36:
1264:
992:
1000:
1369:
93:
6214:
5284:-component). This exact sequence is similar to the ones coming from the Hopf fibration; the difference is that it works for all even-dimensional spheres, albeit at the expense of ignoring 2-torsion. Combining the results for odd and even dimensional spheres shows that much of the odd torsion of unstable homotopy groups is determined by the odd torsion of the stable homotopy groups.
1576:
140:
152:
6065:. In principle this gives an effective algorithm for computing all homotopy groups of any finite simply connected simplicial complex, but in practice it is too cumbersome to use for computing anything other than the first few nontrivial homotopy groups as the simplicial complex becomes much more complicated every time one kills a homotopy group.
1439:
8733:
are the products of cyclic groups of the infinite or prime power orders shown in the table. (For largely historical reasons, stable homotopy groups are usually given as products of cyclic groups of prime power order, while tables of unstable homotopy groups often give them as products of the smallest
6102:
to the calculation of homology groups of its repeated loop spaces. The Serre spectral sequence relates the homology of a space to that of its loop space, so can sometimes be used to calculate the homology of loop spaces. The Serre spectral sequence tends to have many non-zero differentials, which are
1446:
Any continuous mapping from a circle to an ordinary sphere can be continuously deformed to a one-point mapping, and so its homotopy class is trivial. One way to visualize this is to imagine a rubber-band wrapped around a frictionless ball: the band can always be slid off the ball. The homotopy group
6228:
The motivic Adams spectral sequence converges to the motivic stable homotopy groups of spheres. By comparing the motivic one over the complex numbers with the classical one, Isaksen gives rigorous proof of computations up to the 59-stem. In particular, Isaksen computes the Coker J of the 56-stem is
1235:
has exactly the same homotopy groups as a solitary point (as does a
Euclidean space of any dimension), and the real plane with a point removed has the same groups as a circle, so groups alone are not enough to distinguish spaces. Although the loss of discrimination power is unfortunate, it can also
1512:
All the interesting cases of homotopy groups of spheres involve mappings from a higher-dimensional sphere onto one of lower dimension. Unfortunately, the only example which can easily be visualized is not interesting: there are no nontrivial mappings from the ordinary sphere to the circle. Hence,
1376:
Mappings from a 2-sphere to a 2-sphere can be visualized as wrapping a plastic bag around a ball and then sealing it. The sealed bag is topologically equivalent to a 2-sphere, as is the surface of the ball. The bag can be wrapped more than once by twisting it and wrapping it back over the ball.
1244:
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.
4791:
with a nonstandard trivialization of the normal 2-plane bundle. Until the advent of more sophisticated algebraic methods in the early 1950s (Serre) the
Pontrjagin isomorphism was the main tool for computing the homotopy groups of spheres. In 1954 the Pontrjagin isomorphism was generalized by
6249:
page on positive stems. Wang and Xu develops a method using the KahnβPriddy map to deduce Adams differentials for the sphere spectrum inductively. They give detailed argument for several Adams differentials and compute the 60 and 61-stem. A geometric corollary of their result is the sphere
1686:
which can be used to calculate some of the groups. An important method for calculating the various groups is the concept of stable algebraic topology, which finds properties that are independent of the dimensions. Typically these only hold for larger dimensions. The first such result was
791:, pronounced "modulo", means to take the topological space on the left (the disk) and in it join together as one all the points on the right (the circle). The region is 2-dimensional, which is why topology calls the resulting topological space a 2-sphere. Generalized,
381:
under addition. For example, every point on a circle can be mapped continuously onto a point of another circle; as the first point is moved around the first circle, the second point may cycle several times around the second circle, depending on the particular
3629:
1036:, where "closed curves" are continuous maps from the circle into the complex plane, and where two closed curves produce the same integral result if they are homotopic in the topological space consisting of the plane minus the points of singularity.
1027:
is a continuous path between continuous maps; two maps connected by a homotopy are said to be homotopic. The idea common to all these concepts is to discard variations that do not affect outcomes of interest. An important practical example is the
6222:
A variation of this last approach uses a backwards version of the AdamsβNovikov spectral sequence for BrownβPeterson cohomology: the limit is known, and the initial terms involve unknown stable homotopy groups of spheres that one is trying to
3428:
5267:
437:
turned out to be a central question in algebraic topology that has contributed to development of many of its fundamental techniques and has served as a stimulating focus of research. One of the main discoveries is that the homotopy groups
6045:. This is usually done by constructing suitable fibrations and taking the associated long exact sequences of homotopy groups; spectral sequences are a systematic way of organizing the complicated information that this process generates.
5683:-homomorphism is the subgroup of "well understood" or "easy" elements of the stable homotopy groups. These well understood elements account for most elements of the stable homotopy groups of spheres in small dimensions. The quotient of
5991:
used the composition product and Toda brackets to label many of the elements of homotopy groups. There are also higher Toda brackets of several elements, defined when suitable lower Toda brackets vanish. This parallels the theory of
4526:
1286:
around one's finger: it can be wrapped once, twice, three times and so on. The wrapping can be in either of two directions, and wrappings in opposite directions will cancel out after a deformation. The homotopy group
1128:" β two pointed spheres joined at their distinguished point. The two maps to be added map the upper and lower spheres separately, agreeing on the distinguished point, and composition with the pinch gives the sum map.
4364:
1381:
and so the bag is allowed to pass through itself.) The twist can be in one of two directions and opposite twists can cancel out by deformation. The total number of twists after cancellation is an integer, called the
6318:
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,
4251:
3793:
6103:
hard to control, and too many ambiguities appear for higher homotopy groups. Consequently, it has been superseded by more powerful spectral sequences with fewer non-zero differentials, which give more information.
155:
This picture mimics part of the Hopf fibration, an interesting mapping from the three-dimensional sphere to the two-dimensional sphere. This mapping is the generator of the third homotopy group of the 2-sphere.
1994:, are all trivial. It therefore came as a great surprise historically that the corresponding homotopy groups are not trivial in general. This is the case that is of real importance: the higher homotopy groups
5821:
6808:
can be reduced to a question about stable homotopy groups of spheres. For example, knowledge of stable homotopy groups of degree up to 48 has been used to settle the
Kervaire invariant problem in dimension
4588:
6888:. The stable homotopy groups are highlighted in blue, the unstable ones in red. Each homotopy group is the product of the cyclic groups of the orders given in the table, using the following conventions:
4103:
3281:
854:, and sweep each point on it to one point above (the North Pole), producing the northern hemisphere, and to one point below (the South Pole), producing the southern hemisphere. For each positive integer
4047:
3988:
6182:-component of the stable homotopy groups. The initial terms of the Adams spectral sequence are themselves quite hard to compute: this is sometimes done using an auxiliary spectral sequence called the
1583:
is a nontrivial mapping of the 3-sphere to the 2-sphere, and generates the third homotopy group of the 2-sphere. Each colored circle maps to the corresponding point on the 2-sphere shown bottom right.
6041:
separately. The first few homotopy groups of spheres can be computed using ad hoc variations of the ideas above; beyond this point, most methods for computing homotopy groups of spheres are based on
6716:
1315:
under addition: a homotopy class is identified with an integer by counting the number of times a mapping in the homotopy class wraps around the circle. This integer can also be thought of as the
6000:. Every element of the stable homotopy groups of spheres can be expressed using composition products and higher Toda brackets in terms of certain well known elements, called Hopf elements.
1107:
fixed), so that two maps are in the same class if they are homotopic. Just as one point is distinguished, so one class is distinguished: all maps (or curves) homotopic to the constant map
5676:
3515:
979:. For some spaces the choice matters, but for a sphere all points are equivalent so the choice is a matter of convenience. For spheres constructed as a repeated suspension, the point
983:, which is on the equator of all the levels of suspension, works well; for the disk with collapsed rim, the point resulting from the collapse of the rim is another obvious choice.
6240:
The KahnβPriddy map induces a map of Adams spectral sequences from the suspension spectrum of infinite real projective space to the sphere spectrum. It is surjective on the Adams
1124:
with the introduction of addition, defined via an "equator pinch". This pinch maps the equator of a pointed sphere (here a circle) to the distinguished point, producing a "
9508:
6535:
6511:
2949:
5987:
of these elements. The Toda bracket is not quite an element of a stable homotopy group, because it is only defined up to addition of products of certain other elements.
2026:
The following table gives an idea of the complexity of the higher homotopy groups even for spheres of dimension 8 or less. In this table, the entries are either a) the
3310:
5090:
10164:
1649:
introduced the notion of homotopy and used the notion of a homotopy group, without using the language of group theory. A more rigorous approach was adopted by
10724:
10124:
10075:
1466:
This result generalizes to higher dimensions. All mappings from a lower-dimensional sphere into a sphere of higher dimension are similarly trivial: if
4427:
2971:
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).
1459:
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
11370:
6575:
11081:
Gheorghe, Bogdan; Wang, Guozhen; Xu, Zhouli (2021), "The special fiber of the motivic deformation of the stable homotopy category is algebraic",
4951:. In the case of odd torsion there are more precise results; in this case there is a big difference between odd and even dimensional spheres. If
4259:
4146:
3688:
11298:
11203:
11167:
10980:
10880:
10849:
10645:
10606:
10490:
6789:
above, and therefore the stable homotopy groups of spheres, are used in the classification of possible smooth structures on a topological or
1388:
of the mapping. As in the case mappings from the circle to the circle, this degree identifies the homotopy group with the group of integers,
6110:
can be used to compute many homotopy groups of spheres; it is based on some fibrations used by Toda in his calculations of homotopy groups.
2018:, are surprisingly complex and difficult to compute, and the effort to compute them has generated a significant amount of new mathematics.
6190:
1282:
The simplest case concerns the ways that a circle (1-sphere) can be wrapped around another circle. This can be visualized by wrapping a
5760:
268:
of mappings are summarized. An "addition" operation defined on these equivalence classes makes the set of equivalence classes into an
4140:
are zero. Thus the long exact sequences again break into families of split short exact sequences, implying two families of relations.
4537:
1678:
and Heinz Hopf, on the grounds that the groups were commutative so could not be the right generalizations of the fundamental group.)
11015:
10756:
10409:
79:
57:
4055:
3230:
6297:
is a map between motivic spheres. The
GheorgheβWangβXu theorem identifies the motivic Adams spectral sequence for the cofiber of
6017:
3999:
3940:
6568:
480:
5735:, and these are also considered to be "well understood".) Tables of homotopy groups of spheres sometimes omit the "easy" part
4998:. This is in some sense the best possible result, as these groups are known to have elements of this order for some values of
6641:
6431:
3655:
3084:
1692:
1491:
6202:
2968:
The groups below the jagged black line are constant along the diagonals (as indicated by the red, green and blue coloring).
10872:
10704:
10440:
1165:, and otherwise follows the same procedure. The null homotopic class acts as the identity of the group addition, and for
6487:
787:
is punctured and spread flat it produces a disk; this construction repairs the puncture, like pulling a drawstring. The
6062:
3083:
The fact that the groups below the jagged line in the table above are constant along the diagonals is explained by the
11195:
10699:
10435:
6460:
6206:
10430:
6821:
1695:, published in 1937. Stable algebraic topology flourished between 1945 and 1966 with many important results. In 1953
10236:; Neisendorfer, Joseph A. (November 1979), "The double suspension and exponents of the homotopy groups of spheres",
4804:. In more recent work the argument is usually reversed, with cobordism groups computed in terms of homotopy groups.
11159:
10972:
10508:
Isaksen, Daniel C.; Wang, Guozhen; Xu, Zhouli (2023), "Stable homotopy groups of spheres: from dimension 0 to 90",
6797:
4801:
1557:
can be lifted to a map into the real line and the nullhomotopy descends to the downstairs space (via composition).
1016:
552:
in three-dimensional spaceβthe surface, not the solid ballβis just one example of what a sphere means in topology.
11357:
3624:{\displaystyle 0\rightarrow \pi _{i}(S^{3})\rightarrow \pi _{i}(S^{2})\rightarrow \pi _{i-1}(S^{1})\rightarrow 0.}
6790:
6628:-spheres, up to orientation-preserving diffeomorphism; the non-trivial elements of this group are represented by
1622:
1542:
6061:
to compute the first non-trivial homotopy group and then killing (eliminating) it with a fibration involving an
147:
is a nontrivial mapping of the 3-sphere to the 2-sphere, and generates the third homotopy group of the 2-sphere.
11403:
6021:
5649:
5429:
4971:
2052:
1184:
484:
50:
44:
521:
The study of homotopy groups of spheres builds on a great deal of background material, here briefly reviewed.
5076:
The results above about odd torsion only hold for odd-dimensional spheres: for even-dimensional spheres, the
817:, and the construction joins its ends to make a circle. An equivalent description is that the boundary of an
11032:
Wang, Guozhen; Xu, Zhouli (2017), "The triviality of the 61-stem in the stable homotopy groups of spheres",
6735:
6114:
6069:
4651:
363:
that measures how many times the sphere is wrapped around itself. This degree identifies the homotopy group
10694:
5833:, where multiplication is given by composition of representing maps, and any element of non-zero degree is
10084:
6837:
6334:
1660:
1655:
61:
10743:, Graduate Studies in Mathematics, vol. 5, Providence, Rhode Island: American Mathematical Society,
11249:
11034:
10548:
10238:
6920:
6183:
6107:
5827:
4851:
3641:
1372:
Illustration of how a 2-sphere can be wrapped twice around another 2-sphere. Edges should be identified.
1300:
117:
96:
Illustration of how a 2-sphere can be wrapped twice around another 2-sphere. Edges should be identified.
6072:
was used by Serre to prove some of the results mentioned previously. He used the fact that taking the
9978:
6935:
6583:
6579:
5752:
3506:
1320:
324:
is homotopic (i.e., continuously deformable) to a constant mapping, i.e., a mapping that maps all of
10089:
1758:
10666:
6483:
5481:
1696:
1460:
1324:
1263:
1232:
1121:
991:
788:
747:
11378:
6518:
6494:
2932:
1762:
11266:
11112:
11092:
11069:
11043:
10915:
10640:, Contemporary Mathematics, vol. 181, Providence, R.I.: Amer. Math. Soc., pp. 299β316,
10565:
10517:
10354:
10292:
10255:
10233:
10229:
10115:
10025:
9470:
6801:
6475:
6468:
5838:
5485:
3423:{\displaystyle \cdots \to \pi _{i}(F)\to \pi _{i}(E)\to \pi _{i}(B)\to \pi _{i-1}(F)\to \cdots .}
1858:
1704:
1208:
1125:
522:
105:
17:
6574:
The fact that the third stable homotopy group of spheres is cyclic of order 24, first proved by
5262:{\displaystyle \pi _{2m+k}(S^{2m})(p)=\pi _{2m+k-1}(S^{2m-1})(p)\oplus \pi _{2m+k}(S^{4m-1})(p)}
2964:) appear to be chaotic, but in fact there are many patterns, some obvious and some very subtle.
3193:, and have been computed in numerous cases, although the general pattern is still elusive. For
11294:
11199:
11163:
11011:
10976:
10944:
10876:
10845:
10805:
10752:
10738:
10641:
10633:
10602:
10486:
10405:
10372:
10331:
10318:
10209:
10197:
10173:
10159:
9439:
6931:
6829:
6825:
6198:
6098:)-fold loop space by the Hurewicz theorem. This reduces the calculation of homotopy groups of
6042:
5842:
4900:
4813:
1700:
1664:
1650:
1368:
1335:
1331:
1100:
1040:
1008:
513:. Several important patterns have been established, yet much remains unknown and unexplained.
510:
506:
265:
244:
125:
92:
11244:
4862:. The 2-components are hardest to calculate, and in several ways behave differently from the
1682:
is also credited with the introduction of homotopy groups in his 1935 paper and also for the
11408:
11282:
11258:
11102:
11083:
11053:
11003:
10907:
10839:
10795:
10785:
10744:
10594:
10557:
10527:
10476:
10397:
10364:
10308:
10284:
10247:
10133:
10094:
10055:
10034:
8769:
6621:
6193:
is a more powerful version of the Adams spectral sequence replacing ordinary cohomology mod
6165:
6058:
5701:-homomorphism is considered to be the "hard" part of the stable homotopy groups of spheres (
5540:
5465:
3088:
1883:
1875:
1688:
1683:
1675:
1118:
1033:
530:
121:
11213:
11177:
11134:
11065:
11025:
10990:
10956:
10927:
10890:
10859:
10817:
10766:
10686:
10655:
10616:
10577:
10500:
10460:
10419:
10384:
10304:
10267:
10221:
10185:
10147:
10106:
6938:) of the cyclic groups of those orders. Powers indicate repeated products. (Note that when
11338:
11209:
11173:
11130:
11061:
11021:
10986:
10952:
10923:
10886:
10855:
10835:
10813:
10762:
10682:
10651:
10612:
10590:
10573:
10539:
10496:
10456:
10415:
10380:
10300:
10263:
10217:
10181:
10143:
10102:
6833:
6402:
6256:
has a unique smooth structure, and it is the last odd dimensional one β the only ones are
5077:
4796:
to an isomorphism expressing other groups of cobordism classes (e.g. of all manifolds) as
4720:
3651:
1679:
1532:
1384:
1029:
607:
360:
202:
10119:
2974:
The second and third rows of the table are the same starting in the third column (i.e.,
999:
11228:
10824:
10716:
8773:
6761:
6587:
6435:
6415:
5993:
5387:
5356:
4854:
with a finite abelian group. In particular the homotopy groups are determined by their
4797:
4738:
4610:
4594:
3991:
3921:
3301:
3217:
2926:
1879:
1671:
1646:
1618:
1580:
1316:
1056:
1020:
534:
396:
144:
133:
129:
10313:
10073:(1984), "Relations amongst Toda brackets and the Kervaire invariant in dimension 62",
844:
This construction, though simple, is of great theoretical importance. Take the circle
11397:
11334:
11270:
11153:
11149:
11116:
11073:
10662:
10629:
10070:
10060:
10039:
8312:
7022:
6947:
6824:
says that the stable homotopy groups of the spheres can be expressed in terms of the
6629:
6538:. The geometry near a critical point of such a map can be described by an element of
6342:
4664:
2027:
1826:
1754:
1448:
1216:
1201:
1180:
975:
337:
269:
10898:
Serre, Jean-Pierre (1951), "Homologie singulière des espaces fibrés. Applications",
10455:, Pure and Applied Mathematics, vol. 8, New York & London: Academic Press,
5468:, and it is reflected in the stable homotopy groups of spheres via the image of the
4597:
one problem, because such a fibration would imply that the failed relation is true.
1451:, with only one element, the identity element, and so it can be identified with the
260:. This summary does not distinguish between two mappings if one can be continuously
11315:
11278:
11185:
10964:
10773:
10426:
10396:, Undergraduate Texts in Mathematics, Springer-Verlag, New York, pp. 134β136,
10193:
10155:
6904:
6028:
5988:
5967:
4816:
showed that homotopy groups of spheres are all finite except for those of the form
3221:
2035:
1750:
1742:
814:
10392:
Fine, Benjamin; Rosenberger, Gerhard (1997), "8.1 Winding Number and Proof Five",
10138:
4793:
1738:
1535:
which is contractible (it has the homotopy type of a point). In addition, because
1211:
between the associated homotopy groups. In particular, if the map is a continuous
11107:
6012:
is any finite simplicial complex with finite fundamental group, in particular if
11057:
10998:
Walschap, Gerard (2004), "Chapter 3: Homotopy groups and bundles over spheres",
10712:
10543:
10448:
10020:
6849:
Tables of homotopy groups of spheres are most conveniently organized by showing
6606:
6345:
6205:. The initial term is again quite hard to calculate; to do this one can use the
5830:
4521:{\displaystyle \pi _{30}(S^{16})\neq \pi _{30}(S^{31})\oplus \pi _{29}(S^{15}).}
1746:
1283:
101:
10776:(1973), "The nilpotency of elements of the stable homotopy groups of spheres",
10531:
10098:
6091:)-fold repeated loop space, which is equal to the first homology group of the (
6049:"The method of killing homotopy groups", due to Cartan and Serre (
11240:
11007:
10401:
10162:(1952a), "Espaces fibrΓ©s et groupes d'homotopie. I. Constructions gΓ©nΓ©rales",
6966:
6590:
6479:
6438:
6073:
5997:
4109:
1882:, which are generally easier to calculate; in particular, it shows that for a
1848:
1602:
1304:
1224:
822:
11319:
10948:
10831:
American
Mathematical Society Translations, Ser. 2, Vol. 11, pp. 1β114 (1959)
10809:
10376:
10213:
10177:
5637:
The stable homotopy groups of spheres are the direct sum of the image of the
969:
Some theory requires selecting a fixed point on the sphere, calling the pair
11189:
11129:, The Univalent Foundations Program and Institute for Advanced Study, 2013,
10790:
6312:, which allows one to deduce motivic Adams differentials for the cofiber of
6213:
6127:
5834:
5558:
This last case accounts for the elements of unusually large finite order in
4606:
1861:, and any mapping to such a space can be deformed into a one-point mapping.
1766:
1699:
showed that there is a metastable range for the homotopy groups of spheres.
1575:
1378:
1212:
711:
617:
139:
113:
11002:, Graduate Texts in Mathematics, vol. 224, Springer-Verlag, New York,
10322:
3852:
The Hopf fibration may be constructed as follows: pairs of complex numbers
10679:
Proceedings of the
International Congress of Mathematicians (Berlin, 1998)
411:
in a non-trivial fashion, and so is not equivalent to a one-point mapping.
151:
11290:
6076:
of a well behaved space shifts all the homotopy groups down by 1, so the
4359:{\displaystyle \pi _{i}(S^{8})=\pi _{i}(S^{15})\oplus \pi _{i-1}(S^{7}).}
4113:
1598:
1452:
1024:
1012:
665:
553:
526:
261:
164:
10481:
10368:
4246:{\displaystyle \pi _{i}(S^{4})=\pi _{i}(S^{7})\oplus \pi _{i-1}(S^{3}),}
3788:{\displaystyle \pi _{i}(S^{2})=\pi _{i}(S^{3})\oplus \pi _{i-1}(S^{1}).}
610:
found exactly one unit away from the origin. It is called the 2-sphere,
11262:
11245:"Γber die Abbildungen der dreidimensionalen SphΓ€re auf die KugelflΓ€che"
11124:
10919:
10800:
10681:, Documenta Mathematica, Extra Volume, vol. II, pp. 465β472,
10598:
10569:
10296:
10259:
6896:
6035:
1312:
851:
784:
378:
10748:
6016:
is a sphere of dimension at least 2, then its homotopy groups are all
4936:. The case of 2-dimensional spheres is slightly different: the first
4719:, computed by the algebraic sum of their points, corresponding to the
10359:
8778:
6291:
method is so far the most efficient method at the prime 2. The class
685:
549:
479:
up to 90. The stable homotopy groups form the coefficient ring of an
188:
178:
10911:
10561:
10288:
10251:
10200:(1952b), "Espaces fibrΓ©s et groupes d'homotopie. II. Applications",
6734:
is the cyclic subgroup represented by homotopy spheres that bound a
2904:
The first row of this table is straightforward. The homotopy groups
395:. The first such surprise was the discovery of a mapping called the
11097:
11048:
10522:
1707:
to show that most of these groups are finite, the exceptions being
1438:
6376:
1574:
1367:
1096:
998:
990:
556:
defines a sphere rigidly, as a shape. Here are some alternatives.
150:
138:
91:
10347:
Proceedings of the Japan
Academy, Series A, Mathematical Sciences
6596:
Stable homotopy groups of spheres are used to describe the group
5816:{\displaystyle \pi _{\ast }^{S}=\bigoplus _{k\geq 0}\pi _{k}^{S}}
5002:. Furthermore, the stable range can be extended in this case: if
719:
is the region contained by a circle, described by the inequality
5287:
For stable homotopy groups there are more precise results about
498:) are more erratic; nevertheless, they have been tabulated for
10935:
Serre, Jean-Pierre (1952), "Sur la suspension de
Freudenthal",
10587:
Stable homotopy groups of spheres. A computer-assisted approach
10275:
Cohen, Joel M. (1968), "The decomposition of stable homotopy",
4713:
is the cobordism group of framed 0-dimensional submanifolds of
4583:{\displaystyle S^{15}\hookrightarrow S^{31}\rightarrow S^{16},}
3990:, where the bundle projection is a double covering), there are
3075:
These patterns follow from many different theoretical results.
1874:
has also been noted already, and is an easy consequence of the
1442:
A homotopy from a circle around a sphere down to a single point
201:-sphere may be defined geometrically as the set of points in a
11233:
6020:. To compute these groups, they are often factored into their
2951:, which has the same higher homotopy groups, is contractible.
29:
4775:
which corresponds to the framed 1-dimensional submanifold of
3487:
can be deformed to a point inside the higher-dimensional one
1219:), so that the two spaces have the same topology, then their
509:, a technique first applied to homotopy groups of spheres by
8768:
and 0 otherwise. The mod 8 behavior of the table comes from
6229:
0, and therefore by the work of
Kervaire-Milnor, the sphere
4098:{\displaystyle S^{7}\hookrightarrow S^{15}\rightarrow S^{8}}
3276:{\displaystyle S^{1}\hookrightarrow S^{3}\rightarrow S^{2}.}
710:
This construction moves from geometry to pure topology. The
27:
How spheres of various dimensions can wrap around each other
9347:
9345:
9343:
6341:
identify these homotopy groups as certain quotients of the
4415:) and beyond. Although generalizations of the relations to
4042:{\displaystyle S^{3}\hookrightarrow S^{7}\rightarrow S^{4}}
3983:{\displaystyle S^{0}\hookrightarrow S^{1}\rightarrow S^{1}}
2086:). Extended tables of homotopy groups of spheres are given
10829:
Smooth manifolds and their applications in homotopy theory
1207:
A continuous map between two topological spaces induces a
1023:
is a function between spaces that preserves continuity. A
11126:
Homotopy type theoryβunivalent foundations of mathematics
9530:
9528:
6560:
sphere around the critical point maps into a topological
6354:. Under this correspondence, every nontrivial element in
1674:
in 1932. (His first paper was withdrawn on the advice of
616:, for reasons given below. The same idea applies for any
10589:, Lecture Notes in Mathematics, vol. 1423, Berlin:
7911:
5595:
have a cyclic subgroup of order 504, the denominator of
10841:
Complex cobordism and stable homotopy groups of spheres
6711:{\displaystyle \Theta _{n}/bP_{n+1}\to \pi _{n}^{S}/J,}
6474:
The stable homotopy groups of spheres are important in
3091:, which implies that the suspension homomorphism from
1063:
thus begins with continuous maps from a pointed circle
385:
The most interesting and surprising results occur when
11194:, Chicago lectures in mathematics (revised ed.),
9698:
9670:
9668:
9666:
9473:
4613:
established an isomorphism between the homotopy group
3005:). This isomorphism is induced by the Hopf fibration
1377:(There is no requirement for the continuous map to be
9594:
9582:
6644:
6521:
6497:
5763:
5652:
5093:
4540:
4430:
4262:
4149:
4058:
4002:
3943:
3691:
3518:
3313:
3233:
2935:
9953:
9464:
6930:
Where entry is a product, the homotopy group is the
6478:, which studies the structure of singular points of
6178:, and converges to something closely related to the
5498:
a cyclic group of order equal to the denominator of
4869:
In the same paper, Serre found the first place that
1911:, is isomorphic to the first nonzero homology group
1179:) β the homotopy groups of spheres β the groups are
1011:
is its continuity structure, formalized in terms of
1003:
Addition of two circle maps keeping base point fixed
995:
Homotopy of two circle maps keeping base point fixed
9781:
9779:
9617:
9615:
5705:). (Adams also introduced certain order 2 elements
821:-dimensional disk is glued to a point, producing a
136:are surprisingly complex and difficult to compute.
10937:Comptes Rendus de l'AcadΓ©mie des Sciences, SΓ©rie I
10202:Comptes Rendus de l'AcadΓ©mie des Sciences, SΓ©rie I
10165:Comptes Rendus de l'AcadΓ©mie des Sciences, SΓ©rie I
9905:
9502:
6836:, leading to an identification of K-theory of the
6710:
6529:
6505:
6338:
5815:
5670:
5261:
4605:Homotopy groups of spheres are closely related to
4582:
4520:
4421:are often true, they sometimes fail; for example,
4358:
4245:
4097:
4041:
3982:
3787:
3623:
3433:For this specific bundle, each group homomorphism
3422:
3275:
2954:Beyond the first row, the higher homotopy groups (
2943:
1829:. The reason is that a continuous mapping from an
128:, forgetting about their precise geometry. Unlike
10969:Composition methods in homotopy groups of spheres
1117:are called null homotopic. The classes become an
116:can wrap around each other. They are examples of
6919:Where the entry is β, the homotopy group is the
5645:-invariant, a homomorphism from these groups to
5314:-primary component of the stable homotopy group
1601:to the ordinary 2-sphere, and was discovered by
212:located at a unit distance from the origin. The
10971:, Annals of Mathematics Studies, vol. 49,
10621:
9917:
6303:as the algebraic Novikov spectral sequence for
678:)-dimensional space. For example, the 1-sphere
289:falls into three regimes, depending on whether
11231:(1932), "HΓΆherdimensionale Homotopiegruppen",
9941:
9893:
6197:with a generalized cohomology theory, such as
5826:of the stable homotopy groups of spheres is a
1791:up to 90, and, as of 2023, unknown for larger
11321:This week's finds in mathematical physics 102
10717:"Differential topology forty-six years later"
10120:"Configurations, braids, and homotopy groups"
9998:
9351:
6870:The following table shows many of the groups
6817:for which the question was open at the time.)
6320:
5084:in terms of that of odd-dimensional spheres,
3937:Similarly (in addition to the Hopf fibration
2087:
1670:Higher homotopy groups were first defined by
1490:. This can be shown as a consequence of the
1334:) of the homotopy group with the integers is
525:provides the larger context, itself built on
415:The question of computing the homotopy group
132:, which are also topological invariants, the
8:
10778:Journal of the Mathematical Society of Japan
10725:Notices of the American Mathematical Society
10469:Memoirs of the American Mathematical Society
10125:Journal of the American Mathematical Society
6632:). More precisely, there is an injective map
6054:
6050:
5054:, and an epimorphism if equality holds. The
4663:which are "framed", i.e. have a trivialized
3481:to zero, since the lower-dimensional sphere
1191:all maps are null homotopic, then the group
1103:based on homotopy (keeping the "base point"
11390:in MacTutor History of Mathematics archive.
10467:Isaksen, Daniel C. (2019), "Stable Stems",
6775:, in which case the image has index 1 or 2.
5641:-homomorphism, and the kernel of the Adams
5495:is congruent to 2, 4, 5, or 6 modulo 8; and
3505:. Thus the long exact sequence breaks into
1931:-sphere, this immediately implies that for
1200:consists of one element, and is called the
606:This is the set of points in 3-dimensional
233:summarizes the different ways in which the
11377:O'Connor, J. J.; Robertson, E. F. (2001),
11356:O'Connor, J. J.; Robertson, E. F. (1996),
11000:Metric structures in differential geometry
10076:Journal of the London Mathematical Society
6553:, by considering the way in which a small
6327:The computation of the homotopy groups of
4873:-torsion occurs in the homotopy groups of
4681:is homotopic to a differentiable map with
1878:: this theorem links homotopy groups with
1851:. Consequently, its image is contained in
1737:. Others who worked in this area included
1087:, where maps from one pair to another map
124:terms, the structure of spheres viewed as
11106:
11096:
11047:
10799:
10789:
10546:(1963), "Groups of homotopy spheres: I",
10521:
10480:
10358:
10312:
10137:
10088:
10059:
10038:
9491:
9478:
9472:
6756:th stable homotopy group of spheres, and
6697:
6691:
6686:
6667:
6655:
6649:
6643:
6523:
6522:
6520:
6499:
6498:
6496:
5807:
5802:
5786:
5773:
5768:
5762:
5671:{\displaystyle \mathbb {Q} /\mathbb {Z} }
5664:
5663:
5658:
5654:
5653:
5651:
5232:
5210:
5176:
5148:
5120:
5098:
5092:
4571:
4558:
4545:
4539:
4506:
4493:
4477:
4464:
4448:
4435:
4429:
4344:
4325:
4309:
4296:
4280:
4267:
4261:
4231:
4212:
4196:
4183:
4167:
4154:
4148:
4089:
4076:
4063:
4057:
4033:
4020:
4007:
4001:
3974:
3961:
3948:
3942:
3773:
3754:
3738:
3725:
3709:
3696:
3690:
3606:
3587:
3571:
3558:
3542:
3529:
3517:
3390:
3368:
3346:
3324:
3312:
3264:
3251:
3238:
3232:
2937:
2936:
2934:
2055:of such groups (written, for example, as
1847:can always be deformed so that it is not
80:Learn how and when to remove this message
10069:Barratt, Michael G.; Jones, John D. S.;
9821:
9570:
8745:-component that is accounted for by the
6892:The entry "β
" denotes the trivial group.
6212:
4850:), when the group is the product of the
2092:
1605:, who constructed a nontrivial map from
1437:
1262:
293:is less than, equal to, or greater than
43:This article includes a list of general
11371:MacTutor History of Mathematics archive
10002:
9965:
9929:
9869:
9857:
9845:
9833:
9809:
9770:
9734:
9710:
9674:
9657:
9645:
9633:
9546:
9534:
9519:
9451:
9423:
9411:
9399:
9387:
9375:
9363:
9339:
6434:, which states that every non-constant
6201:or, more usually, a piece of it called
5888:is nonzero and 12 times a generator of
5006:is odd then the double suspension from
4699:-dimensional submanifold. For example,
4116:instead of complex numbers. Here, too,
3493:. This corresponds to the vanishing of
11191:A Concise Course in Algebraic Topology
10638:The Δech centennial (Boston, MA, 1993)
9881:
9648:, Stable homotopy groups, pp. 385β393.
6800:, about the existence of manifolds of
5343:, in which case it is cyclic of order
1395:These two results generalize: for all
1095:. These maps (or equivalently, closed
336:. Therefore the homotopy group is the
11283:"Stable Algebraic Topology 1945β1966"
11143:General algebraic topology references
10740:Algebraic curves and Riemann surfaces
9797:
9758:
9722:
9686:
5702:
5679:. Roughly speaking, the image of the
4914:, and has a unique subgroup of order
4877:dimensional spheres, by showing that
4369:The three fibrations have base space
3817:at least 3, the first row shows that
3186:: they are finite abelian groups for
475:and have been computed for values of
7:
10634:"On the computation of stable stems"
10510:Publications mathΓ©matiques de l'IHΓS
9994:
9982:
9785:
9746:
9699:Cohen, Moore & Neisendorfer 1979
9621:
9606:
9558:
6084:is the first homotopy group of its (
5464:. This period 8 pattern is known as
3286:The general theory of fiber bundles
487:. The unstable homotopy groups (for
171:-sphere for brevity, and denoted as
10667:"Toward a global understanding of Ο
9997:. The 2-components can be found in
9979:table of homotopy groups of spheres
6463:that every continuous map from the
6385:that is not Brunnian over the disk
5058:-torsion of the intermediate group
3206:unstable homotopy groups of spheres
1890:, the first nonzero homotopy group
1417:
10394:The Fundamental Theorem of Algebra
9954:Barratt, Jones & Mahowald 1984
9435:
6813:. (This was the smallest value of
6646:
4781:defined by the standard embedding
4593:the first non-trivial case of the
4402:) as mentioned above, but not for
3934:sends any such pair to its ratio.
3892:form a 3-sphere, and their ratios
3849:is at least 3, as observed above.
1587:The first nontrivial example with
1236:make certain computations easier.
49:it lacks sufficient corresponding
25:
11340:Stable homotopy groups of spheres
10431:"Spheres, homotopy groups of the"
10118:; Wong, Yan Loi; Wu, Jie (2006),
10023:(1966), "On the groups J(X) IV",
10001:, and the 3- and 5-components in
8749:-homomorphism is cyclic of order
6907:of that order (generally written
6375:may be represented by a Brunnian
6018:finitely generated abelian groups
5580:. For example, the stable groups
3170:stable homotopy groups of spheres
473:stable homotopy groups of spheres
18:Stable homotopy groups of spheres
6767:. This is an isomorphism unless
6418:(corresponding to an integer of
6057:) involves repeatedly using the
6034:, and calculating each of these
5080:gives the torsion at odd primes
2918:of the 1-sphere are trivial for
1857:with a point removed; this is a
1007:The distinguishing feature of a
112:describe how spheres of various
34:
10332:"Remarks on zeta functions and
10046:Adams, J (1968), "Correction",
8715:Table of stable homotopy groups
6191:AdamsβNovikov spectral sequence
4531:Thus there can be no fibration
505:. Most modern computations use
481:extraordinary cohomology theory
9977:These tables are based on the
9497:
9484:
8734:number of cyclic groups.) For
6679:
6486:. Such singularities arise as
6432:fundamental theorem of algebra
6235:has a unique smooth structure.
5256:
5250:
5247:
5225:
5200:
5194:
5191:
5169:
5138:
5132:
5129:
5113:
4609:classes of manifolds. In 1938
4564:
4551:
4512:
4499:
4483:
4470:
4454:
4441:
4350:
4337:
4315:
4302:
4286:
4273:
4237:
4224:
4202:
4189:
4173:
4160:
4082:
4069:
4026:
4013:
3967:
3954:
3779:
3766:
3744:
3731:
3715:
3702:
3615:
3612:
3599:
3580:
3577:
3564:
3551:
3548:
3535:
3522:
3411:
3408:
3402:
3383:
3380:
3374:
3361:
3358:
3352:
3339:
3336:
3330:
3317:
3257:
3244:
1659:where the related concepts of
1492:cellular approximation theorem
1:
11380:Marie Ennemond Camille Jordan
10873:American Mathematical Society
10869:The wild world of 4-manifolds
10844:(2nd ed.), AMS Chelsea,
10620:Also see the corrections in (
10139:10.1090/S0894-0347-05-00507-2
9999:Isaksen, Wang & Xu (2023)
9595:O'Connor & Robertson 1996
9583:O'Connor & Robertson 2001
8776:, whose image is underlined.
6609:classes of oriented homotopy
6321:Isaksen, Wang & Xu (2023)
6080:th homotopy group of a space
5476:a cyclic group of order 2 if
4389:. A fibration does exist for
2030:0, the infinite cyclic group
1769:. The stable homotopy groups
1039:The first homotopy group, or
11108:10.4310/ACTA.2021.v226.n2.a2
10585:Kochman, Stanley O. (1990),
10061:10.1016/0040-9383(68)90010-4
10040:10.1016/0040-9383(66)90004-8
9894:Gheorghe, Wang & Xu 2021
9503:{\textstyle \pi _{1}(S^{1})}
6903:, the homotopy group is the
6840:with stable homotopy groups.
6530:{\displaystyle \mathbb {R} }
6506:{\displaystyle \mathbb {R} }
6471:to itself has a fixed point.
6391:. For example, the Hopf map
4866:-components for odd primes.
3204:, the groups are called the
2944:{\displaystyle \mathbb {R} }
1676:Pavel Sergeyevich Alexandrov
1541:is simply connected, by the
1099:) are grouped together into
756:, described by the equality
11196:University of Chicago Press
11058:10.4007/annals.2017.186.2.3
10867:Scorpan, Alexandru (2005),
10700:Encyclopedia of Mathematics
10622:Kochman & Mahowald 1995
10436:Encyclopedia of Mathematics
9918:Fine & Rosenberger 1997
9352:Isaksen, Wang & Xu 2023
8719:The stable homotopy groups
6461:Brouwer fixed point theorem
6323:computes up to the 90-stem.
6207:chromatic spectral sequence
5906:is zero because the group
5845:implies Nishida's theorem.
5291:-torsion. For example, if
4858:-components for all primes
4108:constructed using pairs of
3992:generalized Hopf fibrations
3924:, a 2-sphere. The Hopf map
3922:complex plane plus infinity
3457:, induced by the inclusion
1597:concerns mappings from the
399:, which wraps the 3-sphere
275:The problem of determining
177:β generalizes the familiar
11425:
11160:Cambridge University Press
10973:Princeton University Press
10532:10.1007/s10240-023-00139-1
9942:Kervaire & Milnor 1963
6798:Kervaire invariant problem
6335:combinatorial group theory
5354:
4750:represents a generator of
3079:Stable and unstable groups
1653:in his 1895 set of papers
671:as a geometric object in (
110:homotopy groups of spheres
11008:10.1007/978-0-387-21826-7
10402:10.1007/978-1-4612-1928-6
9465:Homotopy type theory 2013
6791:piecewise linear manifold
6430:can be used to prove the
6203:BrownβPeterson cohomology
5870:is nonzero and generates
5361:An important subgroup of
2088:at the end of the article
1645:In the late 19th century
1531:has the real line as its
833:: written in topology as
696:: written in topology as
264:to the other; thus, only
11285:, in I. M. James (ed.),
10693:Mahowald, Mark (2001) ,
10099:10.1112/jlms/s2-30.3.533
6845:Table of homotopy groups
5472:-homomorphism which is:
5430:special orthogonal group
5073:can be strictly larger.
4737:. The projection of the
4650:of cobordism classes of
3845:are isomorphic whenever
3049:do not vanish. However,
2925:, because the universal
1240:Low-dimensional examples
1223:-th homotopy groups are
1149:begins with the pointed
485:stable cohomotopy theory
405:around the usual sphere
10695:"EHP spectral sequence"
10330:Deitmar, Anton (2006),
6736:parallelizable manifold
6620:, this is the group of
6285:The motivic cofiber of
6189:At the odd primes, the
6115:Adams spectral sequence
6070:Serre spectral sequence
6063:EilenbergβMacLane space
5443:, the homotopy groups
4966:, then elements of the
3656:suspension homomorphism
1803:As noted already, when
694:Disk with collapsed rim
471:. These are called the
64:more precise citations.
10737:Miranda, Rick (1995),
9504:
7016:, which is denoted by
6895:Where the entry is an
6838:field with one element
6822:BarrattβPriddy theorem
6712:
6531:
6507:
6333:has been reduced to a
6218:
5817:
5733:≡ 1 or 2 (mod 8)
5672:
5432:. In the stable range
5386:, is the image of the
5263:
4808:Finiteness and torsion
4584:
4522:
4360:
4247:
4099:
4043:
3984:
3789:
3682:, giving isomorphisms
3650:, these sequences are
3625:
3424:
3300:shows that there is a
3277:
3128:is an isomorphism for
2945:
1667:were also introduced.
1584:
1443:
1373:
1330:The identification (a
1279:
1004:
996:
247:continuously into the
156:
148:
118:topological invariants
97:
11359:A history of Topology
11250:Mathematische Annalen
11035:Annals of Mathematics
10900:Annals of Mathematics
10791:10.2969/jmsj/02540707
10628:Kochman, Stanley O.;
10549:Annals of Mathematics
10277:Annals of Mathematics
10239:Annals of Mathematics
9505:
6921:infinite cyclic group
6713:
6532:
6508:
6339:Berrick et al. (2006)
6216:
6184:May spectral sequence
6108:EHP spectral sequence
6004:Computational methods
5818:
5673:
5264:
5035:is an isomorphism of
4852:infinite cyclic group
4585:
4523:
4361:
4248:
4100:
4044:
3985:
3790:
3626:
3507:short exact sequences
3425:
3278:
2946:
1578:
1441:
1371:
1338:as an equality: thus
1319:of a loop around the
1301:infinite cyclic group
1266:
1075:to the pointed space
1002:
994:
952:, and the suspension
831:Suspension of equator
330:to a single point of
154:
142:
95:
11293:, pp. 665β723,
10943:, Paris: 1340β1342,
9471:
9378:, Example 0.3, p. 6.
6760:is the image of the
6642:
6586:of a compact smooth
6519:
6495:
6490:of smooth maps from
5852:is the generator of
5761:
5697:by the image of the
5650:
5304:− 1) − 2
5091:
4955:is an odd prime and
4940:-torsion occurs for
4538:
4428:
4260:
4147:
4056:
4000:
3941:
3689:
3516:
3311:
3231:
3035:the homotopy groups
2933:
1968:The homology groups
1461:space-filling curves
1231:. However, the real
1135:-th homotopy group,
1131:More generally, the
1059:) topological space
903:has as equator the (
537:as a basic example.
237:-dimensional sphere
120:, which reflect, in
11287:History of Topology
10836:Ravenel, Douglas C.
10540:Kervaire, Michel A.
10369:10.3792/pjaa.82.141
10230:Cohen, Frederick R.
10116:Cohen, Frederick R.
9906:Berrick et al. 2006
9426:, pp. 123β125.
6696:
6593:is divisible by 16.
6484:algebraic varieties
6401:corresponds to the
5864:(of order 2), then
5812:
5778:
5576:for such values of
4992:have order at most
3304:of homotopy groups
3302:long exact sequence
1697:George W. Whitehead
1625:the homotopy group
1617:, now known as the
1101:equivalence classes
746:, and its rim (or "
456:are independent of
312:, any mapping from
266:equivalence classes
187:) and the ordinary
11263:10.1007/BF01457962
11155:Algebraic Topology
10599:10.1007/BFb0083795
10208:, Paris: 393β395,
10198:Serre, Jean-Pierre
10172:, Paris: 288β290,
10160:Serre, Jean-Pierre
9882:Wang & Xu 2017
9500:
8741:, the part of the
6802:Kervaire invariant
6708:
6682:
6567:sphere around the
6527:
6503:
6476:singularity theory
6219:
6126:term given by the
6043:spectral sequences
5839:nilpotence theorem
5813:
5798:
5797:
5764:
5668:
5351:The J-homomorphism
5259:
4580:
4518:
4356:
4243:
4095:
4039:
3980:
3785:
3621:
3420:
3273:
3172:, and are denoted
3085:suspension theorem
2941:
1859:contractible space
1705:spectral sequences
1693:suspension theorem
1585:
1525:. This is because
1444:
1374:
1280:
1209:group homomorphism
1185:finitely generated
1126:bouquet of spheres
1119:abstract algebraic
1005:
997:
523:Algebraic topology
507:spectral sequences
377:with the group of
163:-dimensional unit
157:
149:
126:topological spaces
106:algebraic topology
98:
11318:(21 April 1997),
11300:978-0-444-82375-5
11222:Historical papers
11205:978-0-226-51183-2
11169:978-0-521-79540-1
10982:978-0-691-09586-8
10902:, Second Series,
10882:978-0-8218-3749-8
10851:978-0-8218-2967-7
10647:978-0-8218-0296-0
10630:Mahowald, Mark E.
10608:978-3-540-52468-7
10492:978-1-4704-3788-6
10482:10.1090/memo/1269
10279:, Second Series,
10242:, Second Series,
10071:Mahowald, Mark E.
9836:, pp. 67β74.
9326:
9325:
8712:
8711:
8311:
8310:
6932:cartesian product
6830:classifying space
6826:plus construction
6622:smooth structures
6580:Rokhlin's theorem
6199:complex cobordism
5843:complex cobordism
5782:
4814:Jean-Pierre Serre
4690:(1, 0, ..., 0) β
4657:-submanifolds of
2902:
2901:
1701:Jean-Pierre Serre
1665:fundamental group
1543:lifting criterion
1332:group isomorphism
1041:fundamental group
1009:topological space
981:(1, 0, 0, ..., 0)
750:") is the circle
511:Jean-Pierre Serre
353:, every map from
90:
89:
82:
16:(Redirected from
11416:
11389:
11388:
11387:
11368:
11367:
11366:
11352:
11351:
11350:
11345:
11330:
11329:
11328:
11303:
11291:Elsevier Science
11273:
11235:
11216:
11180:
11137:
11119:
11110:
11100:
11084:Acta Mathematica
11076:
11051:
11028:
10993:
10959:
10930:
10893:
10862:
10820:
10803:
10793:
10769:
10733:
10721:
10707:
10689:
10658:
10619:
10580:
10534:
10525:
10503:
10484:
10463:
10443:
10422:
10387:
10362:
10325:
10316:
10270:
10224:
10188:
10150:
10141:
10114:Berrick, A. J.;
10109:
10092:
10064:
10063:
10043:
10042:
10006:
9992:
9986:
9975:
9969:
9963:
9957:
9951:
9945:
9939:
9933:
9927:
9921:
9915:
9909:
9903:
9897:
9891:
9885:
9879:
9873:
9867:
9861:
9855:
9849:
9843:
9837:
9831:
9825:
9819:
9813:
9807:
9801:
9795:
9789:
9783:
9774:
9768:
9762:
9756:
9750:
9744:
9738:
9732:
9726:
9720:
9714:
9708:
9702:
9696:
9690:
9684:
9678:
9672:
9661:
9655:
9649:
9643:
9637:
9631:
9625:
9619:
9610:
9604:
9598:
9592:
9586:
9580:
9574:
9568:
9562:
9556:
9550:
9544:
9538:
9532:
9523:
9517:
9511:
9509:
9507:
9506:
9501:
9496:
9495:
9483:
9482:
9467:, Section 8.1, "
9461:
9455:
9449:
9443:
9433:
9427:
9421:
9415:
9409:
9403:
9397:
9391:
9385:
9379:
9373:
9367:
9361:
9355:
9349:
8779:
8770:Bott periodicity
8767:
8760:
8752:
8748:
8744:
8740:
8732:
8731:
8730:
8313:
8228:
8171:
8004:
7769:
7023:
7019:
7015:
6977:
6964:
6945:
6941:
6926:
6915:
6902:
6887:
6866:
6816:
6812:
6811:2 − 2 = 62
6807:
6804:1 in dimensions
6788:
6774:
6770:
6764:
6759:
6755:
6751:
6750:
6749:
6733:
6717:
6715:
6714:
6709:
6701:
6695:
6690:
6678:
6677:
6659:
6654:
6653:
6627:
6619:
6612:
6604:
6576:Vladimir Rokhlin
6566:
6559:
6552:
6537:
6536:
6534:
6533:
6528:
6526:
6513:
6512:
6510:
6509:
6504:
6502:
6466:
6458:
6429:
6400:
6390:
6384:
6374:
6367:
6353:
6332:
6317:
6311:
6302:
6296:
6290:
6279:
6273:
6267:
6261:
6255:
6248:
6234:
6196:
6181:
6177:
6166:Steenrod algebra
6164:
6160:
6146:
6145:
6125:
6101:
6097:
6090:
6083:
6079:
6059:Hurewicz theorem
6038:
6033:
6024:
6015:
6011:
5986:
5985:
5965:
5954:
5944:
5943:
5942:
5933:are elements of
5932:
5928:
5924:
5917:
5916:
5915:
5905:
5899:
5898:
5897:
5887:
5881:
5880:
5879:
5869:
5863:
5862:
5861:
5851:
5828:supercommutative
5822:
5820:
5819:
5814:
5811:
5806:
5796:
5777:
5772:
5742:
5734:
5727:
5726:
5725:
5713:
5700:
5696:
5695:
5694:
5682:
5678:
5677:
5675:
5674:
5669:
5667:
5662:
5657:
5644:
5640:
5633:
5632:
5630:
5629:
5626:
5623:
5616:
5614:
5613:
5610:
5607:
5594:
5579:
5575:
5553:
5541:Bernoulli number
5538:
5526:
5525:
5523:
5522:
5516:
5513:
5494:
5479:
5471:
5466:Bott periodicity
5463:
5456:
5442:
5427:
5419:
5385:
5378:
5346:
5342:
5335:is divisible by
5334:
5328:vanishes unless
5327:
5326:
5325:
5313:
5309:
5305:
5290:
5283:
5279:
5268:
5266:
5265:
5260:
5246:
5245:
5224:
5223:
5190:
5189:
5168:
5167:
5128:
5127:
5112:
5111:
5083:
5072:
5057:
5053:
5038:
5034:
5019:
5005:
5001:
4997:
4991:
4969:
4965:
4954:
4950:
4939:
4935:
4924:
4917:
4913:
4898:
4894:
4876:
4872:
4865:
4861:
4857:
4849:
4845:
4829:
4790:
4780:
4774:
4768:
4767:
4749:
4736:
4718:
4712:
4698:
4694:
4680:
4662:
4656:
4649:
4643:
4642:
4630:
4601:Framed cobordism
4589:
4587:
4586:
4581:
4576:
4575:
4563:
4562:
4550:
4549:
4527:
4525:
4524:
4519:
4511:
4510:
4498:
4497:
4482:
4481:
4469:
4468:
4453:
4452:
4440:
4439:
4420:
4414:
4407:
4401:
4394:
4388:
4381:
4374:
4365:
4363:
4362:
4357:
4349:
4348:
4336:
4335:
4314:
4313:
4301:
4300:
4285:
4284:
4272:
4271:
4252:
4250:
4249:
4244:
4236:
4235:
4223:
4222:
4201:
4200:
4188:
4187:
4172:
4171:
4159:
4158:
4139:
4127:
4104:
4102:
4101:
4096:
4094:
4093:
4081:
4080:
4068:
4067:
4048:
4046:
4045:
4040:
4038:
4037:
4025:
4024:
4012:
4011:
3989:
3987:
3986:
3981:
3979:
3978:
3966:
3965:
3953:
3952:
3933:
3919:
3918:
3916:
3915:
3907:
3904:
3891:
3889:
3880:
3869:
3848:
3844:
3830:
3816:
3812:
3794:
3792:
3791:
3786:
3778:
3777:
3765:
3764:
3743:
3742:
3730:
3729:
3714:
3713:
3701:
3700:
3681:
3649:
3639:
3630:
3628:
3627:
3622:
3611:
3610:
3598:
3597:
3576:
3575:
3563:
3562:
3547:
3546:
3534:
3533:
3504:
3492:
3486:
3480:
3466:
3456:
3429:
3427:
3426:
3421:
3401:
3400:
3373:
3372:
3351:
3350:
3329:
3328:
3299:
3282:
3280:
3279:
3274:
3269:
3268:
3256:
3255:
3243:
3242:
3203:
3192:
3185:
3184:
3183:
3167:
3156:
3138:
3127:
3108:
3089:Hans Freudenthal
3070:
3063:
3048:
3034:
3024:
3014:
3004:
2997:
2963:
2950:
2948:
2947:
2942:
2940:
2924:
2917:
2831:
2830:
2756:
2755:
2664:
2663:
2583:
2582:
2568:
2567:
2551:
2550:
2518:
2517:
2507:
2506:
2459:
2458:
2448:
2447:
2423:
2422:
2348:
2347:
2337:
2336:
2312:
2311:
2093:
2085:
2076:
2075:
2065:
2050:
2041:
2034:, b) the finite
2033:
2017:
2007:
1993:
1983:
1964:
1937:
1930:
1926:
1910:
1903:
1884:simply-connected
1876:Hurewicz theorem
1873:
1856:
1846:
1836:
1832:
1824:
1810:
1806:
1794:
1790:
1786:
1736:
1720:
1689:Hans Freudenthal
1684:Hurewicz theorem
1636:
1616:
1610:
1596:
1571:
1556:
1550:
1540:
1530:
1524:
1508:
1489:
1475:
1458:
1434:
1415:
1401:
1391:
1364:
1349:
1310:
1299:is therefore an
1298:
1278:
1259:
1230:
1222:
1199:
1190:
1178:
1174:
1168:
1164:
1152:
1148:
1134:
1116:
1106:
1094:
1090:
1086:
1074:
1062:
1054:
1034:complex analysis
982:
972:
964:
958:
951:
949:
948:
934:
933:
922:
921:
909:
902:
900:
899:
886:
885:
874:
873:
861:
857:
849:
839:
820:
812:
806:
800:
782:
780:
779:
768:
767:
755:
745:
743:
742:
731:
730:
718:
705:
683:
677:
668:
663:
661:
660:
647:
646:
635:
634:
622:
615:
601:
599:
598:
587:
586:
575:
574:
561:Implicit surface
543:
531:abstract algebra
504:
497:
478:
470:
459:
455:
436:
432:
410:
404:
394:
376:
359:to itself has a
358:
352:
335:
329:
323:
317:
311:
296:
292:
288:
259:
253:
251:
242:
236:
232:
215:
211:
200:
196:
186:
176:
170:
162:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
11424:
11423:
11419:
11418:
11417:
11415:
11414:
11413:
11404:Homotopy theory
11394:
11393:
11385:
11383:
11376:
11364:
11362:
11355:
11348:
11346:
11343:
11333:
11326:
11324:
11314:
11311:
11301:
11277:
11239:
11227:
11224:
11206:
11184:
11170:
11148:
11145:
11140:
11123:
11080:
11031:
11018:
10997:
10983:
10963:
10934:
10912:10.2307/1969485
10897:
10883:
10866:
10852:
10834:
10825:Pontrjagin, Lev
10772:
10759:
10749:10.1090/gsm/005
10736:
10719:
10713:Milnor, John W.
10711:
10692:
10670:
10661:
10648:
10627:
10609:
10591:Springer-Verlag
10584:
10562:10.2307/1970128
10544:Milnor, John W.
10538:
10507:
10493:
10466:
10453:Homotopy theory
10447:
10427:Fuks, Dmitry B.
10425:
10412:
10391:
10342:
10329:
10289:10.2307/1970586
10274:
10252:10.2307/1971238
10228:
10192:
10154:
10113:
10090:10.1.1.212.1163
10068:
10045:
10021:Adams, J. Frank
10019:
10015:
10010:
10009:
9993:
9989:
9976:
9972:
9964:
9960:
9952:
9948:
9940:
9936:
9928:
9924:
9916:
9912:
9904:
9900:
9892:
9888:
9880:
9876:
9868:
9864:
9856:
9852:
9844:
9840:
9832:
9828:
9820:
9816:
9808:
9804:
9796:
9792:
9784:
9777:
9769:
9765:
9757:
9753:
9745:
9741:
9733:
9729:
9721:
9717:
9709:
9705:
9697:
9693:
9685:
9681:
9673:
9664:
9656:
9652:
9644:
9640:
9632:
9628:
9620:
9613:
9605:
9601:
9593:
9589:
9581:
9577:
9569:
9565:
9557:
9553:
9545:
9541:
9533:
9526:
9518:
9514:
9487:
9474:
9469:
9468:
9462:
9458:
9450:
9446:
9434:
9430:
9422:
9418:
9410:
9406:
9398:
9394:
9386:
9382:
9374:
9370:
9362:
9358:
9350:
9341:
9336:
9331:
9280:
9268:27β
5β
7β
13β
19β
37
9227:
9175:
9122:
9069:
9016:
8967:
8910:
8861:
8820:
8762:
8754:
8750:
8746:
8742:
8735:
8729:
8724:
8723:
8722:
8720:
8717:
8676:
8633:
8590:
8547:
8504:
8461:
8418:
8375:
8263:
8226:
8206:
8169:
8149:
8094:
8039:
8002:
7982:
7927:
7914:
7865:
7807:
7767:
7747:
7689:
7631:
7573:
7515:
7457:
7399:
7341:
7283:
7225:
7167:
7110:
7017:
7014:
7010:
7006:
6998:
6990:
6986:
6976:
6970:
6963:
6957:
6951:
6943:
6939:
6934:(equivalently,
6924:
6914:
6908:
6900:
6881:
6871:
6860:
6850:
6847:
6834:symmetric group
6828:applied to the
6814:
6810:
6805:
6787:
6781:
6772:
6771:is of the form
6768:
6762:
6757:
6753:
6748:
6743:
6742:
6741:
6739:
6732:
6723:
6663:
6645:
6640:
6639:
6625:
6614:
6610:
6603:
6597:
6561:
6554:
6546:
6539:
6517:
6516:
6515:
6493:
6492:
6491:
6488:critical points
6464:
6452:
6445:
6423:
6419:
6411:
6403:Borromean rings
6392:
6386:
6380:
6369:
6361:
6355:
6349:
6328:
6313:
6310:
6304:
6298:
6292:
6286:
6275:
6269:
6263:
6257:
6251:
6247:
6241:
6230:
6217:Borromean rings
6194:
6179:
6168:
6162:
6158:
6152:
6144:
6134:
6133:
6132:
6130:
6124:
6118:
6099:
6092:
6085:
6081:
6077:
6036:
6031:
6022:
6013:
6009:
6006:
5994:Massey products
5971:
5970:
5956:
5946:
5941:
5938:
5937:
5936:
5934:
5930:
5926:
5922:
5914:
5911:
5910:
5909:
5907:
5901:
5896:
5893:
5892:
5891:
5889:
5883:
5878:
5875:
5874:
5873:
5871:
5865:
5860:
5857:
5856:
5855:
5853:
5849:
5759:
5758:
5749:
5743:to save space.
5736:
5729:
5724:
5719:
5718:
5717:
5715:
5712:
5706:
5698:
5693:
5688:
5687:
5686:
5684:
5680:
5648:
5647:
5646:
5642:
5638:
5627:
5624:
5621:
5620:
5618:
5611:
5608:
5606:
5600:
5599:
5597:
5596:
5588:
5581:
5577:
5569:
5559:
5552:β 1 β‘ 3 (mod 4)
5544:
5537:
5528:
5517:
5514:
5512:
5503:
5502:
5500:
5499:
5492:
5477:
5469:
5458:
5457:only depend on
5450:
5444:
5433:
5421:
5413:
5399:
5390:
5380:
5372:
5362:
5359:
5353:
5344:
5336:
5329:
5324:
5319:
5318:
5317:
5315:
5311:
5307:
5292:
5288:
5281:
5280:means take the
5273:
5228:
5206:
5172:
5144:
5116:
5094:
5089:
5088:
5081:
5078:James fibration
5066:
5059:
5055:
5040:
5039:-components if
5036:
5028:
5021:
5013:
5007:
5003:
4999:
4993:
4985:
4975:
4967:
4956:
4952:
4941:
4937:
4926:
4919:
4915:
4904:
4896:
4888:
4878:
4874:
4870:
4863:
4859:
4855:
4847:
4839:
4831:
4823:
4817:
4810:
4798:homotopy groups
4782:
4776:
4766:
4763:
4762:
4761:
4755:
4751:
4741:
4724:
4714:
4706:
4700:
4696:
4682:
4668:
4658:
4654:
4641:
4636:
4635:
4634:
4632:
4624:
4614:
4603:
4567:
4554:
4541:
4536:
4535:
4502:
4489:
4473:
4460:
4444:
4431:
4426:
4425:
4416:
4409:
4403:
4396:
4390:
4383:
4376:
4370:
4340:
4321:
4305:
4292:
4276:
4263:
4258:
4257:
4227:
4208:
4192:
4179:
4163:
4150:
4145:
4144:
4133:
4129:
4121:
4117:
4085:
4072:
4059:
4054:
4053:
4029:
4016:
4003:
3998:
3997:
3970:
3957:
3944:
3939:
3938:
3925:
3914:
3908:
3905:
3903:
3897:
3896:
3894:
3893:
3888:
3882:
3881:| + |
3879:
3873:
3871:
3867:
3860:
3853:
3846:
3838:
3832:
3824:
3818:
3814:
3806:
3799:
3769:
3750:
3734:
3721:
3705:
3692:
3687:
3686:
3675:
3665:
3658:
3645:
3635:
3602:
3583:
3567:
3554:
3538:
3525:
3514:
3513:
3498:
3494:
3488:
3482:
3474:
3468:
3458:
3450:
3440:
3434:
3386:
3364:
3342:
3320:
3309:
3308:
3287:
3260:
3247:
3234:
3229:
3228:
3214:
3212:Hopf fibrations
3194:
3187:
3182:
3177:
3176:
3175:
3173:
3168:are called the
3158:
3150:
3140:
3129:
3121:
3110:
3102:
3092:
3081:
3065:
3057:
3050:
3042:
3036:
3026:
3019:
3006:
2999:
2991:
2981:
2975:
2955:
2931:
2930:
2919:
2911:
2905:
2898:
2892:
2880:
2874:
2868:
2829:
2826:
2825:
2824:
2820:
2814:
2802:
2796:
2790:
2754:
2751:
2750:
2749:
2745:
2741:
2735:
2729:
2717:
2711:
2705:
2674:
2670:
2662:
2659:
2658:
2657:
2653:
2647:
2641:
2635:
2629:
2623:
2617:
2611:
2581:
2578:
2577:
2576:
2574:
2566:
2565:
2561:
2556:
2555:
2554:
2549:
2546:
2545:
2544:
2540:
2534:
2528:
2524:
2516:
2513:
2512:
2511:
2505:
2502:
2501:
2500:
2496:
2490:
2484:
2457:
2454:
2453:
2452:
2446:
2443:
2442:
2441:
2439:
2433:
2429:
2421:
2418:
2417:
2416:
2412:
2406:
2400:
2394:
2388:
2382:
2376:
2370:
2346:
2343:
2342:
2341:
2335:
2332:
2331:
2330:
2328:
2322:
2318:
2310:
2307:
2306:
2305:
2301:
2295:
2289:
2283:
2277:
2271:
2265:
2259:
2185:
2179:
2173:
2167:
2161:
2155:
2149:
2143:
2137:
2131:
2125:
2119:
2113:
2107:
2101:
2084:
2080:
2074:
2071:
2070:
2069:
2067:
2064:
2060:
2056:
2053:direct products
2049:
2043:
2039:
2031:
2024:
2009:
2001:
1995:
1985:
1977:
1969:
1958:
1945:
1939:
1932:
1928:
1920:
1912:
1905:
1897:
1891:
1880:homology groups
1865:
1852:
1838:
1834:
1830:
1818:
1812:
1808:
1804:
1801:
1792:
1788:
1780:
1770:
1730:
1722:
1714:
1708:
1680:Witold Hurewicz
1643:
1630:
1626:
1612:
1606:
1588:
1573:
1565:
1561:
1552:
1546:
1545:, any map from
1536:
1533:universal cover
1526:
1518:
1514:
1510:
1502:
1498:
1483:
1477:
1467:
1456:
1447:is therefore a
1436:
1428:
1424:
1409:
1403:
1396:
1389:
1366:
1358:
1354:
1343:
1339:
1308:
1292:
1288:
1272:
1268:
1261:
1253:
1249:
1242:
1228:
1220:
1198:
1192:
1188:
1176:
1170:
1166:
1154:
1150:
1142:
1136:
1132:
1108:
1104:
1092:
1088:
1076:
1064:
1060:
1048:
1044:
1030:residue theorem
989:
980:
971:(sphere, point)
970:
960:
953:
947:
941:
940:
939:
932:
929:
928:
927:
920:
917:
916:
915:
911:
904:
898:
893:
892:
891:
884:
881:
880:
879:
872:
869:
868:
867:
863:
859:
855:
845:
834:
818:
808:
807:. For example,
802:
792:
778:
775:
774:
773:
766:
763:
762:
761:
757:
751:
741:
738:
737:
736:
729:
726:
725:
724:
720:
714:
697:
679:
672:
666:
659:
654:
653:
652:
645:
642:
641:
640:
633:
630:
629:
628:
624:
623:; the equation
620:
611:
608:Euclidean space
597:
594:
593:
592:
585:
582:
581:
580:
573:
570:
569:
568:
564:
546:
541:
535:homotopy groups
519:
499:
488:
476:
461:
457:
449:
439:
434:
426:
416:
406:
400:
386:
370:
364:
354:
344:
331:
325:
319:
313:
302:
294:
290:
282:
276:
255:
249:
248:
238:
234:
226:
220:
213:
206:
203:Euclidean space
198:
192:
182:
172:
168:
160:
134:homotopy groups
130:homology groups
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
11422:
11420:
11412:
11411:
11406:
11396:
11395:
11392:
11391:
11374:
11353:
11335:Hatcher, Allen
11331:
11310:
11309:External links
11307:
11306:
11305:
11299:
11275:
11257:(1): 637β665,
11237:
11223:
11220:
11219:
11218:
11204:
11182:
11168:
11150:Hatcher, Allen
11144:
11141:
11139:
11138:
11121:
11091:(2): 319β407,
11078:
11042:(2): 501β580,
11029:
11016:
10995:
10981:
10961:
10932:
10906:(3): 425β505,
10895:
10881:
10864:
10850:
10832:
10822:
10784:(4): 707β732,
10770:
10757:
10734:
10709:
10690:
10668:
10663:Mahowald, Mark
10659:
10646:
10625:
10607:
10582:
10556:(3): 504β537,
10536:
10505:
10491:
10464:
10445:
10423:
10410:
10389:
10353:(8): 141β146,
10340:
10327:
10283:(2): 305β320,
10272:
10246:(3): 549β565,
10234:Moore, John C.
10226:
10190:
10152:
10132:(2): 265β326,
10111:
10083:(3): 533β550,
10066:
10016:
10014:
10011:
10008:
10007:
10003:Ravenel (2003)
9987:
9970:
9958:
9946:
9934:
9922:
9910:
9898:
9886:
9874:
9862:
9850:
9838:
9826:
9814:
9802:
9790:
9775:
9763:
9751:
9739:
9727:
9715:
9703:
9691:
9679:
9662:
9650:
9638:
9636:, p. 342.
9626:
9611:
9609:, p. 203.
9599:
9587:
9575:
9563:
9551:
9539:
9537:, p. 349.
9524:
9522:, p. 348.
9512:
9499:
9494:
9490:
9486:
9481:
9477:
9456:
9444:
9428:
9416:
9404:
9392:
9390:, p. 129.
9380:
9368:
9366:, p. xii.
9356:
9338:
9337:
9335:
9332:
9330:
9327:
9324:
9323:
9314:
9311:
9308:
9305:
9296:
9293:
9287:
9281:
9275:
9271:
9270:
9261:
9258:
9255:
9252:
9243:
9240:
9234:
9228:
9222:
9218:
9217:
9208:
9205:
9202:
9199:
9190:
9187:
9181:
9176:
9170:
9166:
9165:
9156:
9153:
9150:
9147:
9138:
9135:
9129:
9123:
9117:
9113:
9112:
9099:
9096:
9093:
9090:
9085:
9082:
9076:
9070:
9064:
9060:
9059:
9050:
9047:
9044:
9041:
9032:
9029:
9023:
9017:
9011:
9007:
9006:
8997:
8994:
8991:
8988:
8983:
8980:
8974:
8968:
8962:
8958:
8957:
8944:
8941:
8938:
8935:
8926:
8923:
8917:
8911:
8905:
8901:
8900:
8891:
8888:
8885:
8882:
8877:
8874:
8868:
8862:
8856:
8852:
8851:
8846:
8843:
8840:
8837:
8832:
8829:
8824:
8821:
8815:
8811:
8810:
8807:
8804:
8801:
8798:
8795:
8792:
8789:
8786:
8774:J-homomorphism
8725:
8716:
8713:
8710:
8709:
8706:
8703:
8700:
8697:
8694:
8691:
8688:
8685:
8682:
8671:
8667:
8666:
8663:
8660:
8657:
8654:
8651:
8648:
8645:
8642:
8639:
8628:
8624:
8623:
8620:
8617:
8614:
8611:
8608:
8605:
8602:
8599:
8596:
8585:
8581:
8580:
8577:
8574:
8571:
8568:
8565:
8562:
8559:
8556:
8553:
8542:
8538:
8537:
8534:
8531:
8528:
8525:
8522:
8519:
8516:
8513:
8510:
8499:
8495:
8494:
8491:
8488:
8485:
8482:
8479:
8476:
8473:
8470:
8467:
8456:
8452:
8451:
8448:
8445:
8442:
8439:
8436:
8433:
8430:
8427:
8424:
8413:
8409:
8408:
8405:
8402:
8399:
8396:
8393:
8390:
8387:
8384:
8381:
8370:
8366:
8365:
8360:
8355:
8350:
8345:
8340:
8335:
8330:
8325:
8320:
8309:
8308:
8305:
8302:
8299:
8296:
8293:
8290:
8287:
8284:
8281:
8278:
8275:
8272:
8269:
8258:
8254:
8253:
8250:
8247:
8244:
8241:
8238:
8235:
8232:
8229:
8224:
8221:
8218:
8215:
8212:
8201:
8197:
8196:
8193:
8190:
8187:
8184:
8181:
8178:
8175:
8172:
8167:
8164:
8161:
8158:
8155:
8144:
8140:
8139:
8136:
8133:
8130:
8127:
8124:
8121:
8118:
8115:
8112:
8109:
8106:
8103:
8100:
8089:
8085:
8084:
8081:
8078:
8075:
8072:
8069:
8066:
8063:
8060:
8057:
8054:
8051:
8048:
8045:
8034:
8030:
8029:
8026:
8023:
8020:
8017:
8014:
8011:
8008:
8005:
8000:
7997:
7994:
7991:
7988:
7977:
7973:
7972:
7969:
7966:
7963:
7960:
7957:
7954:
7951:
7948:
7945:
7942:
7939:
7936:
7933:
7922:
7918:
7917:
7910:
7907:
7904:
7901:
7898:
7895:
7892:
7889:
7886:
7883:
7880:
7877:
7874:
7871:
7860:
7856:
7855:
7852:
7849:
7846:
7843:
7840:
7837:
7834:
7831:
7828:
7825:
7822:
7819:
7816:
7813:
7802:
7798:
7797:
7794:
7791:
7788:
7785:
7782:
7779:
7776:
7773:
7770:
7765:
7762:
7759:
7756:
7753:
7742:
7738:
7737:
7734:
7731:
7728:
7725:
7722:
7719:
7716:
7713:
7710:
7707:
7704:
7701:
7698:
7695:
7684:
7680:
7679:
7676:
7673:
7670:
7667:
7664:
7661:
7658:
7655:
7652:
7649:
7646:
7643:
7640:
7637:
7626:
7622:
7621:
7618:
7615:
7612:
7609:
7606:
7603:
7600:
7597:
7594:
7591:
7588:
7585:
7582:
7579:
7568:
7564:
7563:
7560:
7557:
7554:
7551:
7548:
7545:
7542:
7539:
7536:
7533:
7530:
7527:
7524:
7521:
7510:
7506:
7505:
7502:
7499:
7496:
7493:
7490:
7487:
7484:
7481:
7478:
7475:
7472:
7469:
7466:
7463:
7452:
7448:
7447:
7444:
7441:
7438:
7435:
7432:
7429:
7426:
7423:
7420:
7417:
7414:
7411:
7408:
7405:
7394:
7390:
7389:
7386:
7383:
7380:
7377:
7374:
7371:
7368:
7365:
7362:
7359:
7356:
7353:
7350:
7347:
7336:
7332:
7331:
7328:
7325:
7322:
7319:
7316:
7313:
7310:
7307:
7304:
7301:
7298:
7295:
7292:
7289:
7278:
7274:
7273:
7270:
7267:
7264:
7261:
7258:
7255:
7252:
7249:
7246:
7243:
7240:
7237:
7234:
7231:
7220:
7216:
7215:
7212:
7209:
7206:
7203:
7200:
7197:
7194:
7191:
7188:
7185:
7182:
7179:
7176:
7173:
7162:
7158:
7157:
7154:
7151:
7148:
7145:
7142:
7139:
7136:
7133:
7130:
7127:
7124:
7121:
7118:
7116:
7105:
7101:
7100:
7095:
7090:
7085:
7080:
7075:
7070:
7065:
7060:
7055:
7050:
7045:
7040:
7035:
7030:
7020:in the table.
7012:
7008:
7004:
6996:
6988:
6980:
6979:
6972:
6959:
6953:
6928:
6917:
6910:
6893:
6873:
6852:
6846:
6843:
6842:
6841:
6818:
6794:
6783:
6777:
6776:
6744:
6727:
6720:
6719:
6718:
6707:
6704:
6700:
6694:
6689:
6685:
6681:
6676:
6673:
6670:
6666:
6662:
6658:
6652:
6648:
6634:
6633:
6630:exotic spheres
6613:-spheres (for
6599:
6594:
6572:
6569:critical value
6541:
6525:
6501:
6472:
6447:
6444:The fact that
6442:
6421:
6416:winding number
6410:
6407:
6357:
6325:
6324:
6308:
6282:
6281:
6245:
6237:
6236:
6225:
6224:
6211:
6210:
6187:
6154:
6148:
6135:
6122:
6113:The classical
6111:
6104:
6066:
6005:
6002:
5939:
5912:
5894:
5876:
5858:
5824:
5823:
5810:
5805:
5801:
5795:
5792:
5789:
5785:
5781:
5776:
5771:
5767:
5748:
5747:Ring structure
5745:
5720:
5708:
5689:
5666:
5661:
5656:
5604:
5583:
5561:
5556:
5555:
5532:
5507:
5496:
5489:
5446:
5405:
5395:
5388:J-homomorphism
5364:
5357:J-homomorphism
5355:Main article:
5352:
5349:
5320:
5270:
5269:
5258:
5255:
5252:
5249:
5244:
5241:
5238:
5235:
5231:
5227:
5222:
5219:
5216:
5213:
5209:
5205:
5202:
5199:
5196:
5193:
5188:
5185:
5182:
5179:
5175:
5171:
5166:
5163:
5160:
5157:
5154:
5151:
5147:
5143:
5140:
5137:
5134:
5131:
5126:
5123:
5119:
5115:
5110:
5107:
5104:
5101:
5097:
5061:
5023:
5009:
4977:
4880:
4846:(for positive
4833:
4819:
4809:
4806:
4800:of spaces and
4764:
4753:
4739:Hopf fibration
4702:
4652:differentiable
4637:
4631:and the group
4616:
4611:Lev Pontryagin
4602:
4599:
4595:Hopf invariant
4591:
4590:
4579:
4574:
4570:
4566:
4561:
4557:
4553:
4548:
4544:
4529:
4528:
4517:
4514:
4509:
4505:
4501:
4496:
4492:
4488:
4485:
4480:
4476:
4472:
4467:
4463:
4459:
4456:
4451:
4447:
4443:
4438:
4434:
4367:
4366:
4355:
4352:
4347:
4343:
4339:
4334:
4331:
4328:
4324:
4320:
4317:
4312:
4308:
4304:
4299:
4295:
4291:
4288:
4283:
4279:
4275:
4270:
4266:
4254:
4253:
4242:
4239:
4234:
4230:
4226:
4221:
4218:
4215:
4211:
4207:
4204:
4199:
4195:
4191:
4186:
4182:
4178:
4175:
4170:
4166:
4162:
4157:
4153:
4131:
4119:
4106:
4105:
4092:
4088:
4084:
4079:
4075:
4071:
4066:
4062:
4050:
4049:
4036:
4032:
4028:
4023:
4019:
4015:
4010:
4006:
3977:
3973:
3969:
3964:
3960:
3956:
3951:
3947:
3912:
3901:
3886:
3877:
3865:
3858:
3834:
3820:
3801:
3796:
3795:
3784:
3781:
3776:
3772:
3768:
3763:
3760:
3757:
3753:
3749:
3746:
3741:
3737:
3733:
3728:
3724:
3720:
3717:
3712:
3708:
3704:
3699:
3695:
3671:
3660:
3632:
3631:
3620:
3617:
3614:
3609:
3605:
3601:
3596:
3593:
3590:
3586:
3582:
3579:
3574:
3570:
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3557:
3553:
3550:
3545:
3541:
3537:
3532:
3528:
3524:
3521:
3496:
3470:
3467:, maps all of
3446:
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3404:
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3284:
3283:
3272:
3267:
3263:
3259:
3254:
3250:
3246:
3241:
3237:
3218:Hopf fibration
3216:The classical
3213:
3210:
3178:
3142:
3139:. The groups
3112:
3094:
3080:
3077:
3073:
3072:
3052:
3038:
3016:
2987:
2977:
2972:
2969:
2939:
2927:covering space
2907:
2900:
2899:
2896:
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2102:
2099:
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2082:
2078:
2072:
2062:
2058:
2045:
2023:
2020:
1997:
1973:
1954:
1941:
1916:
1893:
1833:-sphere to an
1814:
1800:
1799:General theory
1797:
1787:are known for
1772:
1759:Daniel Isaksen
1724:
1710:
1656:Analysis situs
1651:Henri PoincarΓ©
1647:Camille Jordan
1642:
1639:
1628:
1619:Hopf fibration
1581:Hopf fibration
1572:
1563:
1559:
1516:
1509:
1500:
1496:
1479:
1435:
1426:
1422:
1405:
1365:
1356:
1352:
1341:
1317:winding number
1290:
1270:
1260:
1251:
1247:
1241:
1238:
1194:
1187:. If for some
1175:(for positive
1138:
1057:path connected
1046:
1021:continuous map
988:
987:Homotopy group
985:
976:pointed sphere
967:
966:
942:
930:
918:
894:
882:
870:
841:
840:
827:
826:
776:
764:
739:
727:
707:
706:
690:
689:
655:
643:
631:
603:
602:
595:
583:
571:
545:
539:
518:
515:
441:
418:
413:
412:
397:Hopf fibration
383:
366:
341:
278:
222:
218:homotopy group
145:Hopf fibration
88:
87:
70:September 2022
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
11421:
11410:
11407:
11405:
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11401:
11399:
11382:
11381:
11375:
11372:
11361:
11360:
11354:
11342:
11341:
11336:
11332:
11323:
11322:
11317:
11313:
11312:
11308:
11302:
11296:
11292:
11288:
11284:
11280:
11279:May, J. Peter
11276:
11272:
11268:
11264:
11260:
11256:
11252:
11251:
11246:
11242:
11238:
11234:
11230:
11226:
11225:
11221:
11215:
11211:
11207:
11201:
11197:
11193:
11192:
11187:
11186:May, J. Peter
11183:
11179:
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11171:
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11157:
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11063:
11059:
11055:
11050:
11045:
11041:
11037:
11036:
11030:
11027:
11023:
11019:
11017:0-387-20430-X
11013:
11009:
11005:
11001:
10996:
10992:
10988:
10984:
10978:
10974:
10970:
10966:
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10865:
10861:
10857:
10853:
10847:
10843:
10842:
10837:
10833:
10830:
10826:
10823:
10819:
10815:
10811:
10807:
10802:
10797:
10792:
10787:
10783:
10779:
10775:
10774:Nishida, Goro
10771:
10768:
10764:
10760:
10758:0-8218-0268-2
10754:
10750:
10746:
10742:
10741:
10735:
10731:
10727:
10726:
10718:
10714:
10710:
10706:
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10575:
10571:
10567:
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10551:
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10515:
10511:
10506:
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10498:
10494:
10488:
10483:
10478:
10474:
10470:
10465:
10462:
10458:
10454:
10450:
10446:
10442:
10438:
10437:
10432:
10428:
10424:
10421:
10417:
10413:
10411:0-387-94657-8
10407:
10403:
10399:
10395:
10390:
10386:
10382:
10378:
10374:
10370:
10366:
10361:
10356:
10352:
10348:
10344:
10339:
10336:-theory over
10335:
10328:
10324:
10320:
10315:
10310:
10306:
10302:
10298:
10294:
10290:
10286:
10282:
10278:
10273:
10269:
10265:
10261:
10257:
10253:
10249:
10245:
10241:
10240:
10235:
10231:
10227:
10223:
10219:
10215:
10211:
10207:
10203:
10199:
10195:
10194:Cartan, Henri
10191:
10187:
10183:
10179:
10175:
10171:
10167:
10166:
10161:
10157:
10156:Cartan, Henri
10153:
10149:
10145:
10140:
10135:
10131:
10127:
10126:
10121:
10117:
10112:
10108:
10104:
10100:
10096:
10091:
10086:
10082:
10078:
10077:
10072:
10067:
10062:
10057:
10053:
10049:
10041:
10036:
10032:
10028:
10027:
10022:
10018:
10017:
10012:
10004:
10000:
9996:
9991:
9988:
9984:
9980:
9974:
9971:
9967:
9962:
9959:
9955:
9950:
9947:
9943:
9938:
9935:
9932:, p. 32.
9931:
9926:
9923:
9919:
9914:
9911:
9907:
9902:
9899:
9895:
9890:
9887:
9883:
9878:
9875:
9871:
9866:
9863:
9859:
9854:
9851:
9847:
9842:
9839:
9835:
9830:
9827:
9823:
9822:Mahowald 2001
9818:
9815:
9811:
9806:
9803:
9799:
9794:
9791:
9787:
9782:
9780:
9776:
9772:
9767:
9764:
9760:
9755:
9752:
9748:
9743:
9740:
9737:, p. 25.
9736:
9731:
9728:
9724:
9719:
9716:
9712:
9707:
9704:
9700:
9695:
9692:
9688:
9683:
9680:
9676:
9671:
9669:
9667:
9663:
9659:
9654:
9651:
9647:
9642:
9639:
9635:
9630:
9627:
9623:
9618:
9616:
9612:
9608:
9603:
9600:
9596:
9591:
9588:
9584:
9579:
9576:
9573:, p. 90.
9572:
9571:Walschap 2004
9567:
9564:
9560:
9555:
9552:
9549:, p. 61.
9548:
9543:
9540:
9536:
9531:
9529:
9525:
9521:
9516:
9513:
9492:
9488:
9479:
9475:
9466:
9460:
9457:
9454:, p. 29.
9453:
9448:
9445:
9441:
9437:
9432:
9429:
9425:
9420:
9417:
9413:
9408:
9405:
9402:, p. 28.
9401:
9396:
9393:
9389:
9384:
9381:
9377:
9372:
9369:
9365:
9360:
9357:
9353:
9348:
9346:
9344:
9340:
9333:
9328:
9322:
9318:
9315:
9312:
9309:
9306:
9304:
9300:
9297:
9294:
9291:
9288:
9285:
9282:
9279:
9273:
9272:
9269:
9265:
9262:
9259:
9256:
9253:
9251:
9247:
9244:
9241:
9238:
9235:
9232:
9229:
9226:
9220:
9219:
9216:
9212:
9209:
9206:
9203:
9200:
9198:
9194:
9191:
9188:
9185:
9182:
9180:
9177:
9174:
9168:
9167:
9164:
9160:
9157:
9154:
9151:
9148:
9146:
9142:
9139:
9136:
9133:
9130:
9127:
9124:
9121:
9115:
9114:
9111:
9107:
9103:
9100:
9097:
9094:
9091:
9089:
9086:
9083:
9080:
9077:
9074:
9071:
9068:
9062:
9061:
9058:
9054:
9051:
9048:
9045:
9042:
9040:
9036:
9033:
9030:
9027:
9024:
9021:
9018:
9015:
9009:
9008:
9005:
9001:
8998:
8995:
8992:
8989:
8987:
8984:
8981:
8978:
8975:
8972:
8969:
8966:
8960:
8959:
8956:
8952:
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8945:
8942:
8939:
8936:
8934:
8930:
8927:
8924:
8921:
8918:
8915:
8912:
8909:
8903:
8902:
8899:
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8780:
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8771:
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8714:
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8689:
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7229:
7224:
7218:
7217:
7213:
7210:
7207:
7204:
7201:
7198:
7195:
7192:
7189:
7186:
7183:
7180:
7177:
7174:
7171:
7166:
7160:
7159:
7155:
7152:
7149:
7146:
7143:
7140:
7137:
7134:
7131:
7128:
7125:
7122:
7119:
7117:
7114:
7109:
7103:
7102:
7099:
7096:
7094:
7091:
7089:
7086:
7084:
7081:
7079:
7076:
7074:
7071:
7069:
7066:
7064:
7061:
7059:
7056:
7054:
7051:
7049:
7046:
7044:
7041:
7039:
7036:
7034:
7031:
7028:
7025:
7024:
7021:
7002:
6994:
6984:
6975:
6968:
6962:
6956:
6949:
6948:common factor
6937:
6933:
6929:
6922:
6918:
6913:
6906:
6898:
6894:
6891:
6890:
6889:
6885:
6880:
6876:
6868:
6864:
6859:
6855:
6844:
6839:
6835:
6831:
6827:
6823:
6819:
6803:
6799:
6795:
6792:
6786:
6779:
6778:
6766:
6765:-homomorphism
6747:
6737:
6730:
6726:
6721:
6705:
6702:
6698:
6692:
6687:
6683:
6674:
6671:
6668:
6664:
6660:
6656:
6650:
6638:
6637:
6636:
6635:
6631:
6623:
6617:
6608:
6602:
6595:
6592:
6589:
6585:
6581:
6577:
6573:
6570:
6564:
6557:
6550:
6544:
6489:
6485:
6481:
6477:
6473:
6470:
6467:-dimensional
6462:
6456:
6450:
6443:
6440:
6437:
6433:
6427:
6417:
6413:
6412:
6408:
6406:
6404:
6399:
6395:
6389:
6383:
6378:
6372:
6365:
6360:
6352:
6347:
6344:
6340:
6336:
6331:
6322:
6316:
6307:
6301:
6295:
6289:
6284:
6283:
6278:
6272:
6266:
6260:
6254:
6244:
6239:
6238:
6233:
6227:
6226:
6221:
6220:
6215:
6208:
6204:
6200:
6192:
6188:
6185:
6175:
6171:
6167:
6161:over the mod
6157:
6151:
6142:
6138:
6129:
6121:
6116:
6112:
6109:
6105:
6095:
6088:
6075:
6071:
6067:
6064:
6060:
6056:
6052:
6048:
6047:
6046:
6044:
6040:
6030:
6026:
6019:
6003:
6001:
5999:
5995:
5990:
5983:
5979:
5975:
5969:
5966:, there is a
5963:
5959:
5952:
5949:
5919:
5904:
5886:
5868:
5846:
5844:
5840:
5836:
5832:
5829:
5808:
5803:
5799:
5793:
5790:
5787:
5783:
5779:
5774:
5769:
5765:
5757:
5756:
5755:
5754:
5746:
5744:
5740:
5732:
5723:
5711:
5704:
5692:
5659:
5635:
5603:
5592:
5586:
5573:
5568:
5564:
5551:
5547:
5542:
5536:
5531:
5521:
5511:
5506:
5497:
5490:
5487:
5483:
5475:
5474:
5473:
5467:
5461:
5454:
5449:
5440:
5436:
5431:
5425:
5417:
5412:
5408:
5403:
5398:
5393:
5389:
5383:
5376:
5371:
5367:
5358:
5350:
5348:
5340:
5332:
5323:
5303:
5299:
5295:
5285:
5277:
5253:
5242:
5239:
5236:
5233:
5229:
5220:
5217:
5214:
5211:
5207:
5203:
5197:
5186:
5183:
5180:
5177:
5173:
5164:
5161:
5158:
5155:
5152:
5149:
5145:
5141:
5135:
5124:
5121:
5117:
5108:
5105:
5102:
5099:
5095:
5087:
5086:
5085:
5079:
5074:
5070:
5064:
5051:
5047:
5043:
5032:
5026:
5017:
5012:
4996:
4989:
4984:
4980:
4973:
4963:
4959:
4948:
4944:
4933:
4929:
4922:
4911:
4907:
4902:
4892:
4887:
4883:
4867:
4853:
4843:
4837:
4827:
4822:
4815:
4807:
4805:
4803:
4799:
4795:
4789:
4785:
4779:
4772:
4759:
4748:
4744:
4740:
4735:
4731:
4727:
4722:
4717:
4710:
4705:
4693:
4689:
4685:
4679:
4675:
4671:
4667:. Every map
4666:
4665:normal bundle
4661:
4653:
4647:
4640:
4628:
4623:
4619:
4612:
4608:
4600:
4598:
4596:
4577:
4572:
4568:
4559:
4555:
4546:
4542:
4534:
4533:
4532:
4515:
4507:
4503:
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4478:
4474:
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4461:
4457:
4449:
4445:
4436:
4432:
4424:
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4422:
4419:
4412:
4406:
4399:
4393:
4386:
4379:
4373:
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4341:
4332:
4329:
4326:
4322:
4318:
4310:
4306:
4297:
4293:
4289:
4281:
4277:
4268:
4264:
4256:
4255:
4240:
4232:
4228:
4219:
4216:
4213:
4209:
4205:
4197:
4193:
4184:
4180:
4176:
4168:
4164:
4155:
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4143:
4142:
4141:
4137:
4125:
4115:
4111:
4090:
4086:
4077:
4073:
4064:
4060:
4052:
4051:
4034:
4030:
4021:
4017:
4008:
4004:
3996:
3995:
3994:
3993:
3975:
3971:
3962:
3958:
3949:
3945:
3935:
3932:
3928:
3923:
3911:
3900:
3885:
3876:
3864:
3857:
3850:
3842:
3837:
3828:
3823:
3813:vanishes for
3810:
3804:
3782:
3774:
3770:
3761:
3758:
3755:
3751:
3747:
3739:
3735:
3726:
3722:
3718:
3710:
3706:
3697:
3693:
3685:
3684:
3683:
3679:
3674:
3669:
3663:
3657:
3653:
3648:
3643:
3638:
3618:
3607:
3603:
3594:
3591:
3588:
3584:
3572:
3568:
3559:
3555:
3543:
3539:
3530:
3526:
3519:
3512:
3511:
3510:
3508:
3502:
3491:
3485:
3478:
3473:
3465:
3461:
3454:
3449:
3444:
3439:
3417:
3414:
3405:
3397:
3394:
3391:
3387:
3377:
3369:
3365:
3355:
3347:
3343:
3333:
3325:
3321:
3314:
3307:
3306:
3305:
3303:
3298:
3294:
3290:
3270:
3265:
3261:
3252:
3248:
3239:
3235:
3227:
3226:
3225:
3223:
3219:
3211:
3209:
3207:
3201:
3197:
3190:
3181:
3171:
3165:
3161:
3154:
3149:
3145:
3136:
3132:
3125:
3119:
3115:
3106:
3101:
3097:
3090:
3086:
3078:
3076:
3068:
3061:
3055:
3046:
3041:
3033:
3029:
3022:
3017:
3013:
3009:
3002:
2995:
2990:
2985:
2980:
2973:
2970:
2967:
2966:
2965:
2962:
2958:
2952:
2928:
2922:
2915:
2910:
2894:
2888:
2885:
2882:
2876:
2870:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2838:
2835:
2834:
2822:
2816:
2810:
2807:
2804:
2798:
2792:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2763:
2760:
2759:
2747:
2737:
2731:
2725:
2722:
2719:
2713:
2707:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2681:
2678:
2677:
2666:
2655:
2649:
2643:
2637:
2631:
2625:
2619:
2613:
2607:
2604:
2601:
2598:
2595:
2592:
2590:
2587:
2586:
2570:
2553:
2542:
2536:
2530:
2520:
2509:
2498:
2492:
2486:
2480:
2477:
2474:
2471:
2468:
2466:
2463:
2462:
2450:
2435:
2425:
2414:
2408:
2402:
2396:
2390:
2384:
2378:
2372:
2366:
2363:
2360:
2357:
2355:
2352:
2351:
2339:
2324:
2314:
2303:
2297:
2291:
2285:
2279:
2273:
2267:
2261:
2255:
2252:
2249:
2246:
2244:
2241:
2240:
2236:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2192:
2189:
2188:
2181:
2175:
2169:
2163:
2157:
2151:
2145:
2139:
2133:
2127:
2121:
2115:
2109:
2103:
2097:
2095:
2094:
2091:
2089:
2054:
2051:), or c) the
2048:
2037:
2036:cyclic groups
2029:
2028:trivial group
2021:
2019:
2016:
2012:
2005:
2000:
1992:
1988:
1981:
1976:
1972:
1966:
1962:
1957:
1953:
1949:
1944:
1935:
1924:
1919:
1915:
1908:
1901:
1896:
1889:
1885:
1881:
1877:
1872:
1868:
1862:
1860:
1855:
1850:
1845:
1841:
1837:-sphere with
1828:
1827:trivial group
1822:
1817:
1807:is less than
1798:
1796:
1784:
1779:
1775:
1768:
1764:
1760:
1756:
1755:Mark Mahowald
1752:
1748:
1744:
1740:
1734:
1728:
1718:
1713:
1706:
1702:
1698:
1694:
1690:
1685:
1681:
1677:
1673:
1668:
1666:
1662:
1658:
1657:
1652:
1648:
1640:
1638:
1634:
1624:
1620:
1615:
1609:
1604:
1600:
1595:
1591:
1582:
1577:
1569:
1560:
1558:
1555:
1549:
1544:
1539:
1534:
1529:
1522:
1506:
1497:
1495:
1493:
1487:
1482:
1474:
1470:
1464:
1462:
1454:
1450:
1449:trivial group
1440:
1432:
1423:
1421:
1419:
1413:
1408:
1399:
1393:
1387:
1386:
1380:
1370:
1362:
1353:
1351:
1347:
1337:
1336:often written
1333:
1328:
1326:
1322:
1318:
1314:
1307:to the group
1306:
1302:
1296:
1285:
1276:
1265:
1257:
1248:
1246:
1239:
1237:
1234:
1226:
1218:
1217:homeomorphism
1214:
1210:
1205:
1203:
1202:trivial group
1197:
1186:
1182:
1173:
1162:
1158:
1146:
1141:
1129:
1127:
1123:
1120:
1115:
1111:
1102:
1098:
1084:
1080:
1072:
1068:
1058:
1052:
1042:
1037:
1035:
1031:
1026:
1022:
1018:
1017:neighborhoods
1014:
1010:
1001:
993:
986:
984:
978:
977:
963:
957:
945:
938:
926:
914:
907:
897:
890:
878:
866:
853:
848:
843:
842:
838:
832:
829:
828:
824:
816:
811:
805:
799:
795:
790:
786:
772:
760:
754:
749:
735:
723:
717:
713:
709:
708:
704:
700:
695:
692:
691:
687:
682:
675:
670:
664:produces the
658:
651:
639:
627:
619:
614:
609:
605:
604:
591:
579:
567:
562:
559:
558:
557:
555:
551:
540:
538:
536:
532:
528:
524:
516:
514:
512:
508:
502:
495:
491:
486:
482:
474:
468:
464:
453:
448:
444:
433:for positive
430:
425:
421:
409:
403:
398:
393:
389:
384:
380:
374:
369:
362:
357:
351:
347:
342:
339:
338:trivial group
334:
328:
322:
316:
310:
306:
300:
299:
298:
286:
281:
273:
271:
270:abelian group
267:
263:
258:
246:
241:
230:
225:
219:
209:
205:of dimension
204:
195:
190:
185:
180:
175:
167:β called the
166:
153:
146:
141:
137:
135:
131:
127:
123:
119:
115:
111:
107:
103:
94:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
11384:, retrieved
11379:
11363:, retrieved
11358:
11347:, retrieved
11339:
11325:, retrieved
11320:
11286:
11254:
11248:
11232:
11229:Δech, Eduard
11190:
11154:
11125:
11088:
11082:
11039:
11033:
10999:
10968:
10965:Toda, Hirosi
10940:
10936:
10903:
10899:
10868:
10840:
10828:
10781:
10777:
10739:
10732:(6): 804β809
10729:
10723:
10698:
10678:
10672:
10637:
10586:
10553:
10547:
10513:
10509:
10472:
10468:
10452:
10449:Hu, Sze-tsen
10434:
10393:
10360:math/0605429
10350:
10346:
10337:
10333:
10280:
10276:
10243:
10237:
10205:
10201:
10169:
10163:
10129:
10123:
10080:
10074:
10051:
10047:
10033:(1): 21β71,
10030:
10024:
9990:
9973:
9966:Deitmar 2006
9961:
9949:
9937:
9930:Hatcher 2002
9925:
9913:
9901:
9889:
9877:
9870:Isaksen 2019
9865:
9858:Kochman 1990
9853:
9848:, Chapter 5.
9846:Ravenel 2003
9841:
9834:Ravenel 2003
9829:
9817:
9810:Ravenel 2003
9805:
9793:
9771:Nishida 1973
9766:
9754:
9742:
9735:Ravenel 2003
9730:
9718:
9713:, p. 4.
9711:Ravenel 2003
9706:
9694:
9682:
9675:Scorpan 2005
9658:Hatcher 2002
9653:
9646:Hatcher 2002
9641:
9634:Hatcher 2002
9629:
9602:
9590:
9578:
9566:
9554:
9547:Hatcher 2002
9542:
9535:Hatcher 2002
9520:Hatcher 2002
9515:
9459:
9452:Hatcher 2002
9447:
9431:
9424:Miranda 1995
9419:
9414:, p. 3.
9412:Hatcher 2002
9407:
9400:Hatcher 2002
9395:
9388:Hatcher 2002
9383:
9376:Hatcher 2002
9371:
9364:Hatcher 2002
9359:
9320:
9316:
9302:
9298:
9289:
9283:
9277:
9267:
9263:
9249:
9245:
9236:
9230:
9224:
9214:
9210:
9196:
9192:
9183:
9178:
9172:
9162:
9158:
9144:
9140:
9131:
9125:
9119:
9109:
9105:
9101:
9087:
9078:
9072:
9066:
9056:
9052:
9038:
9034:
9025:
9019:
9013:
9003:
8999:
8985:
8976:
8970:
8964:
8954:
8950:
8946:
8932:
8928:
8919:
8913:
8907:
8897:
8893:
8879:
8870:
8864:
8858:
8848:
8834:
8826:
8817:
8782:
8763:
8756:
8736:
8726:
8718:
8678:
8673:
8635:
8630:
8592:
8587:
8549:
8544:
8506:
8501:
8463:
8458:
8420:
8415:
8377:
8372:
8362:
8357:
8352:
8347:
8342:
8337:
8332:
8327:
8322:
8316:
8265:
8260:
8208:
8203:
8151:
8146:
8096:
8091:
8041:
8036:
7984:
7979:
7929:
7924:
7912:
7867:
7862:
7809:
7804:
7749:
7744:
7691:
7686:
7633:
7628:
7575:
7570:
7517:
7512:
7459:
7454:
7401:
7396:
7343:
7338:
7285:
7280:
7227:
7222:
7169:
7164:
7112:
7107:
7097:
7092:
7087:
7082:
7077:
7072:
7067:
7062:
7057:
7052:
7047:
7042:
7037:
7032:
7026:
7000:
6992:
6982:
6981:
6973:
6960:
6954:
6911:
6905:cyclic group
6883:
6878:
6874:
6869:
6862:
6857:
6853:
6848:
6784:
6745:
6728:
6724:
6615:
6600:
6562:
6555:
6548:
6542:
6459:implies the
6454:
6448:
6425:
6409:Applications
6397:
6393:
6387:
6381:
6370:
6363:
6358:
6350:
6346:braid groups
6329:
6326:
6314:
6305:
6299:
6293:
6287:
6276:
6270:
6264:
6258:
6252:
6242:
6231:
6173:
6169:
6155:
6149:
6140:
6136:
6119:
6093:
6086:
6007:
5989:Hiroshi Toda
5981:
5977:
5973:
5968:Toda bracket
5961:
5957:
5950:
5947:
5920:
5918:is trivial.
5902:
5884:
5866:
5848:Example: If
5847:
5825:
5750:
5738:
5730:
5721:
5709:
5690:
5636:
5601:
5590:
5584:
5571:
5566:
5562:
5557:
5549:
5545:
5534:
5529:
5519:
5509:
5504:
5459:
5452:
5447:
5438:
5434:
5428:denotes the
5423:
5415:
5410:
5406:
5401:
5396:
5391:
5381:
5374:
5369:
5365:
5360:
5338:
5330:
5321:
5306:for a prime
5301:
5297:
5293:
5286:
5275:
5271:
5075:
5068:
5062:
5049:
5045:
5041:
5030:
5024:
5015:
5010:
4994:
4987:
4982:
4978:
4961:
4957:
4946:
4942:
4931:
4927:
4920:
4909:
4905:
4890:
4885:
4881:
4868:
4841:
4835:
4825:
4820:
4811:
4787:
4783:
4777:
4770:
4757:
4746:
4742:
4733:
4729:
4725:
4715:
4708:
4703:
4691:
4687:
4683:
4677:
4673:
4669:
4659:
4645:
4638:
4626:
4621:
4617:
4604:
4592:
4530:
4417:
4410:
4404:
4397:
4391:
4384:
4377:
4371:
4368:
4135:
4123:
4107:
3936:
3930:
3926:
3909:
3898:
3883:
3874:
3862:
3855:
3851:
3840:
3835:
3826:
3821:
3808:
3802:
3797:
3677:
3672:
3667:
3661:
3646:
3636:
3633:
3500:
3489:
3483:
3476:
3471:
3463:
3459:
3452:
3447:
3442:
3437:
3432:
3296:
3292:
3288:
3285:
3222:fiber bundle
3215:
3205:
3199:
3195:
3188:
3179:
3169:
3163:
3159:
3152:
3147:
3143:
3134:
3130:
3123:
3117:
3113:
3104:
3099:
3095:
3082:
3074:
3066:
3059:
3053:
3044:
3039:
3031:
3027:
3023:= 2, 3, 4, 5
3020:
3011:
3007:
3000:
2993:
2988:
2983:
2978:
2960:
2956:
2953:
2920:
2913:
2908:
2903:
2836:
2761:
2679:
2588:
2464:
2353:
2242:
2190:
2046:
2042:(written as
2025:
2014:
2010:
2003:
1998:
1990:
1986:
1979:
1974:
1970:
1967:
1960:
1955:
1951:
1947:
1942:
1933:
1922:
1917:
1913:
1906:
1899:
1894:
1887:
1870:
1866:
1863:
1853:
1843:
1839:
1820:
1815:
1802:
1782:
1777:
1773:
1763:Guozhen Wang
1751:J. Peter May
1743:Hiroshi Toda
1732:
1726:
1716:
1711:
1669:
1654:
1644:
1632:
1613:
1607:
1593:
1589:
1586:
1567:
1553:
1547:
1537:
1527:
1520:
1511:
1504:
1485:
1480:
1472:
1468:
1465:
1445:
1430:
1411:
1406:
1397:
1394:
1383:
1375:
1360:
1345:
1329:
1294:
1281:
1274:
1267:Elements of
1255:
1243:
1206:
1195:
1171:
1160:
1156:
1144:
1139:
1130:
1113:
1109:
1082:
1078:
1070:
1066:
1050:
1038:
1006:
974:
968:
961:
955:
943:
936:
924:
912:
905:
895:
888:
876:
864:
846:
836:
830:
815:line segment
809:
803:
797:
793:
770:
758:
752:
733:
721:
715:
702:
698:
693:
680:
673:
656:
649:
637:
625:
612:
589:
577:
565:
560:
548:An ordinary
547:
520:
500:
493:
489:
472:
466:
462:
451:
446:
442:
428:
423:
419:
414:
407:
401:
391:
387:
372:
367:
355:
349:
345:
332:
326:
320:
314:
308:
304:
284:
279:
274:
256:
252:-dimensional
239:
228:
223:
217:
207:
193:
183:
173:
158:
109:
102:mathematical
99:
76:
67:
48:
11241:Hopf, Heinz
10801:2433/220059
10516:: 107β243,
10044:. See also
9983:Toda (1962)
9463:See, e.g.,
6806:2 − 2
6780:The groups
6773:2 − 2
6607:h-cobordism
6480:smooth maps
6441:has a zero.
6025:-components
5831:graded ring
5491:trivial if
4110:quaternions
1747:Frank Adams
1672:Eduard Δech
1621:. This map
1284:rubber band
62:introducing
11398:Categories
11386:2007-11-14
11365:2007-11-14
11349:2007-10-20
11327:2007-10-09
11316:Baez, John
11098:1809.09290
11049:1601.02184
10523:2001.04511
10054:(3): 331,
9798:Cohen 1968
9759:Adams 1966
9723:Serre 1952
9687:Serre 1951
9438:, p.
9329:References
9321:3β
25β
11β
41
8759:− 1)
6967:isomorphic
6936:direct sum
6591:4-manifold
6578:, implies
6439:polynomial
6337:question.
6128:Ext groups
6074:loop space
5998:cohomology
5753:direct sum
5703:Adams 1966
5484:to 0 or 1
5341:− 1)
3920:cover the
3890:| = 1
3642:suspension
1927:. For the
1849:surjective
1603:Heinz Hopf
1305:isomorphic
1225:isomorphic
850:to be the
823:CW complex
517:Background
114:dimensions
45:references
11281:(1999a),
11271:123533891
11188:(1999b),
11117:119303902
11074:119147703
10949:0764-4442
10810:0025-5645
10705:EMS Press
10441:EMS Press
10429:(2001) ,
10377:0386-2194
10214:0764-4442
10178:0764-4442
10085:CiteSeerX
9995:Fuks 2001
9786:Toda 1962
9747:Fuks 2001
9622:May 1999a
9607:Δech 1932
9559:Hopf 1931
9476:π
9197:9β
7β
11β
31
9095:16β
2β
9β
5
8240:504β
24β
2
8170:24β
12β
4β
2
8016:240β
24β
4
6684:π
6680:→
6647:Θ
6584:signature
6582:that the
6565:− 1
6558:− 1
6027:for each
5835:nilpotent
5800:π
5791:≥
5784:⨁
5770:∗
5766:π
5482:congruent
5394: : Ο
5310:then the
5240:−
5208:π
5204:⊕
5184:−
5162:−
5146:π
5096:π
4972:component
4812:In 1951,
4794:RenΓ© Thom
4695:a framed
4607:cobordism
4565:→
4552:↪
4491:π
4487:⊕
4462:π
4458:≠
4433:π
4387:= 1, 2, 3
4330:−
4323:π
4319:⊕
4294:π
4265:π
4217:−
4210:π
4206:⊕
4181:π
4152:π
4114:octonions
4083:→
4070:↪
4027:→
4014:↪
3968:→
3955:↪
3759:−
3752:π
3748:⊕
3723:π
3694:π
3616:→
3592:−
3585:π
3581:→
3556:π
3552:→
3527:π
3523:→
3415:⋯
3412:→
3395:−
3388:π
3384:→
3366:π
3362:→
3344:π
3340:→
3322:π
3318:→
3315:⋯
3258:→
3245:↪
2038:of order
1864:The case
1767:Zhouli Xu
1739:JosΓ© Adem
1623:generates
1379:injective
1303:, and is
1213:bijection
1169:equal to
1013:open sets
959:produces
910:)-sphere
908:− 1
801:produces
618:dimension
483:, called
122:algebraic
104:field of
11243:(1931),
11152:(2002),
10967:(1962),
10838:(2003),
10715:(2011),
10665:(1998),
10632:(1995),
10475:(1269),
10451:(1959),
10323:16591550
10048:Topology
10026:Topology
9049:4β
2β
3β
5
8772:via the
8761:divides
8705:ββ
264β
2
8687:264β
4β
2
8521:ββ
480β
2
8252:480β
4β
2
8227:120β
12β
2
8003:2520β
6β
2
7768:120β
12β
2
6946:have no
6545:−1
6451:−1
6343:Brunnian
5984:⟩
5972:⟨
5900:, while
5527:, where
5420:, where
5052:+ 1) β 3
4728: :
4723:of maps
4672: :
1663:and the
1661:homology
1599:3-sphere
1476:, then
1453:subgroup
1313:integers
1227:for all
1153:-sphere
1025:homotopy
862:-sphere
748:boundary
554:Geometry
527:topology
382:mapping.
379:integers
262:deformed
197:). The
11409:Spheres
11214:1702278
11178:1867354
11135:3204653
11066:3702672
11026:2045823
10991:0143217
10957:0046048
10928:0045386
10920:1969485
10891:2136212
10860:0860042
10818:0341485
10767:1326604
10687:1648096
10656:1320997
10617:1052407
10578:0148075
10570:1970128
10501:4046815
10461:0106454
10420:1454356
10385:2279281
10305:0231377
10297:1970586
10268:0554384
10260:1971238
10222:0046046
10186:0046045
10148:2188127
10107:0810962
10013:Sources
9436:Hu 1959
9266:β
8β
4β
2β
9239:β
4β
2β
3
9075:β
4β
2β
3
9057:3β
25β
11
9039:27β
7β
19
8650:24β
8β
2
8289:1056β
8
8234:24β
6β
2
8028:48β
4β
2
7947:24β
6β
2
7003:) = ZΓZ
6983:Example
6897:integer
6832:of the
6752:is the
6436:complex
6039:-groups
5631:
5619:
5615:
5598:
5524:
5501:
5462:(mod 8)
5272:(where
4949:β 3 + 1
4901:torsion
4895:has no
4802:spectra
3917:
3895:
3654:by the
1984:, with
1904:, with
1641:History
1323:in the
1181:abelian
852:equator
785:balloon
783:. If a
669:-sphere
544:-sphere
533:, with
503:< 20
303:0 <
254:sphere
243:can be
100:In the
58:improve
11297:
11269:
11212:
11202:
11176:
11166:
11133:
11115:
11072:
11064:
11024:
11014:
10989:
10979:
10955:
10947:
10926:
10918:
10889:
10879:
10858:
10848:
10816:
10808:
10765:
10755:
10685:
10654:
10644:
10615:
10605:
10576:
10568:
10499:
10489:
10459:
10418:
10408:
10383:
10375:
10321:
10314:224450
10311:
10303:
10295:
10266:
10258:
10220:
10212:
10184:
10176:
10146:
10105:
10087:
9313:4β
2β
3
9307:4β
2β
5
9295:4β
2β
3
9215:3β
5β
17
9163:3β
5β
29
9110:5β
7β
13
9088:8β
3β
23
9084:8β
2β
3
9004:3β
5β
17
8955:5β
7β
13
8849:16β
3β
5
8739:> 5
8708:264β
2
8702:264β
2
8699:264β
2
8696:264β
2
8693:264β
2
8690:264β
2
8684:264β
2
8656:8β
4β
2
8536:480β
2
8533:480β
2
8530:480β
2
8527:480β
2
8524:480β
2
8518:480β
2
8515:480β
2
8512:480β
2
8307:264β
2
8304:264β
2
8301:264β
6
8298:264β
2
8295:264β
2
8292:264β
2
8286:264β
2
8283:132β
2
8280:132β
2
8249:8β
4β
2
8132:240β
2
8120:504β
2
8083:240β
2
8080:240β
2
8077:240β
2
8074:240β
2
8071:120β
2
8068:120β
2
7851:ββ
504
7842:504β
2
7839:504β
2
7836:504β
2
7833:504β
4
7830:504β
2
7605:ββ
120
6722:where
6428:) = Z)
6373:> 2
6274:, and
5882:, and
5837:; the
5486:modulo
5404:)) β Ο
5379:, for
5296:< 2
4908:< 2
4721:degree
4382:, for
3872:|
3798:Since
3634:Since
2923:> 1
2008:, for
1909:> 0
1886:space
1825:, the
1765:, and
1400:> 0
1385:degree
1321:origin
1097:curves
1055:of a (
935:+ β― +
887:+ β― +
858:, the
686:circle
648:+ β― +
550:sphere
361:degree
245:mapped
189:sphere
179:circle
165:sphere
108:, the
47:, but
11344:(PDF)
11267:S2CID
11113:S2CID
11093:arXiv
11070:S2CID
11044:arXiv
10916:JSTOR
10720:(PDF)
10566:JSTOR
10518:arXiv
10355:arXiv
10293:JSTOR
10256:JSTOR
9334:Notes
9319:β
4β
2β
9301:β
2β
9β
9286:β
2β
3
9248:β
4β
2β
9233:β
4β
2
9213:β
4β
2β
9143:β
8β
2β
9134:β
2β
3
9128:β
4β
2
9104:β
4β
2β
9055:β
2β
3β
8949:β
8β
2β
8880:8β
9β
7
8558:24β
2
8469:16β
2
8277:12β
2
8246:24β
2
8243:24β
2
8237:24β
2
8231:24β
2
8223:12β
2
8220:12β
2
8166:12β
2
8065:60β
6
8062:30β
2
8025:16β
2
8022:16β
2
8019:16β
4
8013:24β
4
8010:12β
2
7915:below
7879:84β
2
7827:84β
2
7824:84β
2
7821:12β
2
7787:12β
2
7784:24β
2
7781:24β
2
7778:24β
2
7775:72β
2
7772:72β
2
7764:12β
2
7657:24β
2
7535:24β
3
7361:ββ
12
6995:) = Ο
6457:) = Z
6379:over
6377:braid
6223:find.
6055:1952b
6051:1952a
6029:prime
5945:with
5543:, if
5539:is a
5044:<
4773:) = Z
4760:) = Ξ©
4711:) = Z
4375:with
3870:with
3670:) β Ο
3652:split
3640:is a
3445:) β Ο
3220:is a
3162:>
3157:with
3133:>
3062:) = 0
2986:) = Ο
2959:>
2022:Table
2013:>
1989:>
1963:) = Z
1842:<
1823:) = 0
1703:used
1635:) = Z
1592:>
1570:) = Z
1523:) = 0
1507:) = 0
1488:) = 0
1471:<
1433:) = 0
1418:below
1416:(see
1414:) = Z
1363:) = Z
1348:) = Z
1325:plane
1258:) = Z
1233:plane
1122:group
1091:into
813:is a
789:slash
684:is a
492:<
390:>
343:When
307:<
11295:ISBN
11200:ISBN
11164:ISBN
11012:ISBN
10977:ISBN
10945:ISSN
10877:ISBN
10846:ISBN
10806:ISSN
10753:ISBN
10642:ISBN
10603:ISBN
10487:ISBN
10406:ISBN
10373:ISSN
10319:PMID
10210:ISSN
10174:ISSN
9310:4β
2
9260:4β
2
9254:2β
3
9242:8β
2
9207:2β
3
9155:4β
2
9149:2β
3
9137:2β
3
9098:2β
3
9046:2β
3
9043:2β
3
9031:4β
2
8996:2β
3
8982:2β
3
8943:2β
2
8937:8β
3
8933:3β
11
8925:8β
2
8876:2β
3
8665:8β
2
8662:8β
2
8659:8β
2
8653:8β
2
8647:8β
2
8644:8β
2
8641:8β
2
8613:ββ
2
8475:4β
2
8472:8β
2
8429:ββ
3
8183:6β
2
8174:4β
2
8163:6β
2
8114:6β
2
8111:6β
2
8007:6β
2
7971:6β
2
7968:6β
2
7959:6β
2
7950:6β
2
7891:240
7854:504
7848:504
7845:504
7790:6β
2
7727:ββ
2
7620:240
7617:240
7614:240
7611:240
7608:240
7602:120
7106:<
6997:9+10
6942:and
6820:The
6796:The
6588:spin
6469:ball
6414:The
6368:for
6117:has
6106:The
6068:The
5955:and
5929:and
5925:and
5751:The
5728:for
5451:(SO(
5400:(SO(
4925:and
4128:and
3831:and
3109:to
3064:for
3025:and
3018:For
2998:for
1950:) =
1721:and
1579:The
1183:and
1019:. A
712:disk
529:and
460:for
301:For
216:-th
159:The
143:The
11369:in
11259:doi
11255:104
11103:doi
11089:226
11054:doi
11040:186
11004:doi
10941:234
10908:doi
10796:hdl
10786:doi
10745:doi
10595:doi
10558:doi
10528:doi
10514:137
10477:doi
10473:262
10398:doi
10365:doi
10309:PMC
10285:doi
10248:doi
10244:110
10206:234
10170:234
10134:doi
10095:doi
10056:doi
10035:doi
9981:in
9440:107
9292:β
2
9276:72+
9223:64+
9211:128
9195:β
2β
9186:β
2
9171:56+
9161:β
3β
9118:48+
9108:β
3β
9081:β
2
9065:40+
9037:β
2β
9028:β
2
9022:β
2
9012:32+
9002:β
2β
8986:8β
3
8979:β
2
8973:β
2
8963:24+
8953:β
3β
8931:β
2β
8922:β
2
8916:β
2
8906:16+
8898:3β
5
8896:β
2β
8873:β
2
8867:β
2
8835:8β
3
8766:+ 1
8753:if
8672:19+
8629:18+
8586:17+
8543:16+
8500:15+
8457:14+
8414:13+
8371:12+
8259:19+
8202:18+
8145:17+
8108:30
8090:16+
8059:30
8056:30
8053:30
8035:15+
7999:30
7978:14+
7923:13+
7913:See
7903:12
7861:12+
7803:11+
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