Homotopy groups of spheres - Knowledge (XXG)

25: 1253: 981: 989: 1358: 82: 6203: 5273:-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. 1565: 129: 141: 6054:. 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. 1428: 8722:

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

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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

1435:

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

6217:

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

1224:

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

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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,

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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

6238:

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

1675:

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

780:, 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, 370:

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

3618: 1025:, 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. 1016:

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

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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

6034:. 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. 5672:-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 5980:

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

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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

1117:" — 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. 4353: 1370:

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.

144:

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.

1983:, 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 5810: 6797:

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

4577: 6877:. 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: 4092: 3270: 843:, 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 4036: 3977: 6171:-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 1572:

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

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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

5989:. 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. 1096:

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

5665: 3504: 968:. 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 972:, 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. 6229:

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

1113:

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 "

9497: 6524: 6500: 2938: 5976:

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.

2015:

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

3299: 5079: 10153: 1638:

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

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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",

4940:. 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 4248: 4135: 3677: 11287: 11192: 11156: 10969: 10869: 10838: 10634: 10595: 10479: 6778:

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

1377:

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,

6099:

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.

2007:, are surprisingly complex and difficult to compute, and the effort to compute them has generated a significant amount of new mathematics. 6179: 46: 1271:

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

5749: 257:

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

4129:

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.)

11004: 10745: 10398: 68: 4044: 3219: 6286:

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

6006: 3988: 3929: 6557: 469: 5724:, and these are also considered to be "well understood".) Tables of homotopy groups of spheres sometimes omit the "easy" part 4987:. This is in some sense the best possible result, as these groups are known to have elements of this order for some values of 6630: 6420: 3644: 3073: 1681: 1480: 6191: 2957:

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

10861: 10693: 10429: 1154:, and otherwise follows the same procedure. The null homotopic class acts as the identity of the group addition, and for 6476: 776:

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

11184: 10688: 10424: 6449: 6195: 10419: 6810: 1684:, published in 1937. Stable algebraic topology flourished between 1945 and 1966 with many important results. In 1953 39: 33: 10225:; Neisendorfer, Joseph A. (November 1979), "The double suspension and exponents of the homotopy groups of spheres", 4793:. In more recent work the argument is usually reversed, with cobordism groups computed in terms of homotopy groups. 11148: 10961: 10497:

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).

1005: 541:

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

11346: 3613:{\displaystyle 0\rightarrow \pi _{i}(S^{3})\rightarrow \pi _{i}(S^{2})\rightarrow \pi _{i-1}(S^{1})\rightarrow 0.} 6779: 6617:-spheres, up to orientation-preserving diffeomorphism; the non-trivial elements of this group are represented by 1611: 1531: 6050:

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

136:

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

50: 11392: 6010: 5638: 5418: 4960: 2041: 1173: 473: 510:

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

806:, and the construction joins its ends to make a circle. An equivalent description is that the boundary of an 11021:

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

10683: 5822:, where multiplication is given by composition of representing maps, and any element of non-zero degree is 10073: 6826: 6323: 1649: 1644: 10732:, Graduate Studies in Mathematics, vol. 5, Providence, Rhode Island: American Mathematical Society, 11238: 11023: 10537: 10227: 6909: 6172: 6096: 5816: 4840: 3630: 1361:

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

9967: 6924: 6572: 6568: 5741: 3495: 1309: 313:

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

10078: 1747: 10655: 6472: 5470: 1685: 1449: 1313: 1252: 1221: 1110: 980: 777: 736: 11367: 6507: 6483: 2921: 1751: 11255: 11101: 11081: 11058: 11032: 10904: 10629:, Contemporary Mathematics, vol. 181, Providence, R.I.: Amer. Math. Soc., pp. 299–316, 10554: 10506: 10343: 10281: 10244: 10222: 10218: 10104: 10014: 9459: 6790: 6464: 6457: 5827: 5474: 3412:{\displaystyle \cdots \to \pi _{i}(F)\to \pi _{i}(E)\to \pi _{i}(B)\to \pi _{i-1}(F)\to \cdots .} 1847: 1693: 1197: 1114: 511: 94: 6563:

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

5251:{\displaystyle \pi _{2m+k}(S^{2m})(p)=\pi _{2m+k-1}(S^{2m-1})(p)\oplus \pi _{2m+k}(S^{4m-1})(p)} 2953:) appear to be chaotic, but in fact there are many patterns, some obvious and some very subtle. 3182:, and have been computed in numerous cases, although the general pattern is still elusive. For 11283: 11188: 11152: 11000: 10965: 10933: 10865: 10834: 10794: 10741: 10727: 10630: 10622: 10591: 10475: 10394: 10361: 10320: 10307: 10198: 10186: 10162: 10148: 9428: 6920: 6818: 6814: 6187: 6087:)-fold loop space by the Hurewicz theorem. This reduces the calculation of homotopy groups of 6031: 5831: 4889: 4802: 1689: 1653: 1639: 1357: 1324: 1320: 1089: 1029: 997: 502:. Several important patterns have been established, yet much remains unknown and unexplained. 499: 495: 254: 233: 114: 81: 11233: 4851:. The 2-components are hardest to calculate, and in several ways behave differently from the 1671:

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

11397: 11271: 11247: 11091: 11072: 11042: 10992: 10896: 10828: 10784: 10774: 10733: 10583: 10546: 10516: 10465: 10386: 10353: 10297: 10273: 10236: 10122: 10083: 10044: 10023: 8758: 6610: 6182:

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

6154: 6047: 5690:-homomorphism is considered to be the "hard" part of the stable homotopy groups of spheres ( 5529: 5454: 3077: 1872: 1864: 1677: 1672: 1664: 1107: 1022: 519: 110: 11202: 11166: 11123: 11054: 11014: 10979: 10945: 10916: 10879: 10848: 10806: 10755: 10675: 10644: 10605: 10566: 10489: 10449: 10408: 10373: 10293: 10256: 10210: 10174: 10136: 10095: 6927:) of the cyclic groups of those orders. Powers indicate repeated products. (Note that when 11327: 11198: 11162: 11119: 11050: 11010: 10975: 10941: 10912: 10875: 10844: 10824: 10802: 10751: 10671: 10640: 10601: 10579: 10562: 10528: 10485: 10445: 10404: 10369: 10289: 10252: 10206: 10170: 10132: 10091: 6822: 6391: 6245:

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

4709: 3640: 1668: 1521: 1373: 1018: 596: 349: 191: 10108: 2963:

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

988: 11217: 10813: 10705: 8762: 6750: 6576: 6424: 6404: 5982: 5376: 5345: 4843:

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

4786: 4727: 4599: 4583: 3980: 3910: 3290: 3206: 2915: 1868: 1660: 1635: 1607: 1569: 1305: 1045: 1009: 523: 385: 133: 122: 118: 10302: 10062:(1984), "Relations amongst Toda brackets and the Kervaire invariant in dimension 62", 833:

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

11386: 11323: 11259: 11142: 11138: 11105: 11062: 10651: 10618: 10059: 10049: 10028: 8301: 7011: 6936: 6813:

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

6618: 6527:. The geometry near a critical point of such a map can be described by an element of 6331: 4653: 2016: 1815: 1743: 1437: 1205: 1190: 1169: 964: 326: 258: 10887:

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

10444:, Pure and Applied Mathematics, vol. 8, New York & London: Academic Press, 5457:, and it is reflected in the stable homotopy groups of spheres via the image of the 4586:

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

1440:, with only one element, the identity element, and so it can be identified with the 249:. This summary does not distinguish between two mappings if one can be continuously 11304: 11267: 11174: 10953: 10762: 10415: 10385:, Undergraduate Texts in Mathematics, Springer-Verlag, New York, pp. 134–136, 10182: 10144: 6893: 6017: 5977: 5956: 4805:

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

3210: 2024: 1739: 1731: 803: 10381:

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

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

11046: 10987:

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

6595: 6334: 6194:. The initial term is again quite hard to calculate; to do this one can use the 5819: 4510:{\displaystyle \pi _{30}(S^{16})\neq \pi _{30}(S^{31})\oplus \pi _{29}(S^{15}).} 1735: 1272: 90: 10765:(1973), "The nilpotency of elements of the stable homotopy groups of spheres", 10520: 10087: 6080:)-fold repeated loop space, which is equal to the first homology group of the ( 6038:"The method of killing homotopy groups", due to Cartan and Serre ( 11229: 10996: 10390: 10151:(1952a), "Espaces fibrés et groupes d'homotopie. I. Constructions générales", 6955: 6579: 6468: 6427: 6062: 5986: 4098: 1871:, which are generally easier to calculate; in particular, it shows that for a 1837: 1591: 1293: 1213: 811: 11308: 10937: 10820:

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

10798: 10365: 10202: 10166: 5626:

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

958:

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

11178: 11118:, The Univalent Foundations Program and Institute for Advanced Study, 2013, 10779: 6301:, which allows one to deduce motivic Adams differentials for the cofiber of 6202: 6116: 5823: 5547:

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

4595: 1850:, and any mapping to such a space can be deformed into a one-point mapping. 1755: 1688:

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

1564: 1367: 1201: 700: 606: 128: 102: 10991:, Graduate Texts in Mathematics, vol. 224, Springer-Verlag, New York, 10311: 3841:

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

10668:

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

400:

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

140: 11279: 6065:

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

4348:{\displaystyle \pi _{i}(S^{8})=\pi _{i}(S^{15})\oplus \pi _{i-1}(S^{7}).} 4102: 1587: 1441: 1013: 1001: 654: 542: 515: 250: 153: 10470: 10357: 4235:{\displaystyle \pi _{i}(S^{4})=\pi _{i}(S^{7})\oplus \pi _{i-1}(S^{3}),} 3777:{\displaystyle \pi _{i}(S^{2})=\pi _{i}(S^{3})\oplus \pi _{i-1}(S^{1}).} 599:

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

11251: 11234:"Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche" 11113: 10908: 10789: 10670:, Documenta Mathematica, Extra Volume, vol. II, pp. 465–472, 10587: 10558: 10285: 10248: 6885: 6024: 1301: 840: 773: 367: 10737: 6005:

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

4925:. The case of 2-dimensional spheres is slightly different: the first 4708:, computed by the algebraic sum of their points, corresponding to the 10348: 8767: 6280:

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

674: 538: 468:

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

177: 167: 10900: 10550: 10277: 10240: 10189:(1952b), "Espaces fibrés et groupes d'homotopie. II. Applications", 6723:

is the cyclic subgroup represented by homotopy spheres that bound a

2893:

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

384:. The first such surprise was the discovery of a mapping called the 11086: 11037: 10511: 1696:

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

1427: 6365: 1563: 1356: 1085: 987: 979: 545:

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

139: 127: 80: 10336:

Proceedings of the Japan Academy, Series A, Mathematical Sciences

6585:

Stable homotopy groups of spheres are used to describe the group

5805:{\displaystyle \pi _{\ast }^{S}=\bigoplus _{k\geq 0}\pi _{k}^{S}} 4991:. Furthermore, the stable range can be extended in this case: if 708:

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

5276:

For stable homotopy groups there are more precise results about

487:) are more erratic; nevertheless, they have been tabulated for 10924:

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

10576:

Stable homotopy groups of spheres. A computer-assisted approach

10264:

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

4702:

is the cobordism group of framed 0-dimensional submanifolds of

4572:{\displaystyle S^{15}\hookrightarrow S^{31}\rightarrow S^{16},} 3979:, where the bundle projection is a double covering), there are 3064:

These patterns follow from many different theoretical results.

1863:

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

1431:

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

190:-sphere may be defined geometrically as the set of points in a 11222:

Verhandlungen des Internationalen Mathematikerkongress, Zürich

6009:. To compute these groups, they are often factored into their 2940:, which has the same higher homotopy groups, is contractible. 18: 4764:

which corresponds to the framed 1-dimensional submanifold of

3476:

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

1208:), so that the two spaces have the same topology, then their 498:, a technique first applied to homotopy groups of spheres by 8757:

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

6218:

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

4087:{\displaystyle S^{7}\hookrightarrow S^{15}\rightarrow S^{8}} 3265:{\displaystyle S^{1}\hookrightarrow S^{3}\rightarrow S^{2}.} 699:

This construction moves from geometry to pure topology. The

16:

How spheres of various dimensions can wrap around each other

9336: 9334: 9332: 6330:

identify these homotopy groups as certain quotients of the

4404:) and beyond. Although generalizations of the relations to 4031:{\displaystyle S^{3}\hookrightarrow S^{7}\rightarrow S^{4}} 3972:{\displaystyle S^{0}\hookrightarrow S^{1}\rightarrow S^{1}} 2075:). Extended tables of homotopy groups of spheres are given 10818:

Smooth manifolds and their applications in homotopy theory

1196:

A continuous map between two topological spaces induces a

1012:

is a function between spaces that preserves continuity. A

11115:

Homotopy type theory—univalent foundations of mathematics

9519: 9517: 6549:

sphere around the critical point maps into a topological

6343:. Under this correspondence, every nontrivial element in 1663:

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

605:, for reasons given below. The same idea applies for any 10578:, Lecture Notes in Mathematics, vol. 1423, Berlin: 7900: 5584:

have a cyclic subgroup of order 504, the denominator of

10830:

Complex cobordism and stable homotopy groups of spheres

6700:{\displaystyle \Theta _{n}/bP_{n+1}\to \pi _{n}^{S}/J,} 6463:

The stable homotopy groups of spheres are important in

3080:, which implies that the suspension homomorphism from 1052:

thus begins with continuous maps from a pointed circle

374:

The most interesting and surprising results occur when

11183:, Chicago lectures in mathematics (revised ed.), 9687: 9659: 9657: 9655: 9462: 4602:

established an isomorphism between the homotopy group

2994:). This isomorphism is induced by the Hopf fibration 1366:(There is no requirement for the continuous map to be 9583: 9571: 6633: 6510: 6486: 5752: 5641: 5082: 4529: 4419: 4251: 4138: 4047: 3991: 3932: 3680: 3507: 3302: 3222: 2924: 9942: 9453: 6919:

Where entry is a product, the homotopy group is the

6467:, which studies the structure of singular points of 6167:, and converges to something closely related to the 5487:

a cyclic group of order equal to the denominator of

4858:

In the same paper, Serre found the first place that

1900:, is isomorphic to the first nonzero homology group 1168:) — the homotopy groups of spheres — the groups are 1000:

is its continuity structure, formalized in terms of

992:

Addition of two circle maps keeping base point fixed

984:

Homotopy of two circle maps keeping base point fixed

9770: 9768: 9606: 9604: 5694:). (Adams also introduced certain order 2 elements 810:-dimensional disk is glued to a point, producing a 125:are surprisingly complex and difficult to compute. 10926:Comptes Rendus de l'Académie des Sciences, Série I 10191:Comptes Rendus de l'Académie des Sciences, Série I 10154:Comptes Rendus de l'Académie des Sciences, Série I 9894: 9491: 6825:, leading to an identification of K-theory of the 6699: 6518: 6494: 6327: 5804: 5659: 5250: 4594:Homotopy groups of spheres are closely related to 4571: 4509: 4410:are often true, they sometimes fail; for example, 4347: 4234: 4086: 4030: 3971: 3776: 3612: 3422:For this specific bundle, each group homomorphism 3411: 3264: 2943:Beyond the first row, the higher homotopy groups ( 2932: 1818:. The reason is that a continuous mapping from an 117:, forgetting about their precise geometry. Unlike 10958:Composition methods in homotopy groups of spheres 1106:are called null homotopic. The classes become an 105:can wrap around each other. They are examples of 6908:Where the entry is ∞, the homotopy group is the 5634:-invariant, a homomorphism from these groups to 5303:-primary component of the stable homotopy group 1590:to the ordinary 2-sphere, and was discovered by 201:located at a unit distance from the origin. The 10960:, Annals of Mathematics Studies, vol. 49, 10610: 9906: 6292:as the algebraic Novikov spectral sequence for 667:)-dimensional space. For example, the 1-sphere 278:falls into three regimes, depending on whether 11220:(1932), "Höherdimensionale Homotopiegruppen", 9930: 9882: 6186:with a generalized cohomology theory, such as 5815:of the stable homotopy groups of spheres is a 1780:up to 90, and, as of 2023, unknown for larger 11310:This week's finds in mathematical physics 102 10706:"Differential topology forty-six years later" 10109:"Configurations, braids, and homotopy groups" 9987: 9340: 6859:The following table shows many of the groups 6806:for which the question was open at the time.) 6309: 5073:in terms of that of odd-dimensional spheres, 3926:Similarly (in addition to the Hopf fibration 2076: 1659:Higher homotopy groups were first defined by 1479:. This can be shown as a consequence of the 1323:) of the homotopy group with the integers is 514:provides the larger context, itself built on 404:The question of computing the homotopy group 121:, which are also topological invariants, the 8: 10767:Journal of the Mathematical Society of Japan 10714:Notices of the American Mathematical Society 10458:Memoirs of the American Mathematical Society 10114:Journal of the American Mathematical Society 6621:). More precisely, there is an injective map 6043: 6039: 5043:, and an epimorphism if equality holds. The 4652:which are "framed", i.e. have a trivialized 3470:to zero, since the lower-dimensional sphere 1180:all maps are null homotopic, then the group 1092:based on homotopy (keeping the "base point" 11379:in MacTutor History of Mathematics archive. 10456:Isaksen, Daniel C. (2019), "Stable Stems", 6764:, in which case the image has index 1 or 2. 5630:-homomorphism, and the kernel of the Adams 5484:is congruent to 2, 4, 5, or 6 modulo 8; and 3494:. Thus the long exact sequence breaks into 1920:-sphere, this immediately implies that for 1189:consists of one element, and is called the 595:This is the set of points in 3-dimensional 222:summarizes the different ways in which the 11366:O'Connor, J. J.; Robertson, E. F. (2001), 11345:O'Connor, J. J.; Robertson, E. F. (1996), 10989:Metric structures in differential geometry 10065:Journal of the London Mathematical Society 6542:, by considering the way in which a small 6316:The computation of the homotopy groups of 4862:-torsion occurs in the homotopy groups of 4670:is homotopic to a differentiable map with 1867:: this theorem links homotopy groups with 1840:. Consequently, its image is contained in 1726:. Others who worked in this area included 1076:, where maps from one pair to another map 113:terms, the structure of spheres viewed as 11095: 11085: 11036: 10788: 10778: 10535:(1963), "Groups of homotopy spheres: I", 10510: 10469: 10347: 10301: 10126: 10077: 10048: 10027: 9480: 9467: 9461: 6745:th stable homotopy group of spheres, and 6686: 6680: 6675: 6656: 6644: 6638: 6632: 6512: 6511: 6509: 6488: 6487: 6485: 5796: 5791: 5775: 5762: 5757: 5751: 5660:{\displaystyle \mathbb {Q} /\mathbb {Z} } 5653: 5652: 5647: 5643: 5642: 5640: 5221: 5199: 5165: 5137: 5109: 5087: 5081: 4560: 4547: 4534: 4528: 4495: 4482: 4466: 4453: 4437: 4424: 4418: 4333: 4314: 4298: 4285: 4269: 4256: 4250: 4220: 4201: 4185: 4172: 4156: 4143: 4137: 4078: 4065: 4052: 4046: 4022: 4009: 3996: 3990: 3963: 3950: 3937: 3931: 3762: 3743: 3727: 3714: 3698: 3685: 3679: 3595: 3576: 3560: 3547: 3531: 3518: 3506: 3379: 3357: 3335: 3313: 3301: 3253: 3240: 3227: 3221: 2926: 2925: 2923: 2044:of such groups (written, for example, as 1836:can always be deformed so that it is not 69:Learn how and when to remove this message 10058:Barratt, Michael G.; Jones, John D. S.; 9810: 9559: 8734:-component that is accounted for by the 6881:The entry "⋅" denotes the trivial group. 6201: 4839:), when the group is the product of the 2081: 1594:, who constructed a nontrivial map from 1426: 1251: 282:is less than, equal to, or greater than 32:This article includes a list of general 11360:MacTutor History of Mathematics archive 9991: 9954: 9918: 9858: 9846: 9834: 9822: 9798: 9759: 9723: 9699: 9663: 9646: 9634: 9622: 9535: 9523: 9508: 9440: 9412: 9400: 9388: 9376: 9364: 9352: 9328: 6423:, which states that every non-constant 6190:or, more usually, a piece of it called 5877:is nonzero and 12 times a generator of 4995:is odd then the double suspension from 4688:-dimensional submanifold. For example, 4105:instead of complex numbers. Here, too, 3482:. This corresponds to the vanishing of 11180:A Concise Course in Algebraic Topology 10627:The Čech centennial (Boston, MA, 1993) 9870: 9637:, Stable homotopy groups, pp. 385–393. 6789:, about the existence of manifolds of 5332:, in which case it is cyclic of order 1384:These two results generalize: for all 1084:. These maps (or equivalently, closed 325:. Therefore the homotopy group is the 11272:"Stable Algebraic Topology 1945–1966" 11132:General algebraic topology references 10729:Algebraic curves and Riemann surfaces 9786: 9747: 9711: 9675: 5691: 5668:. Roughly speaking, the image of the 4903:, and has a unique subgroup of order 4866:dimensional spheres, by showing that 4358:The three fibrations have base space 3806:at least 3, the first row shows that 3175:: they are finite abelian groups for 464:and have been computed for values of 7: 10623:"On the computation of stable stems" 10499:Publications mathématiques de l'IHÉS 9983: 9971: 9774: 9735: 9688:Cohen, Moore & Neisendorfer 1979 9610: 9595: 9547: 6073:is the first homotopy group of its ( 5453:. This period 8 pattern is known as 3275:The general theory of fiber bundles 476:. The unstable homotopy groups (for 160:-sphere for brevity, and denoted as 10656:"Toward a global understanding of π 9986:. The 2-components can be found in 9968:table of homotopy groups of spheres 6452:that every continuous map from the 6374:that is not Brunnian over the disk 5047:-torsion of the intermediate group 3195:unstable homotopy groups of spheres 1879:, the first nonzero homotopy group 1406: 10383:The Fundamental Theorem of Algebra 9943:Barratt, Jones & Mahowald 1984 9424: 6802:. (This was the smallest value of 6635: 4770:defined by the standard embedding 4582:the first non-trivial case of the 4391:) as mentioned above, but not for 3923:sends any such pair to its ratio. 3881:form a 3-sphere, and their ratios 3838:is at least 3, as observed above. 1576:The first nontrivial example with 1225:make certain computations easier. 38:it lacks sufficient corresponding 14: 11329:Stable homotopy groups of spheres 10420:"Spheres, homotopy groups of the" 10107:; Wong, Yan Loi; Wu, Jie (2006), 10012:(1966), "On the groups J(X) IV", 9990:, and the 3- and 5-components in 8738:-homomorphism is cyclic of order 6896:of that order (generally written 6364:may be represented by a Brunnian 6007:finitely generated abelian groups 5569:. For example, the stable groups 3159:stable homotopy groups of spheres 462:stable homotopy groups of spheres 6756:. This is an isomorphism unless 6407:(corresponding to an integer of 6046:) involves repeatedly using the 6023:, and calculating each of these 5069:gives the torsion at odd primes 2907:of the 1-sphere are trivial for 1846:with a point removed; this is a 996:The distinguishing feature of a 101:describe how spheres of various 23: 10321:"Remarks on zeta functions and 10035:Adams, J (1968), "Correction", 8704:Table of stable homotopy groups 6180:Adams–Novikov spectral sequence 4520:Thus there can be no fibration 494:. Most modern computations use 470:extraordinary cohomology theory 9966:These tables are based on the 9486: 9473: 8723:number of cyclic groups.) For 6668: 6475:. Such singularities arise as 6421:fundamental theorem of algebra 6224:has a unique smooth structure. 5245: 5239: 5236: 5214: 5189: 5183: 5180: 5158: 5127: 5121: 5118: 5102: 4598:classes of manifolds. In 1938 4553: 4540: 4501: 4488: 4472: 4459: 4443: 4430: 4339: 4326: 4304: 4291: 4275: 4262: 4226: 4213: 4191: 4178: 4162: 4149: 4071: 4058: 4015: 4002: 3956: 3943: 3768: 3755: 3733: 3720: 3704: 3691: 3604: 3601: 3588: 3569: 3566: 3553: 3540: 3537: 3524: 3511: 3400: 3397: 3391: 3372: 3369: 3363: 3350: 3347: 3341: 3328: 3325: 3319: 3306: 3246: 3233: 1648:where the related concepts of 1481:cellular approximation theorem 1: 11369:Marie Ennemond Camille Jordan 10862:American Mathematical Society 10858:The wild world of 4-manifolds 10833:(2nd ed.), AMS Chelsea, 10609:Also see the corrections in ( 10128:10.1090/S0894-0347-05-00507-2 9988:Isaksen, Wang & Xu (2023) 9584:O'Connor & Robertson 1996 9572:O'Connor & Robertson 2001 8765:, whose image is underlined. 6598:classes of oriented homotopy 6310:Isaksen, Wang & Xu (2023) 6069:th homotopy group of a space 5465:a cyclic group of order 2 if 4378:. A fibration does exist for 2019:0, the infinite cyclic group 1758:. The stable homotopy groups 1028:The first homotopy group, or 11097:10.4310/ACTA.2021.v226.n2.a2 10574:Kochman, Stanley O. (1990), 10050:10.1016/0040-9383(68)90010-4 10029:10.1016/0040-9383(66)90004-8 9883:Gheorghe, Wang & Xu 2021 9492:{\textstyle \pi _{1}(S^{1})} 6892:, the homotopy group is the 6829:with stable homotopy groups. 6519:{\displaystyle \mathbb {R} } 6495:{\displaystyle \mathbb {R} } 6460:to itself has a fixed point. 6380:. For example, the Hopf map 4855:-components for odd primes. 3193:, the groups are called the 2933:{\displaystyle \mathbb {R} } 1665:Pavel Sergeyevich Alexandrov 1530:is simply connected, by the 1088:) are grouped together into 745:, described by the equality 11185:University of Chicago Press 11047:10.4007/annals.2017.186.2.3 10856:Scorpan, Alexandru (2005), 10689:Encyclopedia of Mathematics 10611:Kochman & Mahowald 1995 10425:Encyclopedia of Mathematics 9907:Fine & Rosenberger 1997 9341:Isaksen, Wang & Xu 2023 8708:The stable homotopy groups 6450:Brouwer fixed point theorem 6312:computes up to the 90-stem. 6196:chromatic spectral sequence 5895:is zero because the group 5834:implies Nishida's theorem. 5280:-torsion. For example, if 4847:-components for all primes 4097:constructed using pairs of 3981:generalized Hopf fibrations 3913:, a 2-sphere. The Hopf map 3911:complex plane plus infinity 3446:, induced by the inclusion 1586:concerns mappings from the 388:, which wraps the 3-sphere 264:The problem of determining 166:— generalizes the familiar 11414: 11149:Cambridge University Press 10962:Princeton University Press 10521:10.1007/s10240-023-00139-1 9931:Kervaire & Milnor 1963 6787:Kervaire invariant problem 6324:combinatorial group theory 5343: 4739:represents a generator of 3068:Stable and unstable groups 1642:in his 1895 set of papers 660:as a geometric object in ( 99:homotopy groups of spheres 10997:10.1007/978-0-387-21826-7 10391:10.1007/978-1-4612-1928-6 9454:Homotopy type theory 2013 6780:piecewise linear manifold 6419:can be used to prove the 6192:Brown–Peterson cohomology 5859:is nonzero and generates 5350:An important subgroup of 2077:at the end of the article 1634:In the late 19th century 1520:has the real line as its 822:: written in topology as 685:: written in topology as 253:to the other; thus, only 11274:, in I. M. James (ed.), 10682:Mahowald, Mark (2001) , 10088:10.1112/jlms/s2-30.3.533 6834:Table of homotopy groups 5461:-homomorphism which is: 5419:special orthogonal group 5062:can be strictly larger. 4726:. The projection of the 4639:of cobordism classes of 3834:are isomorphic whenever 3038:do not vanish. However, 2914:, because the universal 1229:Low-dimensional examples 1212:-th homotopy groups are 1138:begins with the pointed 474:stable cohomotopy theory 394:around the usual sphere 10684:"EHP spectral sequence" 10319:Deitmar, Anton (2006), 6725:parallelizable manifold 6609:, this is the group of 6274:The motivic cofiber of 6178:At the odd primes, the 6104:Adams spectral sequence 6059:Serre spectral sequence 6052:Eilenberg–MacLane space 5432:, the homotopy groups 4955:, then elements of the 3645:suspension homomorphism 1792:As noted already, when 683:Disk with collapsed rim 460:. These are called the 53:more precise citations. 10726:Miranda, Rick (1995), 9493: 7005:, which is denoted by 6884:Where the entry is an 6827:field with one element 6811:Barratt–Priddy theorem 6701: 6520: 6496: 6322:has been reduced to a 6207: 5806: 5722:≡ 1 or 2 (mod 8) 5661: 5421:. In the stable range 5375:, is the image of the 5252: 4797:Finiteness and torsion 4573: 4511: 4349: 4236: 4088: 4032: 3973: 3778: 3671:, giving isomorphisms 3639:, these sequences are 3614: 3413: 3289:shows that there is a 3266: 3117:is an isomorphism for 2934: 1656:were also introduced. 1573: 1432: 1362: 1319:The identification (a 1268: 993: 985: 236:continuously into the 145: 137: 107:topological invariants 86: 11348:A history of Topology 11239:Mathematische Annalen 11024:Annals of Mathematics 10889:Annals of Mathematics 10780:10.2969/jmsj/02540707 10617:Kochman, Stanley O.; 10538:Annals of Mathematics 10266:Annals of Mathematics 10228:Annals of Mathematics 9494: 6910:infinite cyclic group 6702: 6521: 6497: 6328:Berrick et al. (2006) 6205: 6173:May spectral sequence 6097:EHP spectral sequence 5993:Computational methods 5807: 5662: 5253: 5024:is an isomorphism of 4841:infinite cyclic group 4574: 4512: 4350: 4237: 4089: 4033: 3974: 3779: 3615: 3496:short exact sequences 3414: 3267: 2935: 1567: 1430: 1360: 1327:as an equality: thus 1308:of a loop around the 1290:infinite cyclic group 1255: 1064:to the pointed space 991: 983: 941:, and the suspension 820:Suspension of equator 319:to a single point of 143: 131: 84: 11282:, pp. 665–723, 10932:, Paris: 1340–1342, 9460: 9367:, Example 0.3, p. 6. 6749:is the image of the 6631: 6575:of a compact smooth 6508: 6484: 6479:of smooth maps from 5841:is the generator of 5750: 5686:by the image of the 5639: 5293:− 1) − 2 5080: 4944:is an odd prime and 4929:-torsion occurs for 4527: 4417: 4249: 4136: 4045: 3989: 3930: 3678: 3505: 3300: 3220: 3024:the homotopy groups 2922: 1957:The homology groups 1450:space-filling curves 1220:. However, the real 1124:-th homotopy group, 1120:More generally, the 1048:) topological space 892:has as equator the ( 526:as a basic example. 226:-dimensional sphere 109:, which reflect, in 11276:History of Topology 10825:Ravenel, Douglas C. 10529:Kervaire, Michel A. 10358:10.3792/pjaa.82.141 10219:Cohen, Frederick R. 10105:Cohen, Frederick R. 9895:Berrick et al. 2006 9415:, pp. 123–125. 6685: 6582:is divisible by 16. 6473:algebraic varieties 6390:corresponds to the 5853:(of order 2), then 5801: 5767: 5565:for such values of 4981:have order at most 3293:of homotopy groups 3291:long exact sequence 1686:George W. Whitehead 1614:the homotopy group 1606:, now known as the 1090:equivalence classes 735:, and its rim (or " 445:are independent of 301:, any mapping from 255:equivalence classes 176:) and the ordinary 11252:10.1007/BF01457962 11144:Algebraic Topology 10588:10.1007/BFb0083795 10197:, Paris: 393–395, 10187:Serre, Jean-Pierre 10161:, Paris: 288–290, 10149:Serre, Jean-Pierre 9871:Wang & Xu 2017 9489: 8730:, the part of the 6791:Kervaire invariant 6697: 6671: 6556:sphere around the 6516: 6492: 6465:singularity theory 6208: 6115:term given by the 6032:spectral sequences 5828:nilpotence theorem 5802: 5787: 5786: 5753: 5657: 5340:The J-homomorphism 5248: 4569: 4507: 4345: 4232: 4084: 4028: 3969: 3774: 3610: 3409: 3262: 3161:, and are denoted 3074:suspension theorem 2930: 1848:contractible space 1694:spectral sequences 1682:suspension theorem 1574: 1514:. This is because 1433: 1363: 1269: 1198:group homomorphism 1174:finitely generated 1115:bouquet of spheres 1108:abstract algebraic 994: 986: 512:Algebraic topology 496:spectral sequences 366:with the group of 152:-dimensional unit 146: 138: 115:topological spaces 95:algebraic topology 87: 11307:(21 April 1997), 11289:978-0-444-82375-5 11211:Historical papers 11194:978-0-226-51183-2 11158:978-0-521-79540-1 10971:978-0-691-09586-8 10891:, Second Series, 10871:978-0-8218-3749-8 10840:978-0-8218-2967-7 10636:978-0-8218-0296-0 10619:Mahowald, Mark E. 10597:978-3-540-52468-7 10481:978-1-4704-3788-6 10471:10.1090/memo/1269 10268:, Second Series, 10231:, Second Series, 10060:Mahowald, Mark E. 9825:, pp. 67–74. 9315: 9314: 8701: 8700: 8300: 8299: 6921:cartesian product 6819:classifying space 6815:plus construction 6611:smooth structures 6569:Rokhlin's theorem 6188:complex cobordism 5832:complex cobordism 5771: 4803:Jean-Pierre Serre 4679:(1, 0, ..., 0) ⊂ 4646:-submanifolds of 2891: 2890: 1690:Jean-Pierre Serre 1654:fundamental group 1532:lifting criterion 1321:group isomorphism 1030:fundamental group 998:topological space 970:(1, 0, 0, ..., 0) 739:") is the circle 500:Jean-Pierre Serre 342:, every map from 79: 78: 71: 11405: 11378: 11377: 11376: 11357: 11356: 11355: 11341: 11340: 11339: 11334: 11319: 11318: 11317: 11292: 11280:Elsevier Science 11262: 11224: 11205: 11169: 11126: 11108: 11099: 11089: 11073:Acta Mathematica 11065: 11040: 11017: 10982: 10948: 10919: 10882: 10851: 10809: 10792: 10782: 10758: 10722: 10710: 10696: 10678: 10647: 10608: 10569: 10523: 10514: 10492: 10473: 10452: 10432: 10411: 10376: 10351: 10314: 10305: 10259: 10213: 10177: 10139: 10130: 10103:Berrick, A. J.; 10098: 10081: 10053: 10052: 10032: 10031: 9995: 9981: 9975: 9964: 9958: 9952: 9946: 9940: 9934: 9928: 9922: 9916: 9910: 9904: 9898: 9892: 9886: 9880: 9874: 9868: 9862: 9856: 9850: 9844: 9838: 9832: 9826: 9820: 9814: 9808: 9802: 9796: 9790: 9784: 9778: 9772: 9763: 9757: 9751: 9745: 9739: 9733: 9727: 9721: 9715: 9709: 9703: 9697: 9691: 9685: 9679: 9673: 9667: 9661: 9650: 9644: 9638: 9632: 9626: 9620: 9614: 9608: 9599: 9593: 9587: 9581: 9575: 9569: 9563: 9557: 9551: 9545: 9539: 9533: 9527: 9521: 9512: 9506: 9500: 9498: 9496: 9495: 9490: 9485: 9484: 9472: 9471: 9456:, Section 8.1, " 9450: 9444: 9438: 9432: 9422: 9416: 9410: 9404: 9398: 9392: 9386: 9380: 9374: 9368: 9362: 9356: 9350: 9344: 9338: 8768: 8759:Bott periodicity 8756: 8749: 8741: 8737: 8733: 8729: 8721: 8720: 8719: 8302: 8217: 8160: 7993: 7758: 7012: 7008: 7004: 6966: 6953: 6934: 6930: 6915: 6904: 6891: 6876: 6855: 6805: 6801: 6800:2 − 2 = 62 6796: 6793:1 in dimensions 6777: 6763: 6759: 6753: 6748: 6744: 6740: 6739: 6738: 6722: 6706: 6704: 6703: 6698: 6690: 6684: 6679: 6667: 6666: 6648: 6643: 6642: 6616: 6608: 6601: 6593: 6565:Vladimir Rokhlin 6555: 6548: 6541: 6526: 6525: 6523: 6522: 6517: 6515: 6502: 6501: 6499: 6498: 6493: 6491: 6455: 6447: 6418: 6389: 6379: 6373: 6363: 6356: 6342: 6321: 6306: 6300: 6291: 6285: 6279: 6268: 6262: 6256: 6250: 6244: 6237: 6223: 6185: 6170: 6166: 6155:Steenrod algebra 6153: 6149: 6135: 6134: 6114: 6090: 6086: 6079: 6072: 6068: 6048:Hurewicz theorem 6027: 6022: 6013: 6004: 6000: 5975: 5974: 5954: 5943: 5933: 5932: 5931: 5922:are elements of 5921: 5917: 5913: 5906: 5905: 5904: 5894: 5888: 5887: 5886: 5876: 5870: 5869: 5868: 5858: 5852: 5851: 5850: 5840: 5817:supercommutative 5811: 5809: 5808: 5803: 5800: 5795: 5785: 5766: 5761: 5731: 5723: 5716: 5715: 5714: 5702: 5689: 5685: 5684: 5683: 5671: 5667: 5666: 5664: 5663: 5658: 5656: 5651: 5646: 5633: 5629: 5622: 5621: 5619: 5618: 5615: 5612: 5605: 5603: 5602: 5599: 5596: 5583: 5568: 5564: 5542: 5530:Bernoulli number 5527: 5515: 5514: 5512: 5511: 5505: 5502: 5483: 5468: 5460: 5455:Bott periodicity 5452: 5445: 5431: 5416: 5408: 5374: 5367: 5335: 5331: 5324:is divisible by 5323: 5317:vanishes unless 5316: 5315: 5314: 5302: 5298: 5294: 5279: 5272: 5268: 5257: 5255: 5254: 5249: 5235: 5234: 5213: 5212: 5179: 5178: 5157: 5156: 5117: 5116: 5101: 5100: 5072: 5061: 5046: 5042: 5027: 5023: 5008: 4994: 4990: 4986: 4980: 4958: 4954: 4943: 4939: 4928: 4924: 4913: 4906: 4902: 4887: 4883: 4865: 4861: 4854: 4850: 4846: 4838: 4834: 4818: 4779: 4769: 4763: 4757: 4756: 4738: 4725: 4707: 4701: 4687: 4683: 4669: 4651: 4645: 4638: 4632: 4631: 4619: 4590:Framed cobordism 4578: 4576: 4575: 4570: 4565: 4564: 4552: 4551: 4539: 4538: 4516: 4514: 4513: 4508: 4500: 4499: 4487: 4486: 4471: 4470: 4458: 4457: 4442: 4441: 4429: 4428: 4409: 4403: 4396: 4390: 4383: 4377: 4370: 4363: 4354: 4352: 4351: 4346: 4338: 4337: 4325: 4324: 4303: 4302: 4290: 4289: 4274: 4273: 4261: 4260: 4241: 4239: 4238: 4233: 4225: 4224: 4212: 4211: 4190: 4189: 4177: 4176: 4161: 4160: 4148: 4147: 4128: 4116: 4093: 4091: 4090: 4085: 4083: 4082: 4070: 4069: 4057: 4056: 4037: 4035: 4034: 4029: 4027: 4026: 4014: 4013: 4001: 4000: 3978: 3976: 3975: 3970: 3968: 3967: 3955: 3954: 3942: 3941: 3922: 3908: 3907: 3905: 3904: 3896: 3893: 3880: 3878: 3869: 3858: 3837: 3833: 3819: 3805: 3801: 3783: 3781: 3780: 3775: 3767: 3766: 3754: 3753: 3732: 3731: 3719: 3718: 3703: 3702: 3690: 3689: 3670: 3638: 3628: 3619: 3617: 3616: 3611: 3600: 3599: 3587: 3586: 3565: 3564: 3552: 3551: 3536: 3535: 3523: 3522: 3493: 3481: 3475: 3469: 3455: 3445: 3418: 3416: 3415: 3410: 3390: 3389: 3362: 3361: 3340: 3339: 3318: 3317: 3288: 3271: 3269: 3268: 3263: 3258: 3257: 3245: 3244: 3232: 3231: 3192: 3181: 3174: 3173: 3172: 3156: 3145: 3127: 3116: 3097: 3078:Hans Freudenthal 3059: 3052: 3037: 3023: 3013: 3003: 2993: 2986: 2952: 2939: 2937: 2936: 2931: 2929: 2913: 2906: 2820: 2819: 2745: 2744: 2653: 2652: 2572: 2571: 2557: 2556: 2540: 2539: 2507: 2506: 2496: 2495: 2448: 2447: 2437: 2436: 2412: 2411: 2337: 2336: 2326: 2325: 2301: 2300: 2082: 2074: 2065: 2064: 2054: 2039: 2030: 2023:, b) the finite 2022: 2006: 1996: 1982: 1972: 1953: 1926: 1919: 1915: 1899: 1892: 1873:simply-connected 1865:Hurewicz theorem 1862: 1845: 1835: 1825: 1821: 1813: 1799: 1795: 1783: 1779: 1775: 1725: 1709: 1678:Hans Freudenthal 1673:Hurewicz theorem 1625: 1605: 1599: 1585: 1560: 1545: 1539: 1529: 1519: 1513: 1497: 1478: 1464: 1447: 1423: 1404: 1390: 1380: 1353: 1338: 1299: 1288:is therefore an 1287: 1267: 1248: 1219: 1211: 1188: 1179: 1167: 1163: 1157: 1153: 1141: 1137: 1123: 1105: 1095: 1083: 1079: 1075: 1063: 1051: 1043: 1023:complex analysis 971: 961: 953: 947: 940: 938: 937: 923: 922: 911: 910: 898: 891: 889: 888: 875: 874: 863: 862: 850: 846: 838: 828: 809: 801: 795: 789: 771: 769: 768: 757: 756: 744: 734: 732: 731: 720: 719: 707: 694: 672: 666: 657: 652: 650: 649: 636: 635: 624: 623: 611: 604: 590: 588: 587: 576: 575: 564: 563: 550:Implicit surface 532: 520:abstract algebra 493: 486: 467: 459: 448: 444: 425: 421: 399: 393: 383: 365: 348:to itself has a 347: 341: 324: 318: 312: 306: 300: 285: 281: 277: 248: 242: 240: 231: 225: 221: 204: 200: 189: 185: 175: 165: 159: 151: 74: 67: 63: 60: 54: 49:this article by 40:inline citations 27: 26: 19: 11413: 11412: 11408: 11407: 11406: 11404: 11403: 11402: 11393:Homotopy theory 11383: 11382: 11374: 11372: 11365: 11353: 11351: 11344: 11337: 11335: 11332: 11322: 11315: 11313: 11303: 11300: 11290: 11266: 11228: 11216: 11213: 11195: 11173: 11159: 11137: 11134: 11129: 11112: 11069: 11020: 11007: 10986: 10972: 10952: 10923: 10901:10.2307/1969485 10886: 10872: 10855: 10841: 10823: 10814:Pontrjagin, Lev 10761: 10748: 10738:10.1090/gsm/005 10725: 10708: 10702:Milnor, John W. 10700: 10681: 10659: 10650: 10637: 10616: 10598: 10580:Springer-Verlag 10573: 10551:10.2307/1970128 10533:Milnor, John W. 10527: 10496: 10482: 10455: 10442:Homotopy theory 10436: 10416:Fuks, Dmitry B. 10414: 10401: 10380: 10331: 10318: 10278:10.2307/1970586 10263: 10241:10.2307/1971238 10217: 10181: 10143: 10102: 10079:10.1.1.212.1163 10057: 10034: 10010:Adams, J. Frank 10008: 10004: 9999: 9998: 9982: 9978: 9965: 9961: 9953: 9949: 9941: 9937: 9929: 9925: 9917: 9913: 9905: 9901: 9893: 9889: 9881: 9877: 9869: 9865: 9857: 9853: 9845: 9841: 9833: 9829: 9821: 9817: 9809: 9805: 9797: 9793: 9785: 9781: 9773: 9766: 9758: 9754: 9746: 9742: 9734: 9730: 9722: 9718: 9710: 9706: 9698: 9694: 9686: 9682: 9674: 9670: 9662: 9653: 9645: 9641: 9633: 9629: 9621: 9617: 9609: 9602: 9594: 9590: 9582: 9578: 9570: 9566: 9558: 9554: 9546: 9542: 9534: 9530: 9522: 9515: 9507: 9503: 9476: 9463: 9458: 9457: 9451: 9447: 9439: 9435: 9423: 9419: 9411: 9407: 9399: 9395: 9387: 9383: 9375: 9371: 9363: 9359: 9351: 9347: 9339: 9330: 9325: 9320: 9269: 9257:27⋅5⋅7⋅13⋅19⋅37 9216: 9164: 9111: 9058: 9005: 8956: 8899: 8850: 8809: 8751: 8743: 8739: 8735: 8731: 8724: 8718: 8713: 8712: 8711: 8709: 8706: 8665: 8622: 8579: 8536: 8493: 8450: 8407: 8364: 8252: 8215: 8195: 8158: 8138: 8083: 8028: 7991: 7971: 7916: 7903: 7854: 7796: 7756: 7736: 7678: 7620: 7562: 7504: 7446: 7388: 7330: 7272: 7214: 7156: 7099: 7006: 7003: 6999: 6995: 6987: 6979: 6975: 6965: 6959: 6952: 6946: 6940: 6932: 6928: 6923:(equivalently, 6913: 6903: 6897: 6889: 6870: 6860: 6849: 6839: 6836: 6823:symmetric group 6817:applied to the 6803: 6799: 6794: 6776: 6770: 6761: 6760:is of the form 6757: 6751: 6746: 6742: 6737: 6732: 6731: 6730: 6728: 6721: 6712: 6652: 6634: 6629: 6628: 6614: 6603: 6599: 6592: 6586: 6550: 6543: 6535: 6528: 6506: 6505: 6504: 6482: 6481: 6480: 6477:critical points 6453: 6441: 6434: 6412: 6408: 6400: 6392:Borromean rings 6381: 6375: 6369: 6358: 6350: 6344: 6338: 6317: 6302: 6299: 6293: 6287: 6281: 6275: 6264: 6258: 6252: 6246: 6240: 6236: 6230: 6219: 6206:Borromean rings 6183: 6168: 6157: 6151: 6147: 6141: 6133: 6123: 6122: 6121: 6119: 6113: 6107: 6088: 6081: 6074: 6070: 6066: 6025: 6020: 6011: 6002: 5998: 5995: 5983:Massey products 5960: 5959: 5945: 5935: 5930: 5927: 5926: 5925: 5923: 5919: 5915: 5911: 5903: 5900: 5899: 5898: 5896: 5890: 5885: 5882: 5881: 5880: 5878: 5872: 5867: 5864: 5863: 5862: 5860: 5854: 5849: 5846: 5845: 5844: 5842: 5838: 5748: 5747: 5738: 5732:to save space. 5725: 5718: 5713: 5708: 5707: 5706: 5704: 5701: 5695: 5687: 5682: 5677: 5676: 5675: 5673: 5669: 5637: 5636: 5635: 5631: 5627: 5616: 5613: 5610: 5609: 5607: 5600: 5597: 5595: 5589: 5588: 5586: 5585: 5577: 5570: 5566: 5558: 5548: 5541:− 1 ≡ 3 (mod 4) 5533: 5526: 5517: 5506: 5503: 5501: 5492: 5491: 5489: 5488: 5481: 5466: 5458: 5447: 5446:only depend on 5439: 5433: 5422: 5410: 5402: 5388: 5379: 5369: 5361: 5351: 5348: 5342: 5333: 5325: 5318: 5313: 5308: 5307: 5306: 5304: 5300: 5296: 5281: 5277: 5270: 5269:means take the 5262: 5217: 5195: 5161: 5133: 5105: 5083: 5078: 5077: 5070: 5067:James fibration 5055: 5048: 5044: 5029: 5028:-components if 5025: 5017: 5010: 5002: 4996: 4992: 4988: 4982: 4974: 4964: 4956: 4945: 4941: 4930: 4926: 4915: 4908: 4904: 4893: 4885: 4877: 4867: 4863: 4859: 4852: 4848: 4844: 4836: 4828: 4820: 4812: 4806: 4799: 4787:homotopy groups 4771: 4765: 4755: 4752: 4751: 4750: 4744: 4740: 4730: 4713: 4703: 4695: 4689: 4685: 4671: 4657: 4647: 4643: 4630: 4625: 4624: 4623: 4621: 4613: 4603: 4592: 4556: 4543: 4530: 4525: 4524: 4491: 4478: 4462: 4449: 4433: 4420: 4415: 4414: 4405: 4398: 4392: 4385: 4379: 4372: 4365: 4359: 4329: 4310: 4294: 4281: 4265: 4252: 4247: 4246: 4216: 4197: 4181: 4168: 4152: 4139: 4134: 4133: 4122: 4118: 4110: 4106: 4074: 4061: 4048: 4043: 4042: 4018: 4005: 3992: 3987: 3986: 3959: 3946: 3933: 3928: 3927: 3914: 3903: 3897: 3894: 3892: 3886: 3885: 3883: 3882: 3877: 3871: 3870:| + | 3868: 3862: 3860: 3856: 3849: 3842: 3835: 3827: 3821: 3813: 3807: 3803: 3795: 3788: 3758: 3739: 3723: 3710: 3694: 3681: 3676: 3675: 3664: 3654: 3647: 3634: 3624: 3591: 3572: 3556: 3543: 3527: 3514: 3503: 3502: 3487: 3483: 3477: 3471: 3463: 3457: 3447: 3439: 3429: 3423: 3375: 3353: 3331: 3309: 3298: 3297: 3276: 3249: 3236: 3223: 3218: 3217: 3203: 3201:Hopf fibrations 3183: 3176: 3171: 3166: 3165: 3164: 3162: 3157:are called the 3147: 3139: 3129: 3118: 3110: 3099: 3091: 3081: 3070: 3054: 3046: 3039: 3031: 3025: 3015: 3008: 2995: 2988: 2980: 2970: 2964: 2944: 2920: 2919: 2908: 2900: 2894: 2887: 2881: 2869: 2863: 2857: 2818: 2815: 2814: 2813: 2809: 2803: 2791: 2785: 2779: 2743: 2740: 2739: 2738: 2734: 2730: 2724: 2718: 2706: 2700: 2694: 2663: 2659: 2651: 2648: 2647: 2646: 2642: 2636: 2630: 2624: 2618: 2612: 2606: 2600: 2570: 2567: 2566: 2565: 2563: 2555: 2554: 2550: 2545: 2544: 2543: 2538: 2535: 2534: 2533: 2529: 2523: 2517: 2513: 2505: 2502: 2501: 2500: 2494: 2491: 2490: 2489: 2485: 2479: 2473: 2446: 2443: 2442: 2441: 2435: 2432: 2431: 2430: 2428: 2422: 2418: 2410: 2407: 2406: 2405: 2401: 2395: 2389: 2383: 2377: 2371: 2365: 2359: 2335: 2332: 2331: 2330: 2324: 2321: 2320: 2319: 2317: 2311: 2307: 2299: 2296: 2295: 2294: 2290: 2284: 2278: 2272: 2266: 2260: 2254: 2248: 2174: 2168: 2162: 2156: 2150: 2144: 2138: 2132: 2126: 2120: 2114: 2108: 2102: 2096: 2090: 2073: 2069: 2063: 2060: 2059: 2058: 2056: 2053: 2049: 2045: 2042:direct products 2038: 2032: 2028: 2020: 2013: 1998: 1990: 1984: 1974: 1966: 1958: 1947: 1934: 1928: 1921: 1917: 1909: 1901: 1894: 1886: 1880: 1869:homology groups 1854: 1841: 1827: 1823: 1819: 1807: 1801: 1797: 1793: 1790: 1781: 1777: 1769: 1759: 1719: 1711: 1703: 1697: 1669:Witold Hurewicz 1632: 1619: 1615: 1601: 1595: 1577: 1562: 1554: 1550: 1541: 1535: 1534:, any map from 1525: 1522:universal cover 1515: 1507: 1503: 1499: 1491: 1487: 1472: 1466: 1456: 1445: 1436:is therefore a 1425: 1417: 1413: 1398: 1392: 1385: 1378: 1355: 1347: 1343: 1332: 1328: 1297: 1281: 1277: 1261: 1257: 1250: 1242: 1238: 1231: 1217: 1209: 1187: 1181: 1177: 1165: 1159: 1155: 1143: 1139: 1131: 1125: 1121: 1097: 1093: 1081: 1077: 1065: 1053: 1049: 1037: 1033: 1019:residue theorem 978: 969: 960:(sphere, point) 959: 949: 942: 936: 930: 929: 928: 921: 918: 917: 916: 909: 906: 905: 904: 900: 893: 887: 882: 881: 880: 873: 870: 869: 868: 861: 858: 857: 856: 852: 848: 844: 834: 823: 807: 797: 796:. For example, 791: 781: 767: 764: 763: 762: 755: 752: 751: 750: 746: 740: 730: 727: 726: 725: 718: 715: 714: 713: 709: 703: 686: 668: 661: 655: 648: 643: 642: 641: 634: 631: 630: 629: 622: 619: 618: 617: 613: 612:; the equation 609: 600: 597:Euclidean space 586: 583: 582: 581: 574: 571: 570: 569: 562: 559: 558: 557: 553: 535: 530: 524:homotopy groups 508: 488: 477: 465: 450: 446: 438: 428: 423: 415: 405: 395: 389: 375: 359: 353: 343: 333: 320: 314: 308: 302: 291: 283: 279: 271: 265: 244: 238: 237: 227: 223: 215: 209: 202: 195: 192:Euclidean space 187: 181: 171: 161: 157: 149: 123:homotopy groups 119:homology groups 75: 64: 58: 55: 45:Please help to 44: 28: 24: 17: 12: 11: 5: 11411: 11409: 11401: 11400: 11395: 11385: 11384: 11381: 11380: 11363: 11342: 11324:Hatcher, Allen 11320: 11299: 11298:External links 11296: 11295: 11294: 11288: 11264: 11246:(1): 637–665, 11226: 11212: 11209: 11208: 11207: 11193: 11171: 11157: 11139:Hatcher, Allen 11133: 11130: 11128: 11127: 11110: 11080:(2): 319–407, 11067: 11031:(2): 501–580, 11018: 11005: 10984: 10970: 10950: 10921: 10895:(3): 425–505, 10884: 10870: 10853: 10839: 10821: 10811: 10773:(4): 707–732, 10759: 10746: 10723: 10698: 10679: 10657: 10652:Mahowald, Mark 10648: 10635: 10614: 10596: 10571: 10545:(3): 504–537, 10525: 10494: 10480: 10453: 10434: 10412: 10399: 10378: 10342:(8): 141–146, 10329: 10316: 10272:(2): 305–320, 10261: 10235:(3): 549–565, 10223:Moore, John C. 10215: 10179: 10141: 10121:(2): 265–326, 10100: 10072:(3): 533–550, 10055: 10005: 10003: 10000: 9997: 9996: 9992:Ravenel (2003) 9976: 9959: 9947: 9935: 9923: 9911: 9899: 9887: 9875: 9863: 9851: 9839: 9827: 9815: 9803: 9791: 9779: 9764: 9752: 9740: 9728: 9716: 9704: 9692: 9680: 9668: 9651: 9639: 9627: 9625:, p. 342. 9615: 9600: 9598:, p. 203. 9588: 9576: 9564: 9552: 9540: 9528: 9526:, p. 349. 9513: 9511:, p. 348. 9501: 9488: 9483: 9479: 9475: 9470: 9466: 9445: 9433: 9417: 9405: 9393: 9381: 9379:, p. 129. 9369: 9357: 9355:, p. xii. 9345: 9327: 9326: 9324: 9321: 9319: 9316: 9313: 9312: 9303: 9300: 9297: 9294: 9285: 9282: 9276: 9270: 9264: 9260: 9259: 9250: 9247: 9244: 9241: 9232: 9229: 9223: 9217: 9211: 9207: 9206: 9197: 9194: 9191: 9188: 9179: 9176: 9170: 9165: 9159: 9155: 9154: 9145: 9142: 9139: 9136: 9127: 9124: 9118: 9112: 9106: 9102: 9101: 9088: 9085: 9082: 9079: 9074: 9071: 9065: 9059: 9053: 9049: 9048: 9039: 9036: 9033: 9030: 9021: 9018: 9012: 9006: 9000: 8996: 8995: 8986: 8983: 8980: 8977: 8972: 8969: 8963: 8957: 8951: 8947: 8946: 8933: 8930: 8927: 8924: 8915: 8912: 8906: 8900: 8894: 8890: 8889: 8880: 8877: 8874: 8871: 8866: 8863: 8857: 8851: 8845: 8841: 8840: 8835: 8832: 8829: 8826: 8821: 8818: 8813: 8810: 8804: 8800: 8799: 8796: 8793: 8790: 8787: 8784: 8781: 8778: 8775: 8763:J-homomorphism 8714: 8705: 8702: 8699: 8698: 8695: 8692: 8689: 8686: 8683: 8680: 8677: 8674: 8671: 8660: 8656: 8655: 8652: 8649: 8646: 8643: 8640: 8637: 8634: 8631: 8628: 8617: 8613: 8612: 8609: 8606: 8603: 8600: 8597: 8594: 8591: 8588: 8585: 8574: 8570: 8569: 8566: 8563: 8560: 8557: 8554: 8551: 8548: 8545: 8542: 8531: 8527: 8526: 8523: 8520: 8517: 8514: 8511: 8508: 8505: 8502: 8499: 8488: 8484: 8483: 8480: 8477: 8474: 8471: 8468: 8465: 8462: 8459: 8456: 8445: 8441: 8440: 8437: 8434: 8431: 8428: 8425: 8422: 8419: 8416: 8413: 8402: 8398: 8397: 8394: 8391: 8388: 8385: 8382: 8379: 8376: 8373: 8370: 8359: 8355: 8354: 8349: 8344: 8339: 8334: 8329: 8324: 8319: 8314: 8309: 8298: 8297: 8294: 8291: 8288: 8285: 8282: 8279: 8276: 8273: 8270: 8267: 8264: 8261: 8258: 8247: 8243: 8242: 8239: 8236: 8233: 8230: 8227: 8224: 8221: 8218: 8213: 8210: 8207: 8204: 8201: 8190: 8186: 8185: 8182: 8179: 8176: 8173: 8170: 8167: 8164: 8161: 8156: 8153: 8150: 8147: 8144: 8133: 8129: 8128: 8125: 8122: 8119: 8116: 8113: 8110: 8107: 8104: 8101: 8098: 8095: 8092: 8089: 8078: 8074: 8073: 8070: 8067: 8064: 8061: 8058: 8055: 8052: 8049: 8046: 8043: 8040: 8037: 8034: 8023: 8019: 8018: 8015: 8012: 8009: 8006: 8003: 8000: 7997: 7994: 7989: 7986: 7983: 7980: 7977: 7966: 7962: 7961: 7958: 7955: 7952: 7949: 7946: 7943: 7940: 7937: 7934: 7931: 7928: 7925: 7922: 7911: 7907: 7906: 7899: 7896: 7893: 7890: 7887: 7884: 7881: 7878: 7875: 7872: 7869: 7866: 7863: 7860: 7849: 7845: 7844: 7841: 7838: 7835: 7832: 7829: 7826: 7823: 7820: 7817: 7814: 7811: 7808: 7805: 7802: 7791: 7787: 7786: 7783: 7780: 7777: 7774: 7771: 7768: 7765: 7762: 7759: 7754: 7751: 7748: 7745: 7742: 7731: 7727: 7726: 7723: 7720: 7717: 7714: 7711: 7708: 7705: 7702: 7699: 7696: 7693: 7690: 7687: 7684: 7673: 7669: 7668: 7665: 7662: 7659: 7656: 7653: 7650: 7647: 7644: 7641: 7638: 7635: 7632: 7629: 7626: 7615: 7611: 7610: 7607: 7604: 7601: 7598: 7595: 7592: 7589: 7586: 7583: 7580: 7577: 7574: 7571: 7568: 7557: 7553: 7552: 7549: 7546: 7543: 7540: 7537: 7534: 7531: 7528: 7525: 7522: 7519: 7516: 7513: 7510: 7499: 7495: 7494: 7491: 7488: 7485: 7482: 7479: 7476: 7473: 7470: 7467: 7464: 7461: 7458: 7455: 7452: 7441: 7437: 7436: 7433: 7430: 7427: 7424: 7421: 7418: 7415: 7412: 7409: 7406: 7403: 7400: 7397: 7394: 7383: 7379: 7378: 7375: 7372: 7369: 7366: 7363: 7360: 7357: 7354: 7351: 7348: 7345: 7342: 7339: 7336: 7325: 7321: 7320: 7317: 7314: 7311: 7308: 7305: 7302: 7299: 7296: 7293: 7290: 7287: 7284: 7281: 7278: 7267: 7263: 7262: 7259: 7256: 7253: 7250: 7247: 7244: 7241: 7238: 7235: 7232: 7229: 7226: 7223: 7220: 7209: 7205: 7204: 7201: 7198: 7195: 7192: 7189: 7186: 7183: 7180: 7177: 7174: 7171: 7168: 7165: 7162: 7151: 7147: 7146: 7143: 7140: 7137: 7134: 7131: 7128: 7125: 7122: 7119: 7116: 7113: 7110: 7107: 7105: 7094: 7090: 7089: 7084: 7079: 7074: 7069: 7064: 7059: 7054: 7049: 7044: 7039: 7034: 7029: 7024: 7019: 7009:in the table. 7001: 6997: 6993: 6985: 6977: 6969: 6968: 6961: 6948: 6942: 6917: 6906: 6899: 6882: 6862: 6841: 6835: 6832: 6831: 6830: 6807: 6783: 6772: 6766: 6765: 6733: 6716: 6709: 6708: 6707: 6696: 6693: 6689: 6683: 6678: 6674: 6670: 6665: 6662: 6659: 6655: 6651: 6647: 6641: 6637: 6623: 6622: 6619:exotic spheres 6602:-spheres (for 6588: 6583: 6561: 6558:critical value 6530: 6514: 6490: 6461: 6436: 6433:The fact that 6431: 6410: 6405:winding number 6399: 6396: 6346: 6314: 6313: 6297: 6271: 6270: 6234: 6226: 6225: 6214: 6213: 6200: 6199: 6176: 6143: 6137: 6124: 6111: 6102:The classical 6100: 6093: 6055: 5994: 5991: 5928: 5901: 5883: 5865: 5847: 5813: 5812: 5799: 5794: 5790: 5784: 5781: 5778: 5774: 5770: 5765: 5760: 5756: 5737: 5736:Ring structure 5734: 5709: 5697: 5678: 5655: 5650: 5645: 5593: 5572: 5550: 5545: 5544: 5521: 5496: 5485: 5478: 5435: 5394: 5384: 5377:J-homomorphism 5353: 5346:J-homomorphism 5344:Main article: 5341: 5338: 5309: 5259: 5258: 5247: 5244: 5241: 5238: 5233: 5230: 5227: 5224: 5220: 5216: 5211: 5208: 5205: 5202: 5198: 5194: 5191: 5188: 5185: 5182: 5177: 5174: 5171: 5168: 5164: 5160: 5155: 5152: 5149: 5146: 5143: 5140: 5136: 5132: 5129: 5126: 5123: 5120: 5115: 5112: 5108: 5104: 5099: 5096: 5093: 5090: 5086: 5050: 5012: 4998: 4966: 4869: 4835:(for positive 4822: 4808: 4798: 4795: 4789:of spaces and 4753: 4742: 4728:Hopf fibration 4691: 4641:differentiable 4626: 4620:and the group 4605: 4600:Lev Pontryagin 4591: 4588: 4584:Hopf invariant 4580: 4579: 4568: 4563: 4559: 4555: 4550: 4546: 4542: 4537: 4533: 4518: 4517: 4506: 4503: 4498: 4494: 4490: 4485: 4481: 4477: 4474: 4469: 4465: 4461: 4456: 4452: 4448: 4445: 4440: 4436: 4432: 4427: 4423: 4356: 4355: 4344: 4341: 4336: 4332: 4328: 4323: 4320: 4317: 4313: 4309: 4306: 4301: 4297: 4293: 4288: 4284: 4280: 4277: 4272: 4268: 4264: 4259: 4255: 4243: 4242: 4231: 4228: 4223: 4219: 4215: 4210: 4207: 4204: 4200: 4196: 4193: 4188: 4184: 4180: 4175: 4171: 4167: 4164: 4159: 4155: 4151: 4146: 4142: 4120: 4108: 4095: 4094: 4081: 4077: 4073: 4068: 4064: 4060: 4055: 4051: 4039: 4038: 4025: 4021: 4017: 4012: 4008: 4004: 3999: 3995: 3966: 3962: 3958: 3953: 3949: 3945: 3940: 3936: 3901: 3890: 3875: 3866: 3854: 3847: 3823: 3809: 3790: 3785: 3784: 3773: 3770: 3765: 3761: 3757: 3752: 3749: 3746: 3742: 3738: 3735: 3730: 3726: 3722: 3717: 3713: 3709: 3706: 3701: 3697: 3693: 3688: 3684: 3660: 3649: 3621: 3620: 3609: 3606: 3603: 3598: 3594: 3590: 3585: 3582: 3579: 3575: 3571: 3568: 3563: 3559: 3555: 3550: 3546: 3542: 3539: 3534: 3530: 3526: 3521: 3517: 3513: 3510: 3485: 3459: 3456:, maps all of 3435: 3425: 3420: 3419: 3408: 3405: 3402: 3399: 3396: 3393: 3388: 3385: 3382: 3378: 3374: 3371: 3368: 3365: 3360: 3356: 3352: 3349: 3346: 3343: 3338: 3334: 3330: 3327: 3324: 3321: 3316: 3312: 3308: 3305: 3273: 3272: 3261: 3256: 3252: 3248: 3243: 3239: 3235: 3230: 3226: 3207:Hopf fibration 3205:The classical 3202: 3199: 3167: 3131: 3128:. The groups 3101: 3083: 3069: 3066: 3062: 3061: 3041: 3027: 3005: 2976: 2966: 2961: 2958: 2928: 2916:covering space 2896: 2889: 2888: 2885: 2882: 2879: 2876: 2873: 2870: 2867: 2864: 2861: 2858: 2855: 2852: 2849: 2846: 2843: 2840: 2837: 2834: 2831: 2828: 2822: 2821: 2816: 2810: 2807: 2804: 2801: 2798: 2795: 2792: 2789: 2786: 2783: 2780: 2777: 2774: 2771: 2768: 2765: 2762: 2759: 2756: 2753: 2747: 2746: 2741: 2735: 2732: 2728: 2725: 2722: 2719: 2716: 2713: 2710: 2707: 2704: 2701: 2698: 2695: 2692: 2689: 2686: 2683: 2680: 2677: 2674: 2671: 2665: 2664: 2661: 2657: 2654: 2649: 2643: 2640: 2637: 2634: 2631: 2628: 2625: 2622: 2619: 2616: 2613: 2610: 2607: 2604: 2601: 2598: 2595: 2592: 2589: 2586: 2583: 2580: 2574: 2573: 2568: 2561: 2558: 2552: 2548: 2546: 2541: 2536: 2530: 2527: 2524: 2521: 2518: 2515: 2511: 2508: 2503: 2497: 2492: 2486: 2483: 2480: 2477: 2474: 2471: 2468: 2465: 2462: 2459: 2456: 2450: 2449: 2444: 2438: 2433: 2426: 2423: 2420: 2416: 2413: 2408: 2402: 2399: 2396: 2393: 2390: 2387: 2384: 2381: 2378: 2375: 2372: 2369: 2366: 2363: 2360: 2357: 2354: 2351: 2348: 2345: 2339: 2338: 2333: 2327: 2322: 2315: 2312: 2309: 2305: 2302: 2297: 2291: 2288: 2285: 2282: 2279: 2276: 2273: 2270: 2267: 2264: 2261: 2258: 2255: 2252: 2249: 2246: 2243: 2240: 2237: 2234: 2228: 2227: 2224: 2221: 2218: 2215: 2212: 2209: 2206: 2203: 2200: 2197: 2194: 2191: 2188: 2185: 2182: 2176: 2175: 2172: 2169: 2166: 2163: 2160: 2157: 2154: 2151: 2148: 2145: 2142: 2139: 2136: 2133: 2130: 2127: 2124: 2121: 2118: 2115: 2112: 2109: 2106: 2103: 2100: 2097: 2094: 2091: 2088: 2085: 2071: 2067: 2061: 2051: 2047: 2034: 2012: 2009: 1986: 1962: 1943: 1930: 1905: 1882: 1822:-sphere to an 1803: 1789: 1788:General theory 1786: 1776:are known for 1761: 1748:Daniel Isaksen 1713: 1699: 1645:Analysis situs 1640:Henri Poincaré 1636:Camille Jordan 1631: 1628: 1617: 1608:Hopf fibration 1570:Hopf fibration 1561: 1552: 1548: 1505: 1498: 1489: 1485: 1468: 1424: 1415: 1411: 1394: 1354: 1345: 1341: 1330: 1306:winding number 1279: 1259: 1249: 1240: 1236: 1230: 1227: 1183: 1176:. If for some 1164:(for positive 1127: 1046:path connected 1035: 1010:continuous map 977: 976:Homotopy group 974: 965:pointed sphere 956: 955: 931: 919: 907: 883: 871: 859: 830: 829: 816: 815: 765: 753: 728: 716: 696: 695: 679: 678: 644: 632: 620: 592: 591: 584: 572: 560: 534: 528: 507: 504: 430: 407: 402: 401: 386:Hopf fibration 372: 355: 330: 267: 211: 207:homotopy group 134:Hopf fibration 77: 76: 59:September 2022 31: 29: 22: 15: 13: 10: 9: 6: 4: 3: 2: 11410: 11399: 11396: 11394: 11391: 11390: 11388: 11371: 11370: 11364: 11361: 11350: 11349: 11343: 11331: 11330: 11325: 11321: 11312: 11311: 11306: 11302: 11301: 11297: 11291: 11285: 11281: 11277: 11273: 11269: 11268:May, J. Peter 11265: 11261: 11257: 11253: 11249: 11245: 11241: 11240: 11235: 11231: 11227: 11223: 11219: 11215: 11214: 11210: 11204: 11200: 11196: 11190: 11186: 11182: 11181: 11176: 11175:May, J. Peter 11172: 11168: 11164: 11160: 11154: 11150: 11146: 11145: 11140: 11136: 11135: 11131: 11125: 11121: 11117: 11116: 11111: 11107: 11103: 11098: 11093: 11088: 11083: 11079: 11075: 11074: 11068: 11064: 11060: 11056: 11052: 11048: 11044: 11039: 11034: 11030: 11026: 11025: 11019: 11016: 11012: 11008: 11006:0-387-20430-X 11002: 10998: 10994: 10990: 10985: 10981: 10977: 10973: 10967: 10963: 10959: 10955: 10951: 10947: 10943: 10939: 10935: 10931: 10927: 10922: 10918: 10914: 10910: 10906: 10902: 10898: 10894: 10890: 10885: 10881: 10877: 10873: 10867: 10863: 10859: 10854: 10850: 10846: 10842: 10836: 10832: 10831: 10826: 10822: 10819: 10815: 10812: 10808: 10804: 10800: 10796: 10791: 10786: 10781: 10776: 10772: 10768: 10764: 10763:Nishida, Goro 10760: 10757: 10753: 10749: 10747:0-8218-0268-2 10743: 10739: 10735: 10731: 10730: 10724: 10720: 10716: 10715: 10707: 10703: 10699: 10695: 10691: 10690: 10685: 10680: 10677: 10673: 10669: 10665: 10663: 10653: 10649: 10646: 10642: 10638: 10632: 10628: 10624: 10620: 10615: 10612: 10607: 10603: 10599: 10593: 10589: 10585: 10581: 10577: 10572: 10568: 10564: 10560: 10556: 10552: 10548: 10544: 10540: 10539: 10534: 10530: 10526: 10522: 10518: 10513: 10508: 10504: 10500: 10495: 10491: 10487: 10483: 10477: 10472: 10467: 10463: 10459: 10454: 10451: 10447: 10443: 10439: 10435: 10431: 10427: 10426: 10421: 10417: 10413: 10410: 10406: 10402: 10400:0-387-94657-8 10396: 10392: 10388: 10384: 10379: 10375: 10371: 10367: 10363: 10359: 10355: 10350: 10345: 10341: 10337: 10333: 10328: 10325:-theory over 10324: 10317: 10313: 10309: 10304: 10299: 10295: 10291: 10287: 10283: 10279: 10275: 10271: 10267: 10262: 10258: 10254: 10250: 10246: 10242: 10238: 10234: 10230: 10229: 10224: 10220: 10216: 10212: 10208: 10204: 10200: 10196: 10192: 10188: 10184: 10183:Cartan, Henri 10180: 10176: 10172: 10168: 10164: 10160: 10156: 10155: 10150: 10146: 10145:Cartan, Henri 10142: 10138: 10134: 10129: 10124: 10120: 10116: 10115: 10110: 10106: 10101: 10097: 10093: 10089: 10085: 10080: 10075: 10071: 10067: 10066: 10061: 10056: 10051: 10046: 10042: 10038: 10030: 10025: 10021: 10017: 10016: 10011: 10007: 10006: 10001: 9993: 9989: 9985: 9980: 9977: 9973: 9969: 9963: 9960: 9956: 9951: 9948: 9944: 9939: 9936: 9932: 9927: 9924: 9921:, p. 32. 9920: 9915: 9912: 9908: 9903: 9900: 9896: 9891: 9888: 9884: 9879: 9876: 9872: 9867: 9864: 9860: 9855: 9852: 9848: 9843: 9840: 9836: 9831: 9828: 9824: 9819: 9816: 9812: 9811:Mahowald 2001 9807: 9804: 9800: 9795: 9792: 9788: 9783: 9780: 9776: 9771: 9769: 9765: 9761: 9756: 9753: 9749: 9744: 9741: 9737: 9732: 9729: 9726:, p. 25. 9725: 9720: 9717: 9713: 9708: 9705: 9701: 9696: 9693: 9689: 9684: 9681: 9677: 9672: 9669: 9665: 9660: 9658: 9656: 9652: 9648: 9643: 9640: 9636: 9631: 9628: 9624: 9619: 9616: 9612: 9607: 9605: 9601: 9597: 9592: 9589: 9585: 9580: 9577: 9573: 9568: 9565: 9562:, p. 90. 9561: 9560:Walschap 2004 9556: 9553: 9549: 9544: 9541: 9538:, p. 61. 9537: 9532: 9529: 9525: 9520: 9518: 9514: 9510: 9505: 9502: 9481: 9477: 9468: 9464: 9455: 9449: 9446: 9443:, p. 29. 9442: 9437: 9434: 9430: 9426: 9421: 9418: 9414: 9409: 9406: 9402: 9397: 9394: 9391:, p. 28. 9390: 9385: 9382: 9378: 9373: 9370: 9366: 9361: 9358: 9354: 9349: 9346: 9342: 9337: 9335: 9333: 9329: 9322: 9317: 9311: 9307: 9304: 9301: 9298: 9295: 9293: 9289: 9286: 9283: 9280: 9277: 9274: 9271: 9268: 9262: 9261: 9258: 9254: 9251: 9248: 9245: 9242: 9240: 9236: 9233: 9230: 9227: 9224: 9221: 9218: 9215: 9209: 9208: 9205: 9201: 9198: 9195: 9192: 9189: 9187: 9183: 9180: 9177: 9174: 9171: 9169: 9166: 9163: 9157: 9156: 9153: 9149: 9146: 9143: 9140: 9137: 9135: 9131: 9128: 9125: 9122: 9119: 9116: 9113: 9110: 9104: 9103: 9100: 9096: 9092: 9089: 9086: 9083: 9080: 9078: 9075: 9072: 9069: 9066: 9063: 9060: 9057: 9051: 9050: 9047: 9043: 9040: 9037: 9034: 9031: 9029: 9025: 9022: 9019: 9016: 9013: 9010: 9007: 9004: 8998: 8997: 8994: 8990: 8987: 8984: 8981: 8978: 8976: 8973: 8970: 8967: 8964: 8961: 8958: 8955: 8949: 8948: 8945: 8941: 8937: 8934: 8931: 8928: 8925: 8923: 8919: 8916: 8913: 8910: 8907: 8904: 8901: 8898: 8892: 8891: 8888: 8884: 8881: 8878: 8875: 8872: 8870: 8867: 8864: 8861: 8858: 8855: 8852: 8849: 8843: 8842: 8839: 8836: 8833: 8830: 8827: 8825: 8822: 8819: 8817: 8814: 8811: 8808: 8802: 8801: 8797: 8794: 8791: 8788: 8785: 8782: 8779: 8776: 8773: 8770: 8769: 8766: 8764: 8760: 8754: 8747: 8727: 8717: 8703: 8696: 8693: 8690: 8687: 8684: 8681: 8678: 8675: 8672: 8669: 8664: 8658: 8657: 8653: 8650: 8647: 8644: 8641: 8638: 8635: 8632: 8629: 8626: 8621: 8615: 8614: 8610: 8607: 8604: 8601: 8598: 8595: 8592: 8589: 8586: 8583: 8578: 8572: 8571: 8567: 8564: 8561: 8558: 8555: 8552: 8549: 8546: 8543: 8540: 8535: 8529: 8528: 8524: 8521: 8518: 8515: 8512: 8509: 8506: 8503: 8500: 8497: 8492: 8486: 8485: 8481: 8478: 8475: 8472: 8469: 8466: 8463: 8460: 8457: 8454: 8449: 8443: 8442: 8438: 8435: 8432: 8429: 8426: 8423: 8420: 8417: 8414: 8411: 8406: 8400: 8399: 8395: 8392: 8389: 8386: 8383: 8380: 8377: 8374: 8371: 8368: 8363: 8357: 8356: 8353: 8350: 8348: 8345: 8343: 8340: 8338: 8335: 8333: 8330: 8328: 8325: 8323: 8320: 8318: 8315: 8313: 8310: 8307: 8304: 8303: 8295: 8292: 8289: 8286: 8283: 8280: 8277: 8274: 8271: 8268: 8265: 8262: 8259: 8256: 8251: 8245: 8244: 8240: 8237: 8234: 8231: 8228: 8225: 8222: 8219: 8214: 8211: 8208: 8205: 8202: 8199: 8194: 8188: 8187: 8183: 8180: 8177: 8174: 8171: 8168: 8165: 8162: 8157: 8154: 8151: 8148: 8145: 8142: 8137: 8131: 8130: 8126: 8123: 8120: 8117: 8114: 8111: 8108: 8105: 8102: 8099: 8096: 8093: 8090: 8087: 8082: 8076: 8075: 8071: 8068: 8065: 8062: 8059: 8056: 8053: 8050: 8047: 8044: 8041: 8038: 8035: 8032: 8027: 8021: 8020: 8016: 8013: 8010: 8007: 8004: 8001: 7998: 7995: 7990: 7987: 7984: 7981: 7978: 7975: 7970: 7964: 7963: 7959: 7956: 7953: 7950: 7947: 7944: 7941: 7938: 7935: 7932: 7929: 7926: 7923: 7920: 7915: 7909: 7908: 7905: 7897: 7894: 7891: 7888: 7885: 7882: 7879: 7876: 7873: 7870: 7867: 7864: 7861: 7858: 7853: 7847: 7846: 7842: 7839: 7836: 7833: 7830: 7827: 7824: 7821: 7818: 7815: 7812: 7809: 7806: 7803: 7800: 7795: 7789: 7788: 7784: 7781: 7778: 7775: 7772: 7769: 7766: 7763: 7760: 7755: 7752: 7749: 7746: 7743: 7740: 7735: 7729: 7728: 7724: 7721: 7718: 7715: 7712: 7709: 7706: 7703: 7700: 7697: 7694: 7691: 7688: 7685: 7682: 7677: 7671: 7670: 7666: 7663: 7660: 7657: 7654: 7651: 7648: 7645: 7642: 7639: 7636: 7633: 7630: 7627: 7624: 7619: 7613: 7612: 7608: 7605: 7602: 7599: 7596: 7593: 7590: 7587: 7584: 7581: 7578: 7575: 7572: 7569: 7566: 7561: 7555: 7554: 7550: 7547: 7544: 7541: 7538: 7535: 7532: 7529: 7526: 7523: 7520: 7517: 7514: 7511: 7508: 7503: 7497: 7496: 7492: 7489: 7486: 7483: 7480: 7477: 7474: 7471: 7468: 7465: 7462: 7459: 7456: 7453: 7450: 7445: 7439: 7438: 7434: 7431: 7428: 7425: 7422: 7419: 7416: 7413: 7410: 7407: 7404: 7401: 7398: 7395: 7392: 7387: 7381: 7380: 7376: 7373: 7370: 7367: 7364: 7361: 7358: 7355: 7352: 7349: 7346: 7343: 7340: 7337: 7334: 7329: 7323: 7322: 7318: 7315: 7312: 7309: 7306: 7303: 7300: 7297: 7294: 7291: 7288: 7285: 7282: 7279: 7276: 7271: 7265: 7264: 7260: 7257: 7254: 7251: 7248: 7245: 7242: 7239: 7236: 7233: 7230: 7227: 7224: 7221: 7218: 7213: 7207: 7206: 7202: 7199: 7196: 7193: 7190: 7187: 7184: 7181: 7178: 7175: 7172: 7169: 7166: 7163: 7160: 7155: 7149: 7148: 7144: 7141: 7138: 7135: 7132: 7129: 7126: 7123: 7120: 7117: 7114: 7111: 7108: 7106: 7103: 7098: 7092: 7091: 7088: 7085: 7083: 7080: 7078: 7075: 7073: 7070: 7068: 7065: 7063: 7060: 7058: 7055: 7053: 7050: 7048: 7045: 7043: 7040: 7038: 7035: 7033: 7030: 7028: 7025: 7023: 7020: 7017: 7014: 7013: 7010: 6991: 6983: 6973: 6964: 6957: 6951: 6945: 6938: 6937:common factor 6926: 6922: 6918: 6911: 6907: 6902: 6895: 6887: 6883: 6880: 6879: 6878: 6874: 6869: 6865: 6857: 6853: 6848: 6844: 6833: 6828: 6824: 6820: 6816: 6812: 6808: 6792: 6788: 6784: 6781: 6775: 6768: 6767: 6755: 6754:-homomorphism 6736: 6726: 6719: 6715: 6710: 6694: 6691: 6687: 6681: 6676: 6672: 6663: 6660: 6657: 6653: 6649: 6645: 6639: 6627: 6626: 6625: 6624: 6620: 6612: 6606: 6597: 6591: 6584: 6581: 6578: 6574: 6570: 6566: 6562: 6559: 6553: 6546: 6539: 6533: 6478: 6474: 6470: 6466: 6462: 6459: 6456:-dimensional 6451: 6445: 6439: 6432: 6429: 6426: 6422: 6416: 6406: 6402: 6401: 6397: 6395: 6393: 6388: 6384: 6378: 6372: 6367: 6361: 6354: 6349: 6341: 6336: 6333: 6329: 6325: 6320: 6311: 6305: 6296: 6290: 6284: 6278: 6273: 6272: 6267: 6261: 6255: 6249: 6243: 6233: 6228: 6227: 6222: 6216: 6215: 6210: 6209: 6204: 6197: 6193: 6189: 6181: 6177: 6174: 6164: 6160: 6156: 6150:over the mod 6146: 6140: 6131: 6127: 6118: 6110: 6105: 6101: 6098: 6094: 6084: 6077: 6064: 6060: 6056: 6053: 6049: 6045: 6041: 6037: 6036: 6035: 6033: 6029: 6019: 6015: 6008: 5992: 5990: 5988: 5984: 5979: 5972: 5968: 5964: 5958: 5955:, there is a 5952: 5948: 5941: 5938: 5908: 5893: 5875: 5857: 5835: 5833: 5829: 5825: 5821: 5818: 5797: 5792: 5788: 5782: 5779: 5776: 5772: 5768: 5763: 5758: 5754: 5746: 5745: 5744: 5743: 5735: 5733: 5729: 5721: 5712: 5700: 5693: 5681: 5648: 5624: 5592: 5581: 5575: 5562: 5557: 5553: 5540: 5536: 5531: 5525: 5520: 5510: 5500: 5495: 5486: 5479: 5476: 5472: 5464: 5463: 5462: 5456: 5450: 5443: 5438: 5429: 5425: 5420: 5414: 5406: 5401: 5397: 5392: 5387: 5382: 5378: 5372: 5365: 5360: 5356: 5347: 5339: 5337: 5329: 5321: 5312: 5292: 5288: 5284: 5274: 5266: 5242: 5231: 5228: 5225: 5222: 5218: 5209: 5206: 5203: 5200: 5196: 5192: 5186: 5175: 5172: 5169: 5166: 5162: 5153: 5150: 5147: 5144: 5141: 5138: 5134: 5130: 5124: 5113: 5110: 5106: 5097: 5094: 5091: 5088: 5084: 5076: 5075: 5074: 5068: 5063: 5059: 5053: 5040: 5036: 5032: 5021: 5015: 5006: 5001: 4985: 4978: 4973: 4969: 4962: 4952: 4948: 4937: 4933: 4922: 4918: 4911: 4900: 4896: 4891: 4881: 4876: 4872: 4856: 4842: 4832: 4826: 4816: 4811: 4804: 4796: 4794: 4792: 4788: 4784: 4778: 4774: 4768: 4761: 4748: 4737: 4733: 4729: 4724: 4720: 4716: 4711: 4706: 4699: 4694: 4682: 4678: 4674: 4668: 4664: 4660: 4656:. Every map 4655: 4654:normal bundle 4650: 4642: 4636: 4629: 4617: 4612: 4608: 4601: 4597: 4589: 4587: 4585: 4566: 4561: 4557: 4548: 4544: 4535: 4531: 4523: 4522: 4521: 4504: 4496: 4492: 4483: 4479: 4475: 4467: 4463: 4454: 4450: 4446: 4438: 4434: 4425: 4421: 4413: 4412: 4411: 4408: 4401: 4395: 4388: 4382: 4375: 4368: 4362: 4342: 4334: 4330: 4321: 4318: 4315: 4311: 4307: 4299: 4295: 4286: 4282: 4278: 4270: 4266: 4257: 4253: 4245: 4244: 4229: 4221: 4217: 4208: 4205: 4202: 4198: 4194: 4186: 4182: 4173: 4169: 4165: 4157: 4153: 4144: 4140: 4132: 4131: 4130: 4126: 4114: 4104: 4100: 4079: 4075: 4066: 4062: 4053: 4049: 4041: 4040: 4023: 4019: 4010: 4006: 3997: 3993: 3985: 3984: 3983: 3982: 3964: 3960: 3951: 3947: 3938: 3934: 3924: 3921: 3917: 3912: 3900: 3889: 3874: 3865: 3853: 3846: 3839: 3831: 3826: 3817: 3812: 3802:vanishes for 3799: 3793: 3771: 3763: 3759: 3750: 3747: 3744: 3740: 3736: 3728: 3724: 3715: 3711: 3707: 3699: 3695: 3686: 3682: 3674: 3673: 3672: 3668: 3663: 3658: 3652: 3646: 3642: 3637: 3632: 3627: 3607: 3596: 3592: 3583: 3580: 3577: 3573: 3561: 3557: 3548: 3544: 3532: 3528: 3519: 3515: 3508: 3501: 3500: 3499: 3497: 3491: 3480: 3474: 3467: 3462: 3454: 3450: 3443: 3438: 3433: 3428: 3406: 3403: 3394: 3386: 3383: 3380: 3376: 3366: 3358: 3354: 3344: 3336: 3332: 3322: 3314: 3310: 3303: 3296: 3295: 3294: 3292: 3287: 3283: 3279: 3259: 3254: 3250: 3241: 3237: 3228: 3224: 3216: 3215: 3214: 3212: 3208: 3200: 3198: 3196: 3190: 3186: 3179: 3170: 3160: 3154: 3150: 3143: 3138: 3134: 3125: 3121: 3114: 3108: 3104: 3095: 3090: 3086: 3079: 3075: 3067: 3065: 3057: 3050: 3044: 3035: 3030: 3022: 3018: 3011: 3006: 3002: 2998: 2991: 2984: 2979: 2974: 2969: 2962: 2959: 2956: 2955: 2954: 2951: 2947: 2941: 2917: 2911: 2904: 2899: 2883: 2877: 2874: 2871: 2865: 2859: 2853: 2850: 2847: 2844: 2841: 2838: 2835: 2832: 2829: 2827: 2824: 2823: 2811: 2805: 2799: 2796: 2793: 2787: 2781: 2775: 2772: 2769: 2766: 2763: 2760: 2757: 2754: 2752: 2749: 2748: 2736: 2726: 2720: 2714: 2711: 2708: 2702: 2696: 2690: 2687: 2684: 2681: 2678: 2675: 2672: 2670: 2667: 2666: 2655: 2644: 2638: 2632: 2626: 2620: 2614: 2608: 2602: 2596: 2593: 2590: 2587: 2584: 2581: 2579: 2576: 2575: 2559: 2542: 2531: 2525: 2519: 2509: 2498: 2487: 2481: 2475: 2469: 2466: 2463: 2460: 2457: 2455: 2452: 2451: 2439: 2424: 2414: 2403: 2397: 2391: 2385: 2379: 2373: 2367: 2361: 2355: 2352: 2349: 2346: 2344: 2341: 2340: 2328: 2313: 2303: 2292: 2286: 2280: 2274: 2268: 2262: 2256: 2250: 2244: 2241: 2238: 2235: 2233: 2230: 2229: 2225: 2222: 2219: 2216: 2213: 2210: 2207: 2204: 2201: 2198: 2195: 2192: 2189: 2186: 2183: 2181: 2178: 2177: 2170: 2164: 2158: 2152: 2146: 2140: 2134: 2128: 2122: 2116: 2110: 2104: 2098: 2092: 2086: 2084: 2083: 2080: 2078: 2043: 2040:), or c) the 2037: 2026: 2025:cyclic groups 2018: 2017:trivial group 2010: 2008: 2005: 2001: 1994: 1989: 1981: 1977: 1970: 1965: 1961: 1955: 1951: 1946: 1942: 1938: 1933: 1924: 1913: 1908: 1904: 1897: 1890: 1885: 1878: 1874: 1870: 1866: 1861: 1857: 1851: 1849: 1844: 1839: 1834: 1830: 1826:-sphere with 1817: 1816:trivial group 1811: 1806: 1796:is less than 1787: 1785: 1773: 1768: 1764: 1757: 1753: 1749: 1745: 1744:Mark Mahowald 1741: 1737: 1733: 1729: 1723: 1717: 1707: 1702: 1695: 1691: 1687: 1683: 1679: 1674: 1670: 1666: 1662: 1657: 1655: 1651: 1647: 1646: 1641: 1637: 1629: 1627: 1623: 1613: 1609: 1604: 1598: 1593: 1589: 1584: 1580: 1571: 1566: 1558: 1549: 1547: 1544: 1538: 1533: 1528: 1523: 1518: 1511: 1495: 1486: 1484: 1482: 1476: 1471: 1463: 1459: 1453: 1451: 1443: 1439: 1438:trivial group 1429: 1421: 1412: 1410: 1408: 1402: 1397: 1388: 1382: 1376: 1375: 1369: 1359: 1351: 1342: 1340: 1336: 1326: 1325:often written 1322: 1317: 1315: 1311: 1307: 1303: 1296:to the group 1295: 1291: 1285: 1274: 1265: 1254: 1246: 1237: 1235: 1228: 1226: 1223: 1215: 1207: 1206:homeomorphism 1203: 1199: 1194: 1192: 1191:trivial group 1186: 1175: 1171: 1162: 1151: 1147: 1135: 1130: 1118: 1116: 1112: 1109: 1104: 1100: 1091: 1087: 1073: 1069: 1061: 1057: 1047: 1041: 1031: 1026: 1024: 1020: 1015: 1011: 1007: 1006:neighborhoods 1003: 999: 990: 982: 975: 973: 967: 966: 952: 946: 934: 927: 915: 903: 896: 886: 879: 867: 855: 842: 837: 832: 831: 827: 821: 818: 817: 813: 805: 800: 794: 788: 784: 779: 775: 761: 749: 743: 738: 724: 712: 706: 702: 698: 697: 693: 689: 684: 681: 680: 676: 671: 664: 659: 653:produces the 647: 640: 628: 616: 608: 603: 598: 594: 593: 580: 568: 556: 551: 548: 547: 546: 544: 540: 529: 527: 525: 521: 517: 513: 505: 503: 501: 497: 491: 484: 480: 475: 471: 463: 457: 453: 442: 437: 433: 422:for positive 419: 414: 410: 398: 392: 387: 382: 378: 373: 369: 363: 358: 351: 346: 340: 336: 331: 328: 327:trivial group 323: 317: 311: 305: 299: 295: 289: 288: 287: 275: 270: 262: 260: 259:abelian group 256: 252: 247: 235: 230: 219: 214: 208: 198: 194:of dimension 193: 184: 179: 174: 169: 164: 156:— called the 155: 142: 135: 130: 126: 124: 120: 116: 112: 108: 104: 100: 96: 92: 83: 73: 70: 62: 52: 48: 42: 41: 35: 30: 21: 20: 11373:, retrieved 11368: 11352:, retrieved 11347: 11336:, retrieved 11328: 11314:, retrieved 11309: 11275: 11243: 11237: 11221: 11218:Čech, Eduard 11179: 11143: 11114: 11077: 11071: 11028: 11022: 10988: 10957: 10954:Toda, Hirosi 10929: 10925: 10892: 10888: 10857: 10829: 10817: 10770: 10766: 10728: 10721:(6): 804–809 10718: 10712: 10687: 10667: 10661: 10626: 10575: 10542: 10536: 10502: 10498: 10461: 10457: 10441: 10438:Hu, Sze-tsen 10423: 10382: 10349:math/0605429 10339: 10335: 10326: 10322: 10269: 10265: 10232: 10226: 10194: 10190: 10158: 10152: 10118: 10112: 10069: 10063: 10040: 10036: 10022:(1): 21–71, 10019: 10013: 9979: 9962: 9955:Deitmar 2006 9950: 9938: 9926: 9919:Hatcher 2002 9914: 9902: 9890: 9878: 9866: 9859:Isaksen 2019 9854: 9847:Kochman 1990 9842: 9837:, Chapter 5. 9835:Ravenel 2003 9830: 9823:Ravenel 2003 9818: 9806: 9799:Ravenel 2003 9794: 9782: 9760:Nishida 1973 9755: 9743: 9731: 9724:Ravenel 2003 9719: 9707: 9702:, p. 4. 9700:Ravenel 2003 9695: 9683: 9671: 9664:Scorpan 2005 9647:Hatcher 2002 9642: 9635:Hatcher 2002 9630: 9623:Hatcher 2002 9618: 9591: 9579: 9567: 9555: 9543: 9536:Hatcher 2002 9531: 9524:Hatcher 2002 9509:Hatcher 2002 9504: 9448: 9441:Hatcher 2002 9436: 9420: 9413:Miranda 1995 9408: 9403:, p. 3. 9401:Hatcher 2002 9396: 9389:Hatcher 2002 9384: 9377:Hatcher 2002 9372: 9365:Hatcher 2002 9360: 9353:Hatcher 2002 9348: 9309: 9305: 9291: 9287: 9278: 9272: 9266: 9256: 9252: 9238: 9234: 9225: 9219: 9213: 9203: 9199: 9185: 9181: 9172: 9167: 9161: 9151: 9147: 9133: 9129: 9120: 9114: 9108: 9098: 9094: 9090: 9076: 9067: 9061: 9055: 9045: 9041: 9027: 9023: 9014: 9008: 9002: 8992: 8988: 8974: 8965: 8959: 8953: 8943: 8939: 8935: 8921: 8917: 8908: 8902: 8896: 8886: 8882: 8868: 8859: 8853: 8847: 8837: 8823: 8815: 8806: 8771: 8752: 8745: 8725: 8715: 8707: 8667: 8662: 8624: 8619: 8581: 8576: 8538: 8533: 8495: 8490: 8452: 8447: 8409: 8404: 8366: 8361: 8351: 8346: 8341: 8336: 8331: 8326: 8321: 8316: 8311: 8305: 8254: 8249: 8197: 8192: 8140: 8135: 8085: 8080: 8030: 8025: 7973: 7968: 7918: 7913: 7901: 7856: 7851: 7798: 7793: 7738: 7733: 7680: 7675: 7622: 7617: 7564: 7559: 7506: 7501: 7448: 7443: 7390: 7385: 7332: 7327: 7274: 7269: 7216: 7211: 7158: 7153: 7101: 7096: 7086: 7081: 7076: 7071: 7066: 7061: 7056: 7051: 7046: 7041: 7036: 7031: 7026: 7021: 7015: 6989: 6981: 6971: 6970: 6962: 6949: 6943: 6900: 6894:cyclic group 6872: 6867: 6863: 6858: 6851: 6846: 6842: 6837: 6773: 6734: 6717: 6713: 6604: 6589: 6551: 6544: 6537: 6531: 6448:implies the 6443: 6437: 6414: 6398:Applications 6386: 6382: 6376: 6370: 6359: 6352: 6347: 6339: 6335:braid groups 6318: 6315: 6303: 6294: 6288: 6282: 6276: 6265: 6259: 6253: 6247: 6241: 6231: 6220: 6162: 6158: 6144: 6138: 6129: 6125: 6108: 6082: 6075: 5996: 5978:Hiroshi Toda 5970: 5966: 5962: 5957:Toda bracket 5950: 5946: 5939: 5936: 5909: 5907:is trivial. 5891: 5873: 5855: 5837:Example: If 5836: 5814: 5739: 5727: 5719: 5710: 5698: 5679: 5625: 5590: 5579: 5573: 5560: 5555: 5551: 5546: 5538: 5534: 5523: 5518: 5508: 5498: 5493: 5448: 5441: 5436: 5427: 5423: 5417:denotes the 5412: 5404: 5399: 5395: 5390: 5385: 5380: 5370: 5363: 5358: 5354: 5349: 5327: 5319: 5310: 5295:for a prime 5290: 5286: 5282: 5275: 5264: 5260: 5064: 5057: 5051: 5038: 5034: 5030: 5019: 5013: 5004: 4999: 4983: 4976: 4971: 4967: 4950: 4946: 4935: 4931: 4920: 4916: 4909: 4898: 4894: 4879: 4874: 4870: 4857: 4830: 4824: 4814: 4809: 4800: 4776: 4772: 4766: 4759: 4746: 4735: 4731: 4722: 4718: 4714: 4704: 4697: 4692: 4680: 4676: 4672: 4666: 4662: 4658: 4648: 4634: 4627: 4615: 4610: 4606: 4593: 4581: 4519: 4406: 4399: 4393: 4386: 4380: 4373: 4366: 4360: 4357: 4124: 4112: 4096: 3925: 3919: 3915: 3898: 3887: 3872: 3863: 3851: 3844: 3840: 3829: 3824: 3815: 3810: 3797: 3791: 3786: 3666: 3661: 3656: 3650: 3635: 3625: 3622: 3489: 3478: 3472: 3465: 3460: 3452: 3448: 3441: 3436: 3431: 3426: 3421: 3285: 3281: 3277: 3274: 3211:fiber bundle 3204: 3194: 3188: 3184: 3177: 3168: 3158: 3152: 3148: 3141: 3136: 3132: 3123: 3119: 3112: 3106: 3102: 3093: 3088: 3084: 3071: 3063: 3055: 3048: 3042: 3033: 3028: 3020: 3016: 3012:= 2, 3, 4, 5 3009: 3000: 2996: 2989: 2982: 2977: 2972: 2967: 2949: 2945: 2942: 2909: 2902: 2897: 2892: 2825: 2750: 2668: 2577: 2453: 2342: 2231: 2179: 2035: 2031:(written as 2014: 2003: 1999: 1992: 1987: 1979: 1975: 1968: 1963: 1959: 1956: 1949: 1944: 1940: 1936: 1931: 1922: 1911: 1906: 1902: 1895: 1888: 1883: 1876: 1859: 1855: 1852: 1842: 1832: 1828: 1809: 1804: 1791: 1771: 1766: 1762: 1752:Guozhen Wang 1740:J. Peter May 1732:Hiroshi Toda 1721: 1715: 1705: 1700: 1658: 1643: 1633: 1621: 1602: 1596: 1582: 1578: 1575: 1556: 1542: 1536: 1526: 1516: 1509: 1500: 1493: 1474: 1469: 1461: 1457: 1454: 1434: 1419: 1400: 1395: 1386: 1383: 1372: 1364: 1349: 1334: 1318: 1283: 1270: 1263: 1256:Elements of 1244: 1232: 1195: 1184: 1160: 1149: 1145: 1133: 1128: 1119: 1102: 1098: 1071: 1067: 1059: 1055: 1039: 1027: 995: 963: 957: 950: 944: 932: 925: 913: 901: 894: 884: 877: 865: 853: 835: 825: 819: 804:line segment 798: 792: 786: 782: 759: 747: 741: 722: 710: 704: 691: 687: 682: 669: 662: 645: 638: 626: 614: 601: 578: 566: 554: 549: 537:An ordinary 536: 509: 489: 482: 478: 461: 455: 451: 440: 435: 431: 417: 412: 408: 403: 396: 390: 380: 376: 361: 356: 344: 338: 334: 321: 315: 309: 303: 297: 293: 273: 268: 263: 245: 241:-dimensional 228: 217: 212: 206: 196: 182: 172: 162: 147: 98: 91:mathematical 88: 65: 56: 37: 11230:Hopf, Heinz 10790:2433/220059 10505:: 107–243, 10033:. See also 9972:Toda (1962) 9452:See, e.g., 6795:2 − 2 6769:The groups 6762:2 − 2 6596:h-cobordism 6469:smooth maps 6430:has a zero. 6014:-components 5820:graded ring 5480:trivial if 4099:quaternions 1736:Frank Adams 1661:Eduard Čech 1610:. This map 1273:rubber band 51:introducing 11387:Categories 11375:2007-11-14 11354:2007-11-14 11338:2007-10-20 11316:2007-10-09 11305:Baez, John 11087:1809.09290 11038:1601.02184 10512:2001.04511 10043:(3): 331, 9787:Cohen 1968 9748:Adams 1966 9712:Serre 1952 9676:Serre 1951 9427:, p. 9318:References 9310:3⋅25⋅11⋅41 8748:− 1) 6956:isomorphic 6925:direct sum 6580:4-manifold 6567:, implies 6428:polynomial 6326:question. 6117:Ext groups 6063:loop space 5987:cohomology 5742:direct sum 5692:Adams 1966 5473:to 0 or 1 5330:− 1) 3909:cover the 3879:| = 1 3631:suspension 1916:. For the 1838:surjective 1592:Heinz Hopf 1294:isomorphic 1214:isomorphic 839:to be the 812:CW complex 506:Background 103:dimensions 34:references 11270:(1999a), 11260:123533891 11177:(1999b), 11106:119303902 11063:119147703 10938:0764-4442 10799:0025-5645 10694:EMS Press 10430:EMS Press 10418:(2001) , 10366:0386-2194 10203:0764-4442 10167:0764-4442 10074:CiteSeerX 9984:Fuks 2001 9775:Toda 1962 9736:Fuks 2001 9611:May 1999a 9596:Čech 1932 9548:Hopf 1931 9465:π 9186:9⋅7⋅11⋅31 9084:16⋅2⋅9⋅5 8229:504⋅24⋅2 8159:24⋅12⋅4⋅2 8005:240⋅24⋅4 6673:π 6669:→ 6636:Θ 6573:signature 6571:that the 6554:− 1 6547:− 1 6016:for each 5824:nilpotent 5789:π 5780:≥ 5773:⨁ 5759:∗ 5755:π 5471:congruent 5383: : π 5299:then the 5229:− 5197:π 5193:⊕ 5173:− 5151:− 5135:π 5085:π 4961:component 4801:In 1951, 4783:René Thom 4684:a framed 4596:cobordism 4554:→ 4541:↪ 4480:π 4476:⊕ 4451:π 4447:≠ 4422:π 4376:= 1, 2, 3 4319:− 4312:π 4308:⊕ 4283:π 4254:π 4206:− 4199:π 4195:⊕ 4170:π 4141:π 4103:octonions 4072:→ 4059:↪ 4016:→ 4003:↪ 3957:→ 3944:↪ 3748:− 3741:π 3737:⊕ 3712:π 3683:π 3605:→ 3581:− 3574:π 3570:→ 3545:π 3541:→ 3516:π 3512:→ 3404:⋯ 3401:→ 3384:− 3377:π 3373:→ 3355:π 3351:→ 3333:π 3329:→ 3311:π 3307:→ 3304:⋯ 3247:→ 3234:↪ 2027:of order 1853:The case 1756:Zhouli Xu 1728:José Adem 1612:generates 1368:injective 1292:, and is 1202:bijection 1158:equal to 1002:open sets 948:produces 899:)-sphere 897:− 1 790:produces 607:dimension 472:, called 111:algebraic 93:field of 11232:(1931), 11141:(2002), 10956:(1962), 10827:(2003), 10704:(2011), 10654:(1998), 10621:(1995), 10464:(1269), 10440:(1959), 10312:16591550 10037:Topology 10015:Topology 9038:4⋅2⋅3⋅5 8761:via the 8750:divides 8694:∞⋅264⋅2 8676:264⋅4⋅2 8510:∞⋅480⋅2 8241:480⋅4⋅2 8216:120⋅12⋅2 7992:2520⋅6⋅2 7757:120⋅12⋅2 6935:have no 6534:−1 6440:−1 6332:Brunnian 5973:⟩ 5961:⟨ 5889:, while 5516:, where 5409:, where 5041:+ 1) − 3 4717: : 4712:of maps 4661: : 1652:and the 1650:homology 1588:3-sphere 1465:, then 1442:subgroup 1302:integers 1216:for all 1142:-sphere 1014:homotopy 851:-sphere 737:boundary 543:Geometry 516:topology 371:mapping. 368:integers 251:deformed 186:). The 11398:Spheres 11203:1702278 11167:1867354 11124:3204653 11055:3702672 11015:2045823 10980:0143217 10946:0046048 10917:0045386 10909:1969485 10880:2136212 10849:0860042 10807:0341485 10756:1326604 10676:1648096 10645:1320997 10606:1052407 10567:0148075 10559:1970128 10490:4046815 10450:0106454 10409:1454356 10374:2279281 10294:0231377 10286:1970586 10257:0554384 10249:1971238 10211:0046046 10175:0046045 10137:2188127 10096:0810962 10002:Sources 9425:Hu 1959 9255:⋅8⋅4⋅2⋅ 9228:⋅4⋅2⋅3 9064:⋅4⋅2⋅3 9046:3⋅25⋅11 9028:27⋅7⋅19 8639:24⋅8⋅2 8278:1056⋅8 8223:24⋅6⋅2 8017:48⋅4⋅2 7936:24⋅6⋅2 6992:) = Z×Z 6972:Example 6886:integer 6821:of the 6741:is the 6425:complex 6028:-groups 5620:⁠ 5608:⁠ 5604:⁠ 5587:⁠ 5513:⁠ 5490:⁠ 5451:(mod 8) 5261:(where 4938:− 3 + 1 4890:torsion 4884:has no 4791:spectra 3906:⁠ 3884:⁠ 3643:by the 1973:, with 1893:, with 1630:History 1312:in the 1170:abelian 841:equator 774:balloon 772:. If a 658:-sphere 533:-sphere 522:, with 492:< 20 292:0 < 243:sphere 232:can be 89:In the 47:improve 11286: 11258: 11201: 11191: 11165: 11155: 11122: 11104: 11061: 11053: 11013: 11003: 10978: 10968: 10944: 10936: 10915: 10907: 10878: 10868: 10847: 10837: 10805: 10797: 10754: 10744: 10674: 10643: 10633: 10604: 10594: 10565: 10557: 10488: 10478: 10448: 10407: 10397: 10372: 10364: 10310: 10303:224450 10300: 10292: 10284: 10255: 10247: 10209: 10201: 10173: 10165: 10135: 10094: 10076: 9302:4⋅2⋅3 9296:4⋅2⋅5 9284:4⋅2⋅3 9204:3⋅5⋅17 9152:3⋅5⋅29 9099:5⋅7⋅13 9077:8⋅3⋅23 9073:8⋅2⋅3 8993:3⋅5⋅17 8944:5⋅7⋅13 8838:16⋅3⋅5 8728:> 5 8697:264⋅2 8691:264⋅2 8688:264⋅2 8685:264⋅2 8682:264⋅2 8679:264⋅2 8673:264⋅2 8645:8⋅4⋅2 8525:480⋅2 8522:480⋅2 8519:480⋅2 8516:480⋅2 8513:480⋅2 8507:480⋅2 8504:480⋅2 8501:480⋅2 8296:264⋅2 8293:264⋅2 8290:264⋅6 8287:264⋅2 8284:264⋅2 8281:264⋅2 8275:264⋅2 8272:132⋅2 8269:132⋅2 8238:8⋅4⋅2 8121:240⋅2 8109:504⋅2 8072:240⋅2 8069:240⋅2 8066:240⋅2 8063:240⋅2 8060:120⋅2 8057:120⋅2 7840:∞⋅504 7831:504⋅2 7828:504⋅2 7825:504⋅2 7822:504⋅4 7819:504⋅2 7594:∞⋅120 6711:where 6417:) = Z) 6362:> 2 6263:, and 5871:, and 5826:; the 5475:modulo 5393:)) → π 5368:, for 5285:< 2 4897:< 2 4710:degree 4371:, for 3861:| 3787:Since 3623:Since 2912:> 1 1997:, for 1898:> 0 1875:space 1814:, the 1754:, and 1389:> 0 1374:degree 1310:origin 1086:curves 1044:of a ( 924:+ ⋯ + 876:+ ⋯ + 847:, the 675:circle 637:+ ⋯ + 539:sphere 350:degree 234:mapped 178:sphere 168:circle 154:sphere 97:, the 36:, but 11333:(PDF) 11256:S2CID 11102:S2CID 11082:arXiv 11059:S2CID 11033:arXiv 10905:JSTOR 10709:(PDF) 10555:JSTOR 10507:arXiv 10344:arXiv 10282:JSTOR 10245:JSTOR 9323:Notes 9308:⋅4⋅2⋅ 9290:⋅2⋅9⋅ 9275:⋅2⋅3 9237:⋅4⋅2⋅ 9222:⋅4⋅2 9202:⋅4⋅2⋅ 9132:⋅8⋅2⋅ 9123:⋅2⋅3 9117:⋅4⋅2 9093:⋅4⋅2⋅ 9044:⋅2⋅3⋅ 8938:⋅8⋅2⋅ 8869:8⋅9⋅7 8547:24⋅2 8458:16⋅2 8266:12⋅2 8235:24⋅2 8232:24⋅2 8226:24⋅2 8220:24⋅2 8212:12⋅2 8209:12⋅2 8155:12⋅2 8054:60⋅6 8051:30⋅2 8014:16⋅2 8011:16⋅2 8008:16⋅4 8002:24⋅4 7999:12⋅2 7904:below 7868:84⋅2 7816:84⋅2 7813:84⋅2 7810:12⋅2 7776:12⋅2 7773:24⋅2 7770:24⋅2 7767:24⋅2 7764:72⋅2 7761:72⋅2 7753:12⋅2 7646:24⋅2 7524:24⋅3 7350:∞⋅12 6984:) = π 6446:) = Z 6368:over 6366:braid 6212:find. 6044:1952b 6040:1952a 6018:prime 5934:with 5532:, if 5528:is a 5033:< 4762:) = Z 4749:) = Ω 4700:) = Z 4364:with 3859:with 3659:) → π 3641:split 3629:is a 3434:) → π 3209:is a 3151:> 3146:with 3122:> 3051:) = 0 2975:) = π 2948:> 2011:Table 2002:> 1978:> 1952:) = Z 1831:< 1812:) = 0 1692:used 1624:) = Z 1581:> 1559:) = Z 1512:) = 0 1496:) = 0 1477:) = 0 1460:< 1422:) = 0 1407:below 1405:(see 1403:) = Z 1352:) = Z 1337:) = Z 1314:plane 1247:) = Z 1222:plane 1111:group 1080:into 802:is a 778:slash 673:is a 481:< 379:> 332:When 296:< 11284:ISBN 11189:ISBN 11153:ISBN 11001:ISBN 10966:ISBN 10934:ISSN 10866:ISBN 10835:ISBN 10795:ISSN 10742:ISBN 10631:ISBN 10592:ISBN 10476:ISBN 10395:ISBN 10362:ISSN 10308:PMID 10199:ISSN 10163:ISSN 9299:4⋅2 9249:4⋅2 9243:2⋅3 9231:8⋅2 9196:2⋅3 9144:4⋅2 9138:2⋅3 9126:2⋅3 9087:2⋅3 9035:2⋅3 9032:2⋅3 9020:4⋅2 8985:2⋅3 8971:2⋅3 8932:2⋅2 8926:8⋅3 8922:3⋅11 8914:8⋅2 8865:2⋅3 8654:8⋅2 8651:8⋅2 8648:8⋅2 8642:8⋅2 8636:8⋅2 8633:8⋅2 8630:8⋅2 8602:∞⋅2 8464:4⋅2 8461:8⋅2 8418:∞⋅3 8172:6⋅2 8163:4⋅2 8152:6⋅2 8103:6⋅2 8100:6⋅2 7996:6⋅2 7960:6⋅2 7957:6⋅2 7948:6⋅2 7939:6⋅2 7880:240 7843:504 7837:504 7834:504 7779:6⋅2 7716:∞⋅2 7609:240 7606:240 7603:240 7600:240 7597:240 7591:120 7095:< 6986:9+10 6931:and 6809:The 6785:The 6577:spin 6458:ball 6403:The 6357:for 6106:has 6095:The 6057:The 5944:and 5918:and 5914:and 5740:The 5717:for 5440:(SO( 5389:(SO( 4914:and 4117:and 3820:and 3098:to 3053:for 3014:and 3007:For 2987:for 1939:) = 1710:and 1568:The 1172:and 1008:. A 701:disk 518:and 449:for 290:For 205:-th 148:The 132:The 11358:in 11248:doi 11244:104 11092:doi 11078:226 11043:doi 11029:186 10993:doi 10930:234 10897:doi 10785:hdl 10775:doi 10734:doi 10584:doi 10547:doi 10517:doi 10503:137 10466:doi 10462:262 10387:doi 10354:doi 10298:PMC 10274:doi 10237:doi 10233:110 10195:234 10159:234 10123:doi 10084:doi 10045:doi 10024:doi 9970:in 9429:107 9281:⋅2 9265:72+ 9212:64+ 9200:128 9184:⋅2⋅ 9175:⋅2 9160:56+ 9150:⋅3⋅ 9107:48+ 9097:⋅3⋅ 9070:⋅2 9054:40+ 9026:⋅2⋅ 9017:⋅2 9011:⋅2 9001:32+ 8991:⋅2⋅ 8975:8⋅3 8968:⋅2 8962:⋅2 8952:24+ 8942:⋅3⋅ 8920:⋅2⋅ 8911:⋅2 8905:⋅2 8895:16+ 8887:3⋅5 8885:⋅2⋅ 8862:⋅2 8856:⋅2 8824:8⋅3 8755:+ 1 8742:if 8661:19+ 8618:18+ 8575:17+ 8532:16+ 8489:15+ 8446:14+ 8403:13+ 8360:12+ 8248:19+ 8191:18+ 8134:17+ 8097:30 8079:16+ 8048:30 8045:30 8042:30 8024:15+ 7988:30 7967:14+ 7912:13+ 7902:See 7892:12 7850:12+ 7792:11+ 7732:10+ 7634:15 7588:60 7585:30 7582:15 7579:15 7402:12 7377:24 7374:24 7371:24 7368:24 7365:24 7362:24 7359:24 7356:24 7353:24 7347:12 7007:∞⋅2 6958:to 6954:is 6613:on 6607:≠ 4 6594:of 6503:to 6471:or 6337:of 6142:, Z 6120:Ext 5997:If 5985:in 5953:= 0 5942:= 0 5910:If 5830:on 5726:im( 5703:of 5617:504 5576:+11 5537:= 4 5469:is 5430:+ 2 5411:SO( 5373:≥ 2 5322:+ 1 5009:to 4963:of 4953:+ 1 4949:= 2 4934:= 2 4923:− 3 4919:= 2 4912:≥ 3 4907:if 4901:− 3 4892:if 4819:or 4402:= 4 4389:= 0 4369:= 2 4101:or 3633:of 3180:≠ 0 3155:+ 1 3126:+ 1 3076:of 3058:≥ 6 2992:≥ 3 2886:120 2884:Z×Z 2808:120 2482:Z×Z 2066:= Z 2055:or 1925:≥ 2 1680:'s 1600:to 1540:to 1444:of 1409:). 1300:of 1204:(a 1021:of 1004:or 939:= 1 890:= 1 770:= 1 733:≤ 1 665:+ 1 651:= 1 589:= 1 485:+ 2 458:+ 2 307:to 199:+ 1 11389:: 11326:, 11278:, 11254:, 11242:, 11236:, 11199:MR 11197:, 11187:, 11163:MR 11161:, 11151:, 11147:, 11120:MR 11100:, 11090:, 11076:, 11057:, 11051:MR 11049:, 11041:, 11027:, 11011:MR 11009:, 10999:, 10976:MR 10974:, 10964:, 10942:MR 10940:, 10928:, 10913:MR 10911:, 10903:, 10893:54 10876:MR 10874:, 10864:, 10860:, 10845:MR 10843:, 10816:, 10803:MR 10801:, 10793:, 10783:, 10771:25 10769:, 10752:MR 10750:, 10740:, 10719:58 10717:, 10711:, 10692:, 10686:, 10672:MR 10666:, 10664:)" 10641:MR 10639:, 10625:, 10602:MR 10600:, 10590:, 10582:, 10563:MR 10561:, 10553:, 10543:77 10541:, 10531:; 10515:, 10501:, 10486:MR 10484:, 10474:, 10460:, 10446:MR 10428:, 10422:, 10405:MR 10403:, 10393:, 10370:MR 10368:, 10360:, 10352:, 10340:82 10338:, 10334:, 10306:, 10296:, 10290:MR 10288:, 10280:, 10270:87 10253:MR 10251:, 10243:, 10221:; 10207:MR 10205:, 10193:, 10185:; 10171:MR 10169:, 10157:, 10147:; 10133:MR 10131:, 10119:19 10117:, 10111:, 10092:MR 10090:, 10082:, 10070:30 10068:, 10039:, 10018:, 9767:^ 9654:^ 9603:^ 9516:^ 9499:". 9331:^ 9306:32 9253:16 9246:2 9193:⋅ 9190:4 9178:2 9148:16 9141:2 9091:32 9081:8 9042:16 8989:64 8982:3 8979:2 8936:16 8929:2 8883:32 8879:2 8876:3 8873:⋅ 8846:8+ 8834:2 8831:⋅ 8828:⋅ 8820:2 8812:∞ 8805:0+ 8798:7 8795:6 8792:5 8789:4 8786:3 8783:2 8780:1 8777:0 8774:→ 8744:2( 8670:) 8627:) 8611:2 8608:2 8605:2 8599:2 8596:2 8593:2 8590:2 8587:2 8584:) 8568:2 8565:2 8562:2 8559:2 8556:2 8553:2 8550:2 8544:2 8541:) 8498:) 8482:2 8479:2 8476:2 8473:2 8470:2 8467:2 8455:) 8439:3 8436:3 8433:3 8430:3 8427:3 8424:3 8421:3 8415:6 8412:) 8396:⋅ 8393:⋅ 8390:⋅ 8387:⋅ 8384:⋅ 8381:⋅ 8378:⋅ 8375:⋅ 8372:2 8369:) 8308:→ 8263:⋅ 8260:⋅ 8257:) 8206:⋅ 8203:⋅ 8200:) 8184:2 8181:2 8178:2 8175:2 8169:2 8166:2 8149:⋅ 8146:⋅ 8143:) 8127:2 8124:2 8118:2 8115:2 8112:2 8106:2 8094:⋅ 8091:⋅ 8088:) 8039:⋅ 8036:⋅ 8033:) 7985:6 7982:⋅ 7979:⋅ 7976:) 7954:6 7951:6 7945:6 7942:6 7933:6 7930:2 7927:⋅ 7924:⋅ 7921:) 7898:2 7895:2 7889:⋅ 7886:⋅ 7883:⋅ 7877:2 7874:2 7871:2 7865:⋅ 7862:⋅ 7859:) 7807:⋅ 7804:⋅ 7801:) 7785:6 7782:6 7750:2 7747:⋅ 7744:⋅ 7741:) 7725:2 7722:2 7719:2 7713:2 7710:2 7707:2 7704:2 7701:2 7698:2 7695:2 7692:2 7689:⋅ 7686:⋅ 7683:) 7674:9+ 7667:2 7664:2 7661:2 7658:2 7655:2 7652:2 7649:2 7643:2 7640:2 7637:2 7631:⋅ 7628:⋅ 7625:) 7616:8+ 7576:3 7573:⋅ 7570:⋅ 7567:) 7558:7+ 7551:2 7548:2 7545:2 7542:2 7539:2 7536:2 7533:2 7530:2 7527:2 7521:3 7518:2 7515:⋅ 7512:⋅ 7509:) 7500:6+ 7493:⋅ 7490:⋅ 7487:⋅ 7484:⋅ 7481:⋅ 7478:⋅ 7475:⋅ 7472:∞ 7469:2 7466:2 7463:2 7460:2 7457:⋅ 7454:⋅ 7451:) 7442:5+ 7435:⋅ 7432:⋅ 7429:⋅ 7426:⋅ 7423:⋅ 7420:⋅ 7417:⋅ 7414:⋅ 7411:2 7408:2 7405:2 7399:⋅ 7396:⋅ 7393:) 7384:4+ 7344:2 7341:⋅ 7338:⋅ 7335:) 7326:3+ 7319:2 7316:2 7313:2 7310:2 7307:2 7304:2 7301:2 7298:2 7295:2 7292:2 7289:2 7286:2 7283:⋅ 7280:⋅ 7277:) 7268:2+ 7261:2 7258:2 7255:2 7252:2 7249:2 7246:2 7243:2 7240:2 7237:2 7234:2 7231:2 7228:∞ 7225:⋅ 7222:⋅ 7219:) 7210:1+ 7203:∞ 7200:∞ 7197:∞ 7194:∞ 7191:∞ 7188:∞ 7185:∞ 7182:∞ 7179:∞ 7176:∞ 7173:∞ 7170:∞ 7167:∞ 7164:2 7161:) 7152:0+ 7145:⋅ 7142:⋅ 7139:⋅ 7136:⋅ 7133:⋅ 7130:⋅ 7127:⋅ 7124:⋅ 7121:⋅ 7118:⋅ 7115:⋅ 7112:⋅ 7109:⋅ 7104:) 7018:→ 7000:×Z 6996:×Z 6978:19 6974:: 6967:.) 6963:ab 6947:×Z 6939:, 6912:, 6905:). 6888:, 6856:. 6727:, 6720:+1 6714:bP 6394:. 6385:→ 6295:BP 6257:, 6251:, 6136:(Z 6085:−1 6078:−1 6042:, 5969:, 5965:, 5623:. 5606:= 5601:12 5477:8; 5444:)) 5426:≥ 5336:. 5326:2( 5054:+1 5016:+2 4827:−1 4775:⊂ 4734:→ 4721:→ 4675:= 4665:→ 4562:16 4549:31 4536:15 4497:15 4484:29 4468:31 4455:30 4439:16 4426:30 4300:15 4067:15 3918:→ 3794:−1 3653:−1 3608:0. 3498:, 3451:→ 3284:→ 3280:→ 3213:: 3197:. 3191:+1 3187:≤ 3109:+1 3045:+4 3019:≥ 2999:→ 2918:, 2875:0 2872:0 2868:24 2851:Z 2848:0 2845:0 2842:0 2839:0 2836:0 2833:0 2830:0 2797:0 2794:0 2790:24 2773:Z 2770:0 2767:0 2764:0 2761:0 2758:0 2755:0 2731:×Z 2729:24 2723:60 2712:Z 2709:0 2705:24 2688:Z 2685:0 2682:0 2679:0 2676:0 2673:0 2660:×Z 2658:72 2635:30 2611:24 2594:Z 2591:0 2588:0 2585:0 2582:0 2564:×Z 2562:84 2551:×Z 2549:12 2522:15 2514:×Z 2512:24 2484:12 2467:Z 2464:0 2461:0 2458:0 2429:×Z 2427:84 2419:×Z 2417:12 2394:15 2370:12 2353:Z 2350:0 2347:0 2318:×Z 2316:84 2308:×Z 2306:12 2283:15 2259:12 2242:Z 2239:Z 2236:0 2226:0 2223:0 2220:0 2217:0 2214:0 2211:0 2208:0 2205:0 2202:0 2199:0 2196:0 2193:0 2190:0 2187:0 2184:Z 2173:15 2167:14 2161:13 2155:12 2149:11 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