: 2494: 2491: 2488: 2485: 2482: 2479: 2476: 2473: 2470: 2467: 2461: 2460: 2455: 2449: 2444: 2437: 2434: 2431: 2427: 2424: 2419: 2413: 2410: 2407: 2404: 2401: 2398: 2395: 2392: 2389: 2386: 2383: 2380: 2377: 2374: 2371: 2368: 2365: 2362: 2359: 2356: 2350: 2349: 2344: 2338: 2333: 2326: 2323: 2320: 2316: 2313: 2308: 2302: 2299: 2296: 2293: 2290: 2287: 2284: 2281: 2278: 2275: 2272: 2269: 2266: 2263: 2260: 2257: 2254: 2251: 2248: 2245: 2239: 2238: 2235: 2232: 2229: 2226: 2223: 2220: 2217: 2214: 2211: 2208: 2205: 2202: 2199: 2196: 2193: 2187: 2186: 2183: 2180: 2177: 2174: 2171: 2168: 2165: 2162: 2159: 2156: 2153: 2150: 2147: 2144: 2141: 2138: 2135: 2132: 2129: 2126: 2123: 2120: 2117: 2114: 2111: 2108: 2105: 2102: 2099: 2096: 2082: 2078: 2072: 2062: 2058: 2045: 2023: 2020: 1997: 1973: 1954: 1941: 1916: 1893: 1833:-sphere to an 1814: 1800: 1799:General theory 1797: 1787:are known for 1772: 1759:Daniel Isaksen 1724: 1710: 1656:Analysis situs 1651:Henri Poincaré 1647:Camille Jordan 1642: 1639: 1628: 1619:Hopf fibration 1581:Hopf fibration 1572: 1563: 1559: 1516: 1509: 1500: 1496: 1479: 1435: 1426: 1422: 1405: 1365: 1356: 1352: 1341: 1317:winding number 1290: 1270: 1260: 1251: 1247: 1241: 1238: 1194: 1187:. If for some 1175:(for positive 1138: 1057:path connected 1046: 1021:continuous map 988: 987:Homotopy group 985: 976:pointed sphere 967: 966: 942: 930: 918: 894: 882: 870: 841: 840: 827: 826: 776: 764: 739: 727: 707: 706: 690: 689: 655: 643: 631: 603: 602: 595: 583: 571: 545: 539: 518: 515: 441: 418: 413: 412: 397:Hopf fibration 383: 366: 341: 278: 222: 218:homotopy group 145:Hopf fibration 88: 87: 70:September 2022 42: 40: 33: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 11421: 11410: 11407: 11405: 11402: 11401: 11399: 11382: 11381: 11375: 11372: 11361: 11360: 11354: 11342: 11341: 11336: 11332: 11323: 11322: 11317: 11313: 11312: 11308: 11302: 11296: 11292: 11288: 11284: 11280: 11279:May, J. Peter 11276: 11272: 11268: 11264: 11260: 11256: 11252: 11251: 11246: 11242: 11238: 11234: 11230: 11226: 11225: 11221: 11215: 11211: 11207: 11201: 11197: 11193: 11192: 11187: 11186:May, J. Peter 11183: 11179: 11175: 11171: 11165: 11161: 11157: 11156: 11151: 11147: 11146: 11142: 11136: 11132: 11128: 11127: 11122: 11118: 11114: 11109: 11104: 11099: 11094: 11090: 11086: 11085: 11079: 11075: 11071: 11067: 11063: 11059: 11055: 11050: 11045: 11041: 11037: 11036: 11030: 11027: 11023: 11019: 11017:0-387-20430-X 11013: 11009: 11005: 11001: 10996: 10992: 10988: 10984: 10978: 10974: 10970: 10966: 10962: 10958: 10954: 10950: 10946: 10942: 10938: 10933: 10929: 10925: 10921: 10917: 10913: 10909: 10905: 10901: 10896: 10892: 10888: 10884: 10878: 10874: 10870: 10865: 10861: 10857: 10853: 10847: 10843: 10842: 10837: 10833: 10830: 10826: 10823: 10819: 10815: 10811: 10807: 10802: 10797: 10792: 10787: 10783: 10779: 10775: 10774:Nishida, Goro 10771: 10768: 10764: 10760: 10758:0-8218-0268-2 10754: 10750: 10746: 10742: 10741: 10735: 10731: 10727: 10726: 10718: 10714: 10710: 10706: 10702: 10701: 10696: 10691: 10688: 10684: 10680: 10676: 10674: 10664: 10660: 10657: 10653: 10649: 10643: 10639: 10635: 10631: 10626: 10623: 10618: 10614: 10610: 10604: 10600: 10596: 10592: 10588: 10583: 10579: 10575: 10571: 10567: 10563: 10559: 10555: 10551: 10550: 10545: 10541: 10537: 10533: 10529: 10524: 10519: 10515: 10511: 10506: 10502: 10498: 10494: 10488: 10483: 10478: 10474: 10470: 10465: 10462: 10458: 10454: 10450: 10446: 10442: 10438: 10437: 10432: 10428: 10424: 10421: 10417: 10413: 10411:0-387-94657-8 10407: 10403: 10399: 10395: 10390: 10386: 10382: 10378: 10374: 10370: 10366: 10361: 10356: 10352: 10348: 10344: 10339: 10336:-theory over 10335: 10328: 10324: 10320: 10315: 10310: 10306: 10302: 10298: 10294: 10290: 10286: 10282: 10278: 10273: 10269: 10265: 10261: 10257: 10253: 10249: 10245: 10241: 10240: 10235: 10231: 10227: 10223: 10219: 10215: 10211: 10207: 10203: 10199: 10195: 10194:Cartan, Henri 10191: 10187: 10183: 10179: 10175: 10171: 10167: 10166: 10161: 10157: 10156:Cartan, Henri 10153: 10149: 10145: 10140: 10135: 10131: 10127: 10126: 10121: 10117: 10112: 10108: 10104: 10100: 10096: 10091: 10086: 10082: 10078: 10077: 10072: 10067: 10062: 10057: 10053: 10049: 10041: 10036: 10032: 10028: 10027: 10022: 10018: 10017: 10012: 10004: 10000: 9996: 9991: 9988: 9984: 9980: 9974: 9971: 9967: 9962: 9959: 9955: 9950: 9947: 9943: 9938: 9935: 9932:, p. 32. 9931: 9926: 9923: 9919: 9914: 9911: 9907: 9902: 9899: 9895: 9890: 9887: 9883: 9878: 9875: 9871: 9866: 9863: 9859: 9854: 9851: 9847: 9842: 9839: 9835: 9830: 9827: 9823: 9822:Mahowald 2001 9818: 9815: 9811: 9806: 9803: 9799: 9794: 9791: 9787: 9782: 9780: 9776: 9772: 9767: 9764: 9760: 9755: 9752: 9748: 9743: 9740: 9737:, p. 25. 9736: 9731: 9728: 9724: 9719: 9716: 9712: 9707: 9704: 9700: 9695: 9692: 9688: 9683: 9680: 9676: 9671: 9669: 9667: 9663: 9659: 9654: 9651: 9647: 9642: 9639: 9635: 9630: 9627: 9623: 9618: 9616: 9612: 9608: 9603: 9600: 9596: 9591: 9588: 9584: 9579: 9576: 9573:, p. 90. 9572: 9571:Walschap 2004 9567: 9564: 9560: 9555: 9552: 9549:, p. 61. 9548: 9543: 9540: 9536: 9531: 9529: 9525: 9521: 9516: 9513: 9492: 9488: 9479: 9475: 9466: 9460: 9457: 9454:, p. 29. 9453: 9448: 9445: 9441: 9437: 9432: 9429: 9425: 9420: 9417: 9413: 9408: 9405: 9402:, p. 28. 9401: 9396: 9393: 9389: 9384: 9381: 9377: 9372: 9369: 9365: 9360: 9357: 9353: 9348: 9346: 9344: 9340: 9333: 9328: 9322: 9318: 9315: 9312: 9309: 9306: 9304: 9300: 9297: 9294: 9291: 9288: 9285: 9282: 9279: 9273: 9272: 9269: 9265: 9262: 9259: 9256: 9253: 9251: 9247: 9244: 9241: 9238: 9235: 9232: 9229: 9226: 9220: 9219: 9216: 9212: 9209: 9206: 9203: 9200: 9198: 9194: 9191: 9188: 9185: 9182: 9180: 9177: 9174: 9168: 9167: 9164: 9160: 9157: 9154: 9151: 9148: 9146: 9142: 9139: 9136: 9133: 9130: 9127: 9124: 9121: 9115: 9114: 9111: 9107: 9103: 9100: 9097: 9094: 9091: 9089: 9086: 9083: 9080: 9077: 9074: 9071: 9068: 9062: 9061: 9058: 9054: 9051: 9048: 9045: 9042: 9040: 9036: 9033: 9030: 9027: 9024: 9021: 9018: 9015: 9009: 9008: 9005: 9001: 8998: 8995: 8992: 8989: 8987: 8984: 8981: 8978: 8975: 8972: 8969: 8966: 8960: 8959: 8956: 8952: 8948: 8945: 8942: 8939: 8936: 8934: 8930: 8927: 8924: 8921: 8918: 8915: 8912: 8909: 8903: 8902: 8899: 8895: 8892: 8889: 8886: 8883: 8881: 8878: 8875: 8872: 8869: 8866: 8863: 8860: 8854: 8853: 8850: 8847: 8844: 8841: 8838: 8836: 8833: 8830: 8828: 8825: 8822: 8819: 8813: 8812: 8808: 8805: 8802: 8799: 8796: 8793: 8790: 8787: 8784: 8781: 8780: 8777: 8775: 8771: 8765: 8758: 8738: 8728: 8714: 8707: 8704: 8701: 8698: 8695: 8692: 8689: 8686: 8683: 8680: 8675: 8669: 8668: 8664: 8661: 8658: 8655: 8652: 8649: 8646: 8643: 8640: 8637: 8632: 8626: 8625: 8621: 8618: 8615: 8612: 8609: 8606: 8603: 8600: 8597: 8594: 8589: 8583: 8582: 8578: 8575: 8572: 8569: 8566: 8563: 8560: 8557: 8554: 8551: 8546: 8540: 8539: 8535: 8532: 8529: 8526: 8523: 8520: 8517: 8514: 8511: 8508: 8503: 8497: 8496: 8492: 8489: 8486: 8483: 8480: 8477: 8474: 8471: 8468: 8465: 8460: 8454: 8453: 8449: 8446: 8443: 8440: 8437: 8434: 8431: 8428: 8425: 8422: 8417: 8411: 8410: 8406: 8403: 8400: 8397: 8394: 8391: 8388: 8385: 8382: 8379: 8374: 8368: 8367: 8364: 8361: 8359: 8356: 8354: 8351: 8349: 8346: 8344: 8341: 8339: 8336: 8334: 8331: 8329: 8326: 8324: 8321: 8318: 8315: 8314: 8306: 8303: 8300: 8297: 8294: 8291: 8288: 8285: 8282: 8279: 8276: 8273: 8270: 8267: 8262: 8256: 8255: 8251: 8248: 8245: 8242: 8239: 8236: 8233: 8230: 8225: 8222: 8219: 8216: 8213: 8210: 8205: 8199: 8198: 8194: 8191: 8188: 8185: 8182: 8179: 8176: 8173: 8168: 8165: 8162: 8159: 8156: 8153: 8148: 8142: 8141: 8137: 8134: 8131: 8128: 8125: 8122: 8119: 8116: 8113: 8110: 8107: 8104: 8101: 8098: 8093: 8087: 8086: 8082: 8079: 8076: 8073: 8070: 8067: 8064: 8061: 8058: 8055: 8052: 8049: 8046: 8043: 8038: 8032: 8031: 8027: 8024: 8021: 8018: 8015: 8012: 8009: 8006: 8001: 7998: 7995: 7992: 7989: 7986: 7981: 7975: 7974: 7970: 7967: 7964: 7961: 7958: 7955: 7952: 7949: 7946: 7943: 7940: 7937: 7934: 7931: 7926: 7920: 7919: 7916: 7908: 7905: 7902: 7899: 7896: 7893: 7890: 7887: 7884: 7881: 7878: 7875: 7872: 7869: 7864: 7858: 7857: 7853: 7850: 7847: 7844: 7841: 7838: 7835: 7832: 7829: 7826: 7823: 7820: 7817: 7814: 7811: 7806: 7800: 7799: 7795: 7792: 7789: 7786: 7783: 7780: 7777: 7774: 7771: 7766: 7763: 7760: 7757: 7754: 7751: 7746: 7740: 7739: 7735: 7732: 7729: 7726: 7723: 7720: 7717: 7714: 7711: 7708: 7705: 7702: 7699: 7696: 7693: 7688: 7682: 7681: 7677: 7674: 7671: 7668: 7665: 7662: 7659: 7656: 7653: 7650: 7647: 7644: 7641: 7638: 7635: 7630: 7624: 7623: 7619: 7616: 7613: 7610: 7607: 7604: 7601: 7598: 7595: 7592: 7589: 7586: 7583: 7580: 7577: 7572: 7566: 7565: 7561: 7558: 7555: 7552: 7549: 7546: 7543: 7540: 7537: 7534: 7531: 7528: 7525: 7522: 7519: 7514: 7508: 7507: 7503: 7500: 7497: 7494: 7491: 7488: 7485: 7482: 7479: 7476: 7473: 7470: 7467: 7464: 7461: 7456: 7450: 7449: 7445: 7442: 7439: 7436: 7433: 7430: 7427: 7424: 7421: 7418: 7415: 7412: 7409: 7406: 7403: 7398: 7392: 7391: 7387: 7384: 7381: 7378: 7375: 7372: 7369: 7366: 7363: 7360: 7357: 7354: 7351: 7348: 7345: 7340: 7334: 7333: 7329: 7326: 7323: 7320: 7317: 7314: 7311: 7308: 7305: 7302: 7299: 7296: 7293: 7290: 7287: 7282: 7276: 7275: 7271: 7268: 7265: 7262: 7259: 7256: 7253: 7250: 7247: 7244: 7241: 7238: 7235: 7232: 7229: 7224: 7218: 7217: 7213: 7210: 7207: 7204: 7201: 7198: 7195: 7192: 7189: 7186: 7183: 7180: 7177: 7174: 7171: 7166: 7160: 7159: 7155: 7152: 7149: 7146: 7143: 7140: 7137: 7134: 7131: 7128: 7125: 7122: 7119: 7117: 7114: 7109: 7103: 7102: 7099: 7096: 7094: 7091: 7089: 7086: 7084: 7081: 7079: 7076: 7074: 7071: 7069: 7066: 7064: 7061: 7059: 7056: 7054: 7051: 7049: 7046: 7044: 7041: 7039: 7036: 7034: 7031: 7028: 7025: 7024: 7021: 7002: 6994: 6984: 6975: 6968: 6962: 6956: 6949: 6948:common factor 6937: 6933: 6929: 6922: 6918: 6913: 6906: 6898: 6894: 6891: 6890: 6889: 6885: 6880: 6876: 6868: 6864: 6859: 6855: 6844: 6839: 6835: 6831: 6827: 6823: 6819: 6803: 6799: 6795: 6792: 6786: 6779: 6778: 6766: 6765:-homomorphism 6747: 6737: 6730: 6726: 6721: 6705: 6702: 6698: 6692: 6687: 6683: 6674: 6671: 6668: 6664: 6660: 6656: 6650: 6638: 6637: 6636: 6635: 6631: 6623: 6617: 6608: 6602: 6595: 6592: 6589: 6585: 6581: 6577: 6573: 6570: 6564: 6557: 6550: 6544: 6489: 6485: 6481: 6477: 6473: 6470: 6467:-dimensional 6462: 6456: 6450: 6443: 6440: 6437: 6433: 6427: 6417: 6413: 6412: 6408: 6406: 6404: 6399: 6395: 6389: 6383: 6378: 6372: 6365: 6360: 6352: 6347: 6344: 6340: 6336: 6331: 6322: 6316: 6307: 6301: 6295: 6289: 6284: 6283: 6278: 6272: 6266: 6260: 6254: 6244: 6239: 6238: 6233: 6227: 6226: 6221: 6220: 6215: 6208: 6204: 6200: 6192: 6188: 6185: 6175: 6171: 6167: 6161:over the mod 6157: 6151: 6142: 6138: 6129: 6121: 6116: 6112: 6109: 6105: 6095: 6088: 6075: 6071: 6067: 6064: 6060: 6056: 6052: 6048: 6047: 6046: 6044: 6040: 6030: 6026: 6019: 6003: 6001: 5999: 5995: 5990: 5983: 5979: 5975: 5969: 5966:, there is a 5963: 5959: 5952: 5949: 5919: 5904: 5886: 5868: 5846: 5844: 5840: 5836: 5832: 5829: 5808: 5803: 5799: 5793: 5790: 5787: 5783: 5779: 5774: 5769: 5765: 5757: 5756: 5755: 5754: 5746: 5744: 5740: 5732: 5723: 5711: 5704: 5692: 5659: 5635: 5603: 5592: 5586: 5573: 5568: 5564: 5551: 5547: 5542: 5536: 5531: 5521: 5511: 5506: 5497: 5490: 5487: 5483: 5475: 5474: 5473: 5467: 5461: 5454: 5449: 5440: 5436: 5431: 5425: 5417: 5412: 5408: 5403: 5398: 5393: 5389: 5383: 5376: 5371: 5367: 5358: 5350: 5348: 5340: 5332: 5323: 5303: 5299: 5295: 5285: 5277: 5253: 5242: 5239: 5236: 5233: 5229: 5220: 5217: 5214: 5211: 5207: 5203: 5197: 5186: 5183: 5180: 5177: 5173: 5164: 5161: 5158: 5155: 5152: 5149: 5145: 5141: 5135: 5124: 5121: 5117: 5108: 5105: 5102: 5099: 5095: 5087: 5086: 5085: 5079: 5074: 5070: 5064: 5051: 5047: 5043: 5032: 5026: 5017: 5012: 4996: 4989: 4984: 4980: 4973: 4963: 4959: 4948: 4944: 4933: 4929: 4922: 4911: 4907: 4902: 4892: 4887: 4883: 4867: 4853: 4843: 4837: 4827: 4822: 4815: 4807: 4805: 4803: 4799: 4795: 4789: 4785: 4779: 4772: 4759: 4748: 4744: 4740: 4735: 4731: 4727: 4722: 4717: 4710: 4705: 4693: 4689: 4685: 4679: 4675: 4671: 4667:. Every map 4666: 4665:normal bundle 4661: 4653: 4647: 4640: 4628: 4623: 4619: 4612: 4608: 4600: 4598: 4596: 4577: 4572: 4568: 4559: 4555: 4546: 4542: 4534: 4533: 4532: 4515: 4507: 4503: 4494: 4490: 4486: 4478: 4474: 4465: 4461: 4457: 4449: 4445: 4436: 4432: 4424: 4423: 4422: 4419: 4412: 4406: 4399: 4393: 4386: 4379: 4373: 4353: 4345: 4341: 4332: 4329: 4326: 4322: 4318: 4310: 4306: 4297: 4293: 4289: 4281: 4277: 4268: 4264: 4256: 4255: 4240: 4232: 4228: 4219: 4216: 4213: 4209: 4205: 4197: 4193: 4184: 4180: 4176: 4168: 4164: 4155: 4151: 4143: 4142: 4141: 4137: 4125: 4115: 4111: 4090: 4086: 4077: 4073: 4064: 4060: 4052: 4051: 4034: 4030: 4021: 4017: 4008: 4004: 3996: 3995: 3994: 3993: 3975: 3971: 3962: 3958: 3949: 3945: 3935: 3932: 3928: 3923: 3911: 3900: 3885: 3876: 3864: 3857: 3850: 3842: 3837: 3828: 3823: 3813:vanishes for 3810: 3804: 3782: 3774: 3770: 3761: 3758: 3755: 3751: 3747: 3739: 3735: 3726: 3722: 3718: 3710: 3706: 3697: 3693: 3685: 3684: 3683: 3679: 3674: 3669: 3663: 3657: 3653: 3648: 3643: 3638: 3618: 3607: 3603: 3594: 3591: 3588: 3584: 3572: 3568: 3559: 3555: 3543: 3539: 3530: 3526: 3519: 3512: 3511: 3510: 3508: 3502: 3491: 3485: 3478: 3473: 3465: 3461: 3454: 3449: 3444: 3439: 3417: 3414: 3405: 3397: 3394: 3391: 3387: 3377: 3369: 3365: 3355: 3347: 3343: 3333: 3325: 3321: 3314: 3307: 3306: 3305: 3303: 3298: 3294: 3290: 3270: 3265: 3261: 3252: 3248: 3239: 3235: 3227: 3226: 3225: 3223: 3219: 3211: 3209: 3207: 3201: 3197: 3190: 3181: 3171: 3165: 3161: 3154: 3149: 3145: 3136: 3132: 3125: 3119: 3115: 3106: 3101: 3097: 3090: 3086: 3078: 3076: 3068: 3061: 3055: 3046: 3041: 3033: 3029: 3022: 3017: 3013: 3009: 3002: 2995: 2990: 2985: 2980: 2973: 2970: 2967: 2966: 2965: 2962: 2958: 2952: 2928: 2922: 2915: 2910: 2894: 2888: 2885: 2882: 2876: 2870: 2864: 2861: 2858: 2855: 2852: 2849: 2846: 2843: 2840: 2838: 2835: 2834: 2822: 2816: 2810: 2807: 2804: 2798: 2792: 2786: 2783: 2780: 2777: 2774: 2771: 2768: 2765: 2763: 2760: 2759: 2747: 2737: 2731: 2725: 2722: 2719: 2713: 2707: 2701: 2698: 2695: 2692: 2689: 2686: 2683: 2681: 2678: 2677: 2666: 2655: 2649: 2643: 2637: 2631: 2625: 2619: 2613: 2607: 2604: 2601: 2598: 2595: 2592: 2590: 2587: 2586: 2570: 2553: 2542: 2536: 2530: 2520: 2509: 2498: 2492: 2486: 2480: 2477: 2474: 2471: 2468: 2466: 2463: 2462: 2450: 2435: 2425: 2414: 2408: 2402: 2396: 2390: 2384: 2378: 2372: 2366: 2363: 2360: 2357: 2355: 2352: 2351: 2339: 2324: 2314: 2303: 2297: 2291: 2285: 2279: 2273: 2267: 2261: 2255: 2252: 2249: 2246: 2244: 2241: 2240: 2236: 2233: 2230: 2227: 2224: 2221: 2218: 2215: 2212: 2209: 2206: 2203: 2200: 2197: 2194: 2192: 2189: 2188: 2181: 2175: 2169: 2163: 2157: 2151: 2145: 2139: 2133: 2127: 2121: 2115: 2109: 2103: 2097: 2095: 2094: 2091: 2089: 2054: 2051:), or c) the 2048: 2037: 2036:cyclic groups 2029: 2028:trivial group 2021: 2019: 2016: 2012: 2005: 2000: 1992: 1988: 1981: 1976: 1972: 1966: 1962: 1957: 1953: 1949: 1944: 1935: 1924: 1919: 1915: 1908: 1901: 1896: 1889: 1885: 1881: 1877: 1872: 1868: 1862: 1860: 1855: 1850: 1845: 1841: 1837:-sphere with 1828: 1827:trivial group 1822: 1817: 1807:is less than 1798: 1796: 1784: 1779: 1775: 1768: 1764: 1760: 1756: 1755:Mark Mahowald 1752: 1748: 1744: 1740: 1734: 1728: 1718: 1713: 1706: 1702: 1698: 1694: 1690: 1685: 1681: 1677: 1673: 1668: 1666: 1662: 1658: 1657: 1652: 1648: 1640: 1638: 1634: 1624: 1620: 1615: 1609: 1604: 1600: 1595: 1591: 1582: 1577: 1569: 1560: 1558: 1555: 1549: 1544: 1539: 1534: 1529: 1522: 1506: 1497: 1495: 1493: 1487: 1482: 1474: 1470: 1464: 1462: 1454: 1450: 1449:trivial group 1440: 1432: 1423: 1421: 1419: 1413: 1408: 1399: 1393: 1387: 1386: 1380: 1370: 1362: 1353: 1351: 1347: 1337: 1336:often written 1333: 1328: 1326: 1322: 1318: 1314: 1307:to the group 1306: 1302: 1296: 1285: 1276: 1265: 1257: 1248: 1246: 1239: 1237: 1234: 1226: 1218: 1217:homeomorphism 1214: 1210: 1205: 1203: 1202:trivial group 1197: 1186: 1182: 1173: 1162: 1158: 1146: 1141: 1129: 1127: 1123: 1120: 1115: 1111: 1102: 1098: 1084: 1080: 1072: 1068: 1058: 1052: 1042: 1037: 1035: 1031: 1026: 1022: 1018: 1017:neighborhoods 1014: 1010: 1001: 993: 986: 984: 978: 977: 963: 957: 945: 938: 926: 914: 907: 897: 890: 878: 866: 853: 848: 843: 842: 838: 832: 829: 828: 824: 816: 811: 805: 799: 795: 790: 786: 772: 760: 754: 749: 735: 723: 717: 713: 709: 708: 704: 700: 695: 692: 691: 687: 682: 675: 670: 664:produces the 658: 651: 639: 627: 619: 614: 609: 605: 604: 591: 579: 567: 562: 559: 558: 557: 555: 551: 540: 538: 536: 532: 528: 524: 516: 514: 512: 508: 502: 495: 491: 486: 482: 474: 468: 464: 453: 448: 444: 433:for positive 430: 425: 421: 409: 403: 398: 393: 389: 384: 380: 374: 369: 362: 357: 351: 347: 342: 339: 338:trivial group 334: 328: 322: 316: 310: 306: 300: 299: 298: 286: 281: 273: 271: 270:abelian group 267: 263: 258: 246: 241: 230: 225: 219: 209: 205:of dimension 204: 195: 190: 185: 180: 175: 167:— called the 166: 153: 146: 141: 137: 135: 131: 127: 123: 119: 115: 111: 107: 103: 94: 84: 81: 73: 63: 59: 53: 52: 46: 41: 32: 31: 19: 11384:, retrieved 11379: 11363:, retrieved 11358: 11347:, retrieved 11339: 11325:, retrieved 11320: 11286: 11254: 11248: 11232: 11229:Čech, Eduard 11190: 11154: 11125: 11088: 11082: 11039: 11033: 10999: 10968: 10965:Toda, Hirosi 10940: 10936: 10903: 10899: 10868: 10840: 10828: 10781: 10777: 10739: 10732:(6): 804–809 10729: 10723: 10698: 10678: 10672: 10637: 10586: 10553: 10547: 10513: 10509: 10472: 10468: 10452: 10449:Hu, Sze-tsen 10434: 10393: 10360:math/0605429 10350: 10346: 10337: 10333: 10280: 10276: 10243: 10237: 10205: 10201: 10169: 10163: 10129: 10123: 10080: 10074: 10051: 10047: 10033:(1): 21–71, 10030: 10024: 9990: 9973: 9966:Deitmar 2006 9961: 9949: 9937: 9930:Hatcher 2002 9925: 9913: 9901: 9889: 9877: 9870:Isaksen 2019 9865: 9858:Kochman 1990 9853: 9848:, Chapter 5. 9846:Ravenel 2003 9841: 9834:Ravenel 2003 9829: 9817: 9810:Ravenel 2003 9805: 9793: 9771:Nishida 1973 9766: 9754: 9742: 9735:Ravenel 2003 9730: 9718: 9713:, p. 4. 9711:Ravenel 2003 9706: 9694: 9682: 9675:Scorpan 2005 9658:Hatcher 2002 9653: 9646:Hatcher 2002 9641: 9634:Hatcher 2002 9629: 9602: 9590: 9578: 9566: 9554: 9547:Hatcher 2002 9542: 9535:Hatcher 2002 9520:Hatcher 2002 9515: 9459: 9452:Hatcher 2002 9447: 9431: 9424:Miranda 1995 9419: 9414:, p. 3. 9412:Hatcher 2002 9407: 9400:Hatcher 2002 9395: 9388:Hatcher 2002 9383: 9376:Hatcher 2002 9371: 9364:Hatcher 2002 9359: 9320: 9316: 9302: 9298: 9289: 9283: 9277: 9267: 9263: 9249: 9245: 9236: 9230: 9224: 9214: 9210: 9196: 9192: 9183: 9178: 9172: 9162: 9158: 9144: 9140: 9131: 9125: 9119: 9109: 9105: 9101: 9087: 9078: 9072: 9066: 9056: 9052: 9038: 9034: 9025: 9019: 9013: 9003: 8999: 8985: 8976: 8970: 8964: 8954: 8950: 8946: 8932: 8928: 8919: 8913: 8907: 8897: 8893: 8879: 8870: 8864: 8858: 8848: 8834: 8826: 8817: 8782: 8763: 8756: 8736: 8726: 8718: 8678: 8673: 8635: 8630: 8592: 8587: 8549: 8544: 8506: 8501: 8463: 8458: 8420: 8415: 8377: 8372: 8362: 8357: 8352: 8347: 8342: 8337: 8332: 8327: 8322: 8316: 8265: 8260: 8208: 8203: 8151: 8146: 8096: 8091: 8041: 8036: 7984: 7979: 7929: 7924: 7912: 7867: 7862: 7809: 7804: 7749: 7744: 7691: 7686: 7633: 7628: 7575: 7570: 7517: 7512: 7459: 7454: 7401: 7396: 7343: 7338: 7285: 7280: 7227: 7222: 7169: 7164: 7112: 7107: 7097: 7092: 7087: 7082: 7077: 7072: 7067: 7062: 7057: 7052: 7047: 7042: 7037: 7032: 7026: 7000: 6992: 6982: 6981: 6973: 6960: 6954: 6911: 6905:cyclic group 6883: 6878: 6874: 6869: 6862: 6857: 6853: 6848: 6784: 6745: 6728: 6724: 6615: 6600: 6562: 6555: 6548: 6542: 6459:implies the 6454: 6448: 6425: 6409:Applications 6397: 6393: 6387: 6381: 6370: 6363: 6358: 6350: 6346:braid groups 6329: 6326: 6314: 6305: 6299: 6293: 6287: 6276: 6270: 6264: 6258: 6252: 6242: 6231: 6173: 6169: 6155: 6149: 6140: 6136: 6119: 6093: 6086: 6007: 5989:Hiroshi Toda 5981: 5977: 5973: 5968:Toda bracket 5961: 5957: 5950: 5947: 5920: 5918:is trivial. 5902: 5884: 5866: 5848:Example: If 5847: 5825: 5750: 5738: 5730: 5721: 5709: 5690: 5636: 5601: 5590: 5584: 5571: 5566: 5562: 5557: 5549: 5545: 5534: 5529: 5519: 5509: 5504: 5459: 5452: 5447: 5438: 5434: 5428:denotes the 5423: 5415: 5410: 5406: 5401: 5396: 5391: 5381: 5374: 5369: 5365: 5360: 5338: 5330: 5321: 5306:for a prime 5301: 5297: 5293: 5286: 5275: 5271: 5075: 5068: 5062: 5049: 5045: 5041: 5030: 5024: 5015: 5010: 4994: 4987: 4982: 4978: 4961: 4957: 4946: 4942: 4931: 4927: 4920: 4909: 4905: 4890: 4885: 4881: 4868: 4841: 4835: 4825: 4820: 4811: 4787: 4783: 4777: 4770: 4757: 4746: 4742: 4733: 4729: 4725: 4715: 4708: 4703: 4691: 4687: 4683: 4677: 4673: 4669: 4659: 4645: 4638: 4626: 4621: 4617: 4604: 4592: 4530: 4417: 4410: 4404: 4397: 4391: 4384: 4377: 4371: 4368: 4135: 4123: 4107: 3936: 3930: 3926: 3909: 3898: 3883: 3874: 3862: 3855: 3851: 3840: 3835: 3826: 3821: 3808: 3802: 3797: 3677: 3672: 3667: 3661: 3646: 3636: 3633: 3500: 3489: 3483: 3476: 3471: 3463: 3459: 3452: 3447: 3442: 3437: 3432: 3296: 3292: 3288: 3285: 3222:fiber bundle 3215: 3205: 3199: 3195: 3188: 3179: 3169: 3163: 3159: 3152: 3147: 3143: 3134: 3130: 3123: 3117: 3113: 3104: 3099: 3095: 3082: 3074: 3066: 3059: 3053: 3044: 3039: 3031: 3027: 3023:= 2, 3, 4, 5 3020: 3011: 3007: 3000: 2993: 2988: 2983: 2978: 2960: 2956: 2953: 2920: 2913: 2908: 2903: 2836: 2761: 2679: 2588: 2464: 2353: 2242: 2190: 2046: 2042:(written as 2025: 2014: 2010: 2003: 1998: 1990: 1986: 1979: 1974: 1970: 1967: 1960: 1955: 1951: 1947: 1942: 1933: 1922: 1917: 1913: 1906: 1899: 1894: 1887: 1870: 1866: 1863: 1853: 1843: 1839: 1820: 1815: 1802: 1782: 1777: 1773: 1763:Guozhen Wang 1751:J. Peter May 1743:Hiroshi Toda 1732: 1726: 1716: 1711: 1669: 1654: 1644: 1632: 1613: 1607: 1593: 1589: 1586: 1567: 1553: 1547: 1537: 1527: 1520: 1511: 1504: 1485: 1480: 1472: 1468: 1465: 1445: 1430: 1411: 1406: 1397: 1394: 1383: 1375: 1360: 1345: 1329: 1294: 1281: 1274: 1267:Elements of 1255: 1243: 1206: 1195: 1171: 1160: 1156: 1144: 1139: 1130: 1113: 1109: 1082: 1078: 1070: 1066: 1050: 1038: 1006: 974: 968: 961: 955: 943: 936: 924: 912: 905: 895: 888: 876: 864: 846: 836: 830: 815:line segment 809: 803: 797: 793: 770: 758: 752: 733: 721: 715: 702: 698: 693: 680: 673: 656: 649: 637: 625: 612: 589: 577: 565: 560: 548:An ordinary 547: 520: 500: 493: 489: 472: 466: 462: 451: 446: 442: 428: 423: 419: 414: 407: 401: 391: 387: 372: 367: 355: 349: 345: 332: 326: 320: 314: 308: 304: 284: 279: 274: 256: 252:-dimensional 239: 228: 223: 217: 207: 193: 183: 173: 158: 109: 102:mathematical 99: 76: 67: 48: 11241:Hopf, Heinz 10801:2433/220059 10516:: 107–243, 10044:. See also 9983:Toda (1962) 9463:See, e.g., 6806:2 − 2 6780:The groups 6773:2 − 2 6607:h-cobordism 6480:smooth maps 6441:has a zero. 6025:-components 5831:graded ring 5491:trivial if 4110:quaternions 1747:Frank Adams 1672:Eduard Čech 1621:. This map 1284:rubber band 62:introducing 11398:Categories 11386:2007-11-14 11365:2007-11-14 11349:2007-10-20 11327:2007-10-09 11316:Baez, John 11098:1809.09290 11049:1601.02184 10523:2001.04511 10054:(3): 331, 9798:Cohen 1968 9759:Adams 1966 9723:Serre 1952 9687:Serre 1951 9438:, p. 9329:References 9321:3⋅25⋅11⋅41 8759:− 1) 6967:isomorphic 6936:direct sum 6591:4-manifold 6578:, implies 6439:polynomial 6337:question. 6128:Ext groups 6074:loop space 5998:cohomology 5753:direct sum 5703:Adams 1966 5484:to 0 or 1 5341:− 1) 3920:cover the 3890:| = 1 3642:suspension 1927:. For the 1849:surjective 1603:Heinz Hopf 1305:isomorphic 1225:isomorphic 850:to be the 823:CW complex 517:Background 114:dimensions 45:references 11281:(1999a), 11271:123533891 11188:(1999b), 11117:119303902 11074:119147703 10949:0764-4442 10810:0025-5645 10705:EMS Press 10441:EMS Press 10429:(2001) , 10377:0386-2194 10214:0764-4442 10178:0764-4442 10085:CiteSeerX 9995:Fuks 2001 9786:Toda 1962 9747:Fuks 2001 9622:May 1999a 9607:Čech 1932 9559:Hopf 1931 9476:π 9197:9⋅7⋅11⋅31 9095:16⋅2⋅9⋅5 8240:504⋅24⋅2 8170:24⋅12⋅4⋅2 8016:240⋅24⋅4 6684:π 6680:→ 6647:Θ 6584:signature 6582:that the 6565:− 1 6558:− 1 6027:for each 5835:nilpotent 5800:π 5791:≥ 5784:⨁ 5770:∗ 5766:π 5482:congruent 5394: : π 5310:then the 5240:− 5208:π 5204:⊕ 5184:− 5162:− 5146:π 5096:π 4972:component 4812:In 1951, 4794:René Thom 4695:a framed 4607:cobordism 4565:→ 4552:↪ 4491:π 4487:⊕ 4462:π 4458:≠ 4433:π 4387:= 1, 2, 3 4330:− 4323:π 4319:⊕ 4294:π 4265:π 4217:− 4210:π 4206:⊕ 4181:π 4152:π 4114:octonions 4083:→ 4070:↪ 4027:→ 4014:↪ 3968:→ 3955:↪ 3759:− 3752:π 3748:⊕ 3723:π 3694:π 3616:→ 3592:− 3585:π 3581:→ 3556:π 3552:→ 3527:π 3523:→ 3415:⋯ 3412:→ 3395:− 3388:π 3384:→ 3366:π 3362:→ 3344:π 3340:→ 3322:π 3318:→ 3315:⋯ 3258:→ 3245:↪ 2038:of order 1864:The case 1767:Zhouli Xu 1739:José Adem 1623:generates 1379:injective 1303:, and is 1213:bijection 1169:equal to 1013:open sets 959:produces 910:)-sphere 908:− 1 801:produces 618:dimension 483:, called 122:algebraic 104:field of 11243:(1931), 11152:(2002), 10967:(1962), 10838:(2003), 10715:(2011), 10665:(1998), 10632:(1995), 10475:(1269), 10451:(1959), 10323:16591550 10048:Topology 10026:Topology 9049:4⋅2⋅3⋅5 8772:via the 8761:divides 8705:∞⋅264⋅2 8687:264⋅4⋅2 8521:∞⋅480⋅2 8252:480⋅4⋅2 8227:120⋅12⋅2 8003:2520⋅6⋅2 7768:120⋅12⋅2 6946:have no 6545:−1 6451:−1 6343:Brunnian 5984:⟩ 5972:⟨ 5900:, while 5527:, where 5420:, where 5052:+ 1) − 3 4728: : 4723:of maps 4672: : 1663:and the 1661:homology 1599:3-sphere 1476:, then 1453:subgroup 1313:integers 1227:for all 1153:-sphere 1025:homotopy 862:-sphere 748:boundary 554:Geometry 527:topology 382:mapping. 379:integers 262:deformed 197:). The 11409:Spheres 11214:1702278 11178:1867354 11135:3204653 11066:3702672 11026:2045823 10991:0143217 10957:0046048 10928:0045386 10920:1969485 10891:2136212 10860:0860042 10818:0341485 10767:1326604 10687:1648096 10656:1320997 10617:1052407 10578:0148075 10570:1970128 10501:4046815 10461:0106454 10420:1454356 10385:2279281 10305:0231377 10297:1970586 10268:0554384 10260:1971238 10222:0046046 10186:0046045 10148:2188127 10107:0810962 10013:Sources 9436:Hu 1959 9266:⋅8⋅4⋅2⋅ 9239:⋅4⋅2⋅3 9075:⋅4⋅2⋅3 9057:3⋅25⋅11 9039:27⋅7⋅19 8650:24⋅8⋅2 8289:1056⋅8 8234:24⋅6⋅2 8028:48⋅4⋅2 7947:24⋅6⋅2 7003:) = Z×Z 6983:Example 6897:integer 6832:of the 6752:is the 6436:complex 6039:-groups 5631:⁠ 5619:⁠ 5615:⁠ 5598:⁠ 5524:⁠ 5501:⁠ 5462:(mod 8) 5272:(where 4949:− 3 + 1 4901:torsion 4895:has no 4802:spectra 3917:⁠ 3895:⁠ 3654:by the 1984:, with 1904:, with 1641:History 1323:in the 1181:abelian 852:equator 785:balloon 783:. If a 669:-sphere 544:-sphere 533:, with 503:< 20 303:0 < 254:sphere 243:can be 100:In the 58:improve 11297: 11269: 11212: 11202: 11176: 11166: 11133: 11115: 11072: 11064: 11024: 11014: 10989: 10979: 10955: 10947: 10926: 10918: 10889: 10879: 10858: 10848: 10816: 10808: 10765: 10755: 10685: 10654: 10644: 10615: 10605: 10576: 10568: 10499: 10489: 10459: 10418: 10408: 10383: 10375: 10321: 10314:224450 10311: 10303: 10295: 10266: 10258: 10220: 10212: 10184: 10176: 10146: 10105: 10087: 9313:4⋅2⋅3 9307:4⋅2⋅5 9295:4⋅2⋅3 9215:3⋅5⋅17 9163:3⋅5⋅29 9110:5⋅7⋅13 9088:8⋅3⋅23 9084:8⋅2⋅3 9004:3⋅5⋅17 8955:5⋅7⋅13 8849:16⋅3⋅5 8739:> 5 8708:264⋅2 8702:264⋅2 8699:264⋅2 8696:264⋅2 8693:264⋅2 8690:264⋅2 8684:264⋅2 8656:8⋅4⋅2 8536:480⋅2 8533:480⋅2 8530:480⋅2 8527:480⋅2 8524:480⋅2 8518:480⋅2 8515:480⋅2 8512:480⋅2 8307:264⋅2 8304:264⋅2 8301:264⋅6 8298:264⋅2 8295:264⋅2 8292:264⋅2 8286:264⋅2 8283:132⋅2 8280:132⋅2 8249:8⋅4⋅2 8132:240⋅2 8120:504⋅2 8083:240⋅2 8080:240⋅2 8077:240⋅2 8074:240⋅2 8071:120⋅2 8068:120⋅2 7851:∞⋅504 7842:504⋅2 7839:504⋅2 7836:504⋅2 7833:504⋅4 7830:504⋅2 7605:∞⋅120 6722:where 6428:) = Z) 6373:> 2 6274:, and 5882:, and 5837:; the 5486:modulo 5404:)) → π 5379:, for 5296:< 2 4908:< 2 4721:degree 4382:, for 3872:| 3798:Since 3634:Since 2923:> 1 2008:, for 1909:> 0 1886:space 1825:, the 1765:, and 1400:> 0 1385:degree 1321:origin 1097:curves 1055:of a ( 935:+ ⋯ + 887:+ ⋯ + 858:, the 686:circle 648:+ ⋯ + 550:sphere 361:degree 245:mapped 189:sphere 179:circle 165:sphere 108:, the 47:, but 11344:(PDF) 11267:S2CID 11113:S2CID 11093:arXiv 11070:S2CID 11044:arXiv 10916:JSTOR 10720:(PDF) 10566:JSTOR 10518:arXiv 10355:arXiv 10293:JSTOR 10256:JSTOR 9334:Notes 9319:⋅4⋅2⋅ 9301:⋅2⋅9⋅ 9286:⋅2⋅3 9248:⋅4⋅2⋅ 9233:⋅4⋅2 9213:⋅4⋅2⋅ 9143:⋅8⋅2⋅ 9134:⋅2⋅3 9128:⋅4⋅2 9104:⋅4⋅2⋅ 9055:⋅2⋅3⋅ 8949:⋅8⋅2⋅ 8880:8⋅9⋅7 8558:24⋅2 8469:16⋅2 8277:12⋅2 8246:24⋅2 8243:24⋅2 8237:24⋅2 8231:24⋅2 8223:12⋅2 8220:12⋅2 8166:12⋅2 8065:60⋅6 8062:30⋅2 8025:16⋅2 8022:16⋅2 8019:16⋅4 8013:24⋅4 8010:12⋅2 7915:below 7879:84⋅2 7827:84⋅2 7824:84⋅2 7821:12⋅2 7787:12⋅2 7784:24⋅2 7781:24⋅2 7778:24⋅2 7775:72⋅2 7772:72⋅2 7764:12⋅2 7657:24⋅2 7535:24⋅3 7361:∞⋅12 6995:) = π 6457:) = Z 6379:over 6377:braid 6223:find. 6055:1952b 6051:1952a 6029:prime 5945:with 5543:, if 5539:is a 5044:< 4773:) = Z 4760:) = Ω 4711:) = Z 4375:with 3870:with 3670:) → π 3652:split 3640:is a 3445:) → π 3220:is a 3162:> 3157:with 3133:> 3062:) = 0 2986:) = π 2959:> 2022:Table 2013:> 1989:> 1963:) = Z 1842:< 1823:) = 0 1703:used 1635:) = Z 1592:> 1570:) = Z 1523:) = 0 1507:) = 0 1488:) = 0 1471:< 1433:) = 0 1418:below 1416:(see 1414:) = Z 1363:) = Z 1348:) = Z 1325:plane 1258:) = Z 1233:plane 1122:group 1091:into 813:is a 789:slash 684:is a 492:< 390:> 343:When 307:< 11295:ISBN 11200:ISBN 11164:ISBN 11012:ISBN 10977:ISBN 10945:ISSN 10877:ISBN 10846:ISBN 10806:ISSN 10753:ISBN 10642:ISBN 10603:ISBN 10487:ISBN 10406:ISBN 10373:ISSN 10319:PMID 10210:ISSN 10174:ISSN 9310:4⋅2 9260:4⋅2 9254:2⋅3 9242:8⋅2 9207:2⋅3 9155:4⋅2 9149:2⋅3 9137:2⋅3 9098:2⋅3 9046:2⋅3 9043:2⋅3 9031:4⋅2 8996:2⋅3 8982:2⋅3 8943:2⋅2 8937:8⋅3 8933:3⋅11 8925:8⋅2 8876:2⋅3 8665:8⋅2 8662:8⋅2 8659:8⋅2 8653:8⋅2 8647:8⋅2 8644:8⋅2 8641:8⋅2 8613:∞⋅2 8475:4⋅2 8472:8⋅2 8429:∞⋅3 8183:6⋅2 8174:4⋅2 8163:6⋅2 8114:6⋅2 8111:6⋅2 8007:6⋅2 7971:6⋅2 7968:6⋅2 7959:6⋅2 7950:6⋅2 7891:240 7854:504 7848:504 7845:504 7790:6⋅2 7727:∞⋅2 7620:240 7617:240 7614:240 7611:240 7608:240 7602:120 7106:< 6997:9+10 6942:and 6820:The 6796:The 6588:spin 6469:ball 6414:The 6368:for 6117:has 6106:The 6068:The 5955:and 5929:and 5925:and 5751:The 5728:for 5451:(SO( 5400:(SO( 4925:and 4128:and 3831:and 3109:to 3064:for 3025:and 3018:For 2998:for 1950:) = 1721:and 1579:The 1183:and 1019:. A 712:disk 529:and 460:for 301:For 216:-th 159:The 143:The 11369:in 11259:doi 11255:104 11103:doi 11089:226 11054:doi 11040:186 11004:doi 10941:234 10908:doi 10796:hdl 10786:doi 10745:doi 10595:doi 10558:doi 10528:doi 10514:137 10477:doi 10473:262 10398:doi 10365:doi 10309:PMC 10285:doi 10248:doi 10244:110 10206:234 10170:234 10134:doi 10095:doi 10056:doi 10035:doi 9981:in 9440:107 9292:⋅2 9276:72+ 9223:64+ 9211:128 9195:⋅2⋅ 9186:⋅2 9171:56+ 9161:⋅3⋅ 9118:48+ 9108:⋅3⋅ 9081:⋅2 9065:40+ 9037:⋅2⋅ 9028:⋅2 9022:⋅2 9012:32+ 9002:⋅2⋅ 8986:8⋅3 8979:⋅2 8973:⋅2 8963:24+ 8953:⋅3⋅ 8931:⋅2⋅ 8922:⋅2 8916:⋅2 8906:16+ 8898:3⋅5 8896:⋅2⋅ 8873:⋅2 8867:⋅2 8835:8⋅3 8766:+ 1 8753:if 8672:19+ 8629:18+ 8586:17+ 8543:16+ 8500:15+ 8457:14+ 8414:13+ 8371:12+ 8259:19+ 8202:18+ 8145:17+ 8108:30 8090:16+ 8059:30 8056:30 8053:30 8035:15+ 7999:30 7978:14+ 7923:13+ 7913:See 7903:12 7861:12+ 7803:11+ 7743:10+ 7645:15 7599:60 7596:30 7593:15 7590:15 7413:12 7388:24 7385:24 7382:24 7379:24 7376:24 7373:24 7370:24 7367:24 7364:24 7358:12 7018:∞⋅2 6969:to 6965:is 6624:on 6618:≠ 4 6605:of 6514:to 6482:or 6348:of 6153:, Z 6131:Ext 6008:If 5996:in 5964:= 0 5953:= 0 5921:If 5841:on 5737:im( 5714:of 5628:504 5587:+11 5548:= 4 5480:is 5441:+ 2 5422:SO( 5384:≥ 2 5333:+ 1 5020:to 4974:of 4964:+ 1 4960:= 2 4945:= 2 4934:− 3 4930:= 2 4923:≥ 3 4918:if 4912:− 3 4903:if 4830:or 4413:= 4 4400:= 0 4380:= 2 4112:or 3644:of 3191:≠ 0 3166:+ 1 3137:+ 1 3087:of 3069:≥ 6 3003:≥ 3 2897:120 2895:Z×Z 2819:120 2493:Z×Z 2077:= Z 2066:or 1936:≥ 2 1691:'s 1611:to 1551:to 1455:of 1420:). 1311:of 1215:(a 1032:of 1015:or 950:= 1 901:= 1 781:= 1 744:≤ 1 676:+ 1 662:= 1 600:= 1 496:+ 2 469:+ 2 318:to 210:+ 1 11400:: 11337:, 11289:, 11265:, 11253:, 11247:, 11210:MR 11208:, 11198:, 11174:MR 11172:, 11162:, 11158:, 11131:MR 11111:, 11101:, 11087:, 11068:, 11062:MR 11060:, 11052:, 11038:, 11022:MR 11020:, 11010:, 10987:MR 10985:, 10975:, 10953:MR 10951:, 10939:, 10924:MR 10922:, 10914:, 10904:54 10887:MR 10885:, 10875:, 10871:, 10856:MR 10854:, 10827:, 10814:MR 10812:, 10804:, 10794:, 10782:25 10780:, 10763:MR 10761:, 10751:, 10730:58 10728:, 10722:, 10703:, 10697:, 10683:MR 10677:, 10675:)" 10652:MR 10650:, 10636:, 10613:MR 10611:, 10601:, 10593:, 10574:MR 10572:, 10564:, 10554:77 10552:, 10542:; 10526:, 10512:, 10497:MR 10495:, 10485:, 10471:, 10457:MR 10439:, 10433:, 10416:MR 10414:, 10404:, 10381:MR 10379:, 10371:, 10363:, 10351:82 10349:, 10345:, 10317:, 10307:, 10301:MR 10299:, 10291:, 10281:87 10264:MR 10262:, 10254:, 10232:; 10218:MR 10216:, 10204:, 10196:; 10182:MR 10180:, 10168:, 10158:; 10144:MR 10142:, 10130:19 10128:, 10122:, 10103:MR 10101:, 10093:, 10081:30 10079:, 10050:, 10029:, 9778:^ 9665:^ 9614:^ 9527:^ 9510:". 9342:^ 9317:32 9264:16 9257:2 9204:⋅ 9201:4 9189:2 9159:16 9152:2 9102:32 9092:8 9053:16 9000:64 8993:3 8990:2 8947:16 8940:2 8894:32 8890:2 8887:3 8884:⋅ 8857:8+ 8845:2 8842:⋅ 8839:⋅ 8831:2 8823:∞ 8816:0+ 8809:7 8806:6 8803:5 8800:4 8797:3 8794:2 8791:1 8788:0 8785:→ 8755:2( 8681:) 8638:) 8622:2 8619:2 8616:2 8610:2 8607:2 8604:2 8601:2 8598:2 8595:) 8579:2 8576:2 8573:2 8570:2 8567:2 8564:2 8561:2 8555:2 8552:) 8509:) 8493:2 8490:2 8487:2 8484:2 8481:2 8478:2 8466:) 8450:3 8447:3 8444:3 8441:3 8438:3 8435:3 8432:3 8426:6 8423:) 8407:⋅ 8404:⋅ 8401:⋅ 8398:⋅ 8395:⋅ 8392:⋅ 8389:⋅ 8386:⋅ 8383:2 8380:) 8319:→ 8274:⋅ 8271:⋅ 8268:) 8217:⋅ 8214:⋅ 8211:) 8195:2 8192:2 8189:2 8186:2 8180:2 8177:2 8160:⋅ 8157:⋅ 8154:) 8138:2 8135:2 8129:2 8126:2 8123:2 8117:2 8105:⋅ 8102:⋅ 8099:) 8050:⋅ 8047:⋅ 8044:) 7996:6 7993:⋅ 7990:⋅ 7987:) 7965:6 7962:6 7956:6 7953:6 7944:6 7941:2 7938:⋅ 7935:⋅ 7932:) 7909:2 7906:2 7900:⋅ 7897:⋅ 7894:⋅ 7888:2 7885:2 7882:2 7876:⋅ 7873:⋅ 7870:) 7818:⋅ 7815:⋅ 7812:) 7796:6 7793:6 7761:2 7758:⋅ 7755:⋅ 7752:) 7736:2 7733:2 7730:2 7724:2 7721:2 7718:2 7715:2 7712:2 7709:2 7706:2 7703:2 7700:⋅ 7697:⋅ 7694:) 7685:9+ 7678:2 7675:2 7672:2 7669:2 7666:2 7663:2 7660:2 7654:2 7651:2 7648:2 7642:⋅ 7639:⋅ 7636:) 7627:8+ 7587:3 7584:⋅ 7581:⋅ 7578:) 7569:7+ 7562:2 7559:2 7556:2 7553:2 7550:2 7547:2 7544:2 7541:2 7538:2 7532:3 7529:2 7526:⋅ 7523:⋅ 7520:) 7511:6+ 7504:⋅ 7501:⋅ 7498:⋅ 7495:⋅ 7492:⋅ 7489:⋅ 7486:⋅ 7483:∞ 7480:2 7477:2 7474:2 7471:2 7468:⋅ 7465:⋅ 7462:) 7453:5+ 7446:⋅ 7443:⋅ 7440:⋅ 7437:⋅ 7434:⋅ 7431:⋅ 7428:⋅ 7425:⋅ 7422:2 7419:2 7416:2 7410:⋅ 7407:⋅ 7404:) 7395:4+ 7355:2 7352:⋅ 7349:⋅ 7346:) 7337:3+ 7330:2 7327:2 7324:2 7321:2 7318:2 7315:2 7312:2 7309:2 7306:2 7303:2 7300:2 7297:2 7294:⋅ 7291:⋅ 7288:) 7279:2+ 7272:2 7269:2 7266:2 7263:2 7260:2 7257:2 7254:2 7251:2 7248:2 7245:2 7242:2 7239:∞ 7236:⋅ 7233:⋅ 7230:) 7221:1+ 7214:∞ 7211:∞ 7208:∞ 7205:∞ 7202:∞ 7199:∞ 7196:∞ 7193:∞ 7190:∞ 7187:∞ 7184:∞ 7181:∞ 7178:∞ 7175:2 7172:) 7163:0+ 7156:⋅ 7153:⋅ 7150:⋅ 7147:⋅ 7144:⋅ 7141:⋅ 7138:⋅ 7135:⋅ 7132:⋅ 7129:⋅ 7126:⋅ 7123:⋅ 7120:⋅ 7115:) 7029:→ 7011:×Z 7007:×Z 6989:19 6985:: 6978:.) 6974:ab 6958:×Z 6950:, 6923:, 6916:). 6899:, 6867:. 6738:, 6731:+1 6725:bP 6405:. 6396:→ 6306:BP 6268:, 6262:, 6147:(Z 6096:−1 6089:−1 6053:, 5980:, 5976:, 5634:. 5617:= 5612:12 5488:8; 5455:)) 5437:≥ 5347:. 5337:2( 5065:+1 5027:+2 4838:−1 4786:⊂ 4745:→ 4732:→ 4686:= 4676:→ 4573:16 4560:31 4547:15 4508:15 4495:29 4479:31 4466:30 4450:16 4437:30 4311:15 4078:15 3929:→ 3805:−1 3664:−1 3619:0. 3509:, 3462:→ 3295:→ 3291:→ 3224:: 3208:. 3202:+1 3198:≤ 3120:+1 3056:+4 3030:≥ 3010:→ 2929:, 2886:0 2883:0 2879:24 2862:Z 2859:0 2856:0 2853:0 2850:0 2847:0 2844:0 2841:0 2808:0 2805:0 2801:24 2784:Z 2781:0 2778:0 2775:0 2772:0 2769:0 2766:0 2742:×Z 2740:24 2734:60 2723:Z 2720:0 2716:24 2699:Z 2696:0 2693:0 2690:0 2687:0 2684:0 2671:×Z 2669:72 2646:30 2622:24 2605:Z 2602:0 2599:0 2596:0 2593:0 2575:×Z 2573:84 2562:×Z 2560:12 2533:15 2525:×Z 2523:24 2495:12 2478:Z 2475:0 2472:0 2469:0 2440:×Z 2438:84 2430:×Z 2428:12 2405:15 2381:12 2364:Z 2361:0 2358:0 2329:×Z 2327:84 2319:×Z 2317:12 2294:15 2270:12 2253:Z 2250:Z 2247:0 2237:0 2234:0 2231:0 2228:0 2225:0 2222:0 2219:0 2216:0 2213:0 2210:0 2207:0 2204:0 2201:0 2198:0 2195:Z 2184:15 2178:14 2172:13 2166:12 2160:11 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