25:
1253:
981:
989:
1358:
82:
6203:
1565:
129:
141:
5273:-component). This exact sequence is similar to the ones coming from the Hopf fibration; the difference is that it works for all even-dimensional spheres, albeit at the expense of ignoring 2-torsion. Combining the results for odd and even dimensional spheres shows that much of the odd torsion of unstable homotopy groups is determined by the odd torsion of the stable homotopy groups.
1428:
6054:. In principle this gives an effective algorithm for computing all homotopy groups of any finite simply connected simplicial complex, but in practice it is too cumbersome to use for computing anything other than the first few nontrivial homotopy groups as the simplicial complex becomes much more complicated every time one kills a homotopy group.
8722:
are the products of cyclic groups of the infinite or prime power orders shown in the table. (For largely historical reasons, stable homotopy groups are usually given as products of cyclic groups of prime power order, while tables of unstable homotopy groups often give them as products of the smallest
6091:
to the calculation of homology groups of its repeated loop spaces. The Serre spectral sequence relates the homology of a space to that of its loop space, so can sometimes be used to calculate the homology of loop spaces. The Serre spectral sequence tends to have many non-zero differentials, which are
1435:
Any continuous mapping from a circle to an ordinary sphere can be continuously deformed to a one-point mapping, and so its homotopy class is trivial. One way to visualize this is to imagine a rubber-band wrapped around a frictionless ball: the band can always be slid off the ball. The homotopy group
6217:
The motivic Adams spectral sequence converges to the motivic stable homotopy groups of spheres. By comparing the motivic one over the complex numbers with the classical one, Isaksen gives rigorous proof of computations up to the 59-stem. In particular, Isaksen computes the Coker J of the 56-stem is
1224:
has exactly the same homotopy groups as a solitary point (as does a
Euclidean space of any dimension), and the real plane with a point removed has the same groups as a circle, so groups alone are not enough to distinguish spaces. Although the loss of discrimination power is unfortunate, it can also
1501:
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,
1365:
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.
1233:
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.
4780:
with a nonstandard trivialization of the normal 2-plane bundle. Until the advent of more sophisticated algebraic methods in the early 1950s (Serre) the
Pontrjagin isomorphism was the main tool for computing the homotopy groups of spheres. In 1954 the Pontrjagin isomorphism was generalized by
6238:
page on positive stems. Wang and Xu develops a method using the 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
1675:
which can be used to calculate some of the groups. An important method for calculating the various groups is the concept of stable algebraic topology, which finds properties that are independent of the dimensions. Typically these only hold for larger dimensions. The first such result was
780:, pronounced "modulo", means to take the topological space on the left (the disk) and in it join together as one all the points on the right (the circle). The region is 2-dimensional, which is why topology calls the resulting topological space a 2-sphere. Generalized,
370:
under addition. For example, every point on a circle can be mapped continuously onto a point of another circle; as the first point is moved around the first circle, the second point may cycle several times around the second circle, depending on the particular
3618:
1025:, where "closed curves" are continuous maps from the circle into the complex plane, and where two closed curves produce the same integral result if they are homotopic in the topological space consisting of the plane minus the points of singularity.
1016:
is a continuous path between continuous maps; two maps connected by a homotopy are said to be homotopic. The idea common to all these concepts is to discard variations that do not affect outcomes of interest. An important practical example is the
6211:
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
3417:
5256:
426:
turned out to be a central question in algebraic topology that has contributed to development of many of its fundamental techniques and has served as a stimulating focus of research. One of the main discoveries is that the homotopy groups
6034:. This is usually done by constructing suitable fibrations and taking the associated long exact sequences of homotopy groups; spectral sequences are a systematic way of organizing the complicated information that this process generates.
5672:-homomorphism is the subgroup of "well understood" or "easy" elements of the stable homotopy groups. These well understood elements account for most elements of the stable homotopy groups of spheres in small dimensions. The quotient of
5980:
used the composition product and Toda brackets to label many of the elements of homotopy groups. There are also higher Toda brackets of several elements, defined when suitable lower Toda brackets vanish. This parallels the theory of
4515:
1275:
around one's finger: it can be wrapped once, twice, three times and so on. The wrapping can be in either of two directions, and wrappings in opposite directions will cancel out after a deformation. The homotopy group
1117:" 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.
4353:
1370:
and so the bag is allowed to pass through itself.) The twist can be in one of two directions and opposite twists can cancel out by deformation. The total number of twists after cancellation is an integer, called the
6307:
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,
4240:
3782:
6092:
hard to control, and too many ambiguities appear for higher homotopy groups. Consequently, it has been superseded by more powerful spectral sequences with fewer non-zero differentials, which give more information.
144:
This picture mimics part of the Hopf fibration, an interesting mapping from the three-dimensional sphere to the two-dimensional sphere. This mapping is the generator of the third homotopy group of the 2-sphere.
1983:, are all trivial. It therefore came as a great surprise historically that the corresponding homotopy groups are not trivial in general. This is the case that is of real importance: the higher homotopy groups
5810:
6797:
can be reduced to a question about stable homotopy groups of spheres. For example, knowledge of stable homotopy groups of degree up to 48 has been used to settle the
Kervaire invariant problem in dimension
4577:
6877:. The stable homotopy groups are highlighted in blue, the unstable ones in red. Each homotopy group is the product of the cyclic groups of the orders given in the table, using the following conventions:
4092:
3270:
843:, and sweep each point on it to one point above (the North Pole), producing the northern hemisphere, and to one point below (the South Pole), producing the southern hemisphere. For each positive integer
4036:
3977:
6171:-component of the stable homotopy groups. The initial terms of the Adams spectral sequence are themselves quite hard to compute: this is sometimes done using an auxiliary spectral sequence called the
1572:
is a nontrivial mapping of the 3-sphere to the 2-sphere, and generates the third homotopy group of the 2-sphere. Each colored circle maps to the corresponding point on the 2-sphere shown bottom right.
6030:
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
6705:
1304:
under addition: a homotopy class is identified with an integer by counting the number of times a mapping in the homotopy class wraps around the circle. This integer can also be thought of as the
5989:. Every element of the stable homotopy groups of spheres can be expressed using composition products and higher Toda brackets in terms of certain well known elements, called Hopf elements.
1096:
fixed), so that two maps are in the same class if they are homotopic. Just as one point is distinguished, so one class is distinguished: all maps (or curves) homotopic to the constant map
5665:
3504:
968:. For some spaces the choice matters, but for a sphere all points are equivalent so the choice is a matter of convenience. For spheres constructed as a repeated suspension, the point
972:, which is on the equator of all the levels of suspension, works well; for the disk with collapsed rim, the point resulting from the collapse of the rim is another obvious choice.
6229:
The 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
1113:
with the introduction of addition, defined via an "equator pinch". This pinch maps the equator of a pointed sphere (here a circle) to the distinguished point, producing a "
9497:
6524:
6500:
2938:
5976:
of these elements. The Toda bracket is not quite an element of a stable homotopy group, because it is only defined up to addition of products of certain other elements.
2015:
The following table gives an idea of the complexity of the higher homotopy groups even for spheres of dimension 8 or less. In this table, the entries are either a) the
3299:
5079:
1638:
introduced the notion of homotopy and used the notion of a homotopy group, without using the language of group theory. A more rigorous approach was adopted by
1455:
This result generalizes to higher dimensions. All mappings from a lower-dimensional sphere into a sphere of higher dimension are similarly trivial: if
4416:
2960:
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).
1448:
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
1747:
11070:
Gheorghe, Bogdan; Wang, Guozhen; Xu, Zhouli (2021), "The special fiber of the motivic deformation of the stable homotopy category is algebraic",
4940:. In the case of odd torsion there are more precise results; in this case there is a big difference between odd and even dimensional spheres. If
10153:
4248:
1751:
4135:
3677:
6778:
above, and therefore the stable homotopy groups of spheres, are used in the classification of possible smooth structures on a topological or
1377:
of the mapping. As in the case mappings from the circle to the circle, this degree identifies the homotopy group with the group of integers,
10713:
10113:
6099:
can be used to compute many homotopy groups of spheres; it is based on some fibrations used by Toda in his calculations of homotopy groups.
2007:, are surprisingly complex and difficult to compute, and the effort to compute them has generated a significant amount of new mathematics.
1271:
The simplest case concerns the ways that a circle (1-sphere) can be wrapped around another circle. This can be visualized by wrapping a
10064:
5749:
257:
of mappings are summarized. An "addition" operation defined on these equivalence classes makes the set of equivalence classes into an
4129:
are zero. Thus the long exact sequences again break into families of split short exact sequences, implying two families of relations.
4526:
1667:
and Heinz Hopf, on the grounds that the groups were commutative so could not be the right generalizations of the fundamental group.)
4044:
3219:
11359:
6564:
6286:
is a map between motivic spheres. The
GheorgheβWangβXu theorem identifies the motivic Adams spectral sequence for the cofiber of
3988:
3929:
5724:, and these are also considered to be "well understood".) Tables of homotopy groups of spheres sometimes omit the "easy" part
4987:. This is in some sense the best possible result, as these groups are known to have elements of this order for some values of
11287:
11192:
11156:
10969:
10869:
10838:
10634:
10595:
10479:
6630:
2957:
The groups below the jagged black line are constant along the diagonals (as indicated by the red, green and blue coloring).
6179:
1154:, and otherwise follows the same procedure. The null homotopic class acts as the identity of the group addition, and for
776:
is punctured and spread flat it produces a disk; this construction repairs the puncture, like pulling a drawstring. The
3072:
The fact that the groups below the jagged line in the table above are constant along the diagonals is explained by the
11004:
10745:
10419:
10398:
1684:, published in 1937. Stable algebraic topology flourished between 1945 and 1966 with many important results. In 1953
68:
46:
10225:; Neisendorfer, Joseph A. (November 1979), "The double suspension and exponents of the homotopy groups of spheres",
4793:. In more recent work the argument is usually reversed, with cobordism groups computed in terms of homotopy groups.
39:
10497:
Isaksen, Daniel C.; Wang, Guozhen; Xu, Zhouli (2023), "Stable homotopy groups of spheres: from dimension 0 to 90",
6006:
1546:
can be lifted to a map into the real line and the nullhomotopy descends to the downstairs space (via composition).
541:
in three-dimensional spaceβthe surface, not the solid ballβis just one example of what a sphere means in topology.
11346:
3613:{\displaystyle 0\rightarrow \pi _{i}(S^{3})\rightarrow \pi _{i}(S^{2})\rightarrow \pi _{i-1}(S^{1})\rightarrow 0.}
3290:
6617:-spheres, up to orientation-preserving diffeomorphism; the non-trivial elements of this group are represented by
6557:
469:
6050:
to compute the first non-trivial homotopy group and then killing (eliminating) it with a fibration involving an
136:
is a nontrivial mapping of the 3-sphere to the 2-sphere, and generates the third homotopy group of the 2-sphere.
6420:
5638:
3644:
3073:
1681:
1480:
510:
The study of homotopy groups of spheres builds on a great deal of background material, here briefly reviewed.
10861:
10693:
10429:
6191:
5065:
The results above about odd torsion only hold for odd-dimensional spheres: for even-dimensional spheres, the
806:, and the construction joins its ends to make a circle. An equivalent description is that the boundary of an
11021:
Wang, Guozhen; Xu, Zhouli (2017), "The triviality of the 61-stem in the stable homotopy groups of spheres",
352:
that measures how many times the sphere is wrapped around itself. This degree identifies the homotopy group
10683:
6476:
5822:, where multiplication is given by composition of representing maps, and any element of non-zero degree is
11184:
10732:, Graduate Studies in Mathematics, vol. 5, Providence, Rhode Island: American Mathematical Society,
10688:
10424:
6449:
6195:
6051:
9418:
1361:
Illustration of how a 2-sphere can be wrapped twice around another 2-sphere. Edges should be identified.
85:
Illustration of how a 2-sphere can be wrapped twice around another 2-sphere. Edges should be identified.
11148:
10961:
6810:
6786:
4790:
1005:
6061:
was used by Serre to prove some of the results mentioned previously. He used the fact that taking the
9967:
6779:
1611:
1531:
313:
is homotopic (i.e., continuously deformable) to a constant mapping, i.e., a mapping that maps all of
10078:
11392:
10655:
6010:
5418:
4960:
2041:
1173:
473:
33:
11367:
6507:
6483:
2921:
10629:, Contemporary Mathematics, vol. 181, Providence, R.I.: Amer. Math. Soc., pp. 299β316,
9459:
6724:
6103:
6058:
4640:
3412:{\displaystyle \cdots \to \pi _{i}(F)\to \pi _{i}(E)\to \pi _{i}(B)\to \pi _{i-1}(F)\to \cdots .}
6563:
The fact that the third stable homotopy group of spheres is cyclic of order 24, first proved by
5251:{\displaystyle \pi _{2m+k}(S^{2m})(p)=\pi _{2m+k-1}(S^{2m-1})(p)\oplus \pi _{2m+k}(S^{4m-1})(p)}
2953:) appear to be chaotic, but in fact there are many patterns, some obvious and some very subtle.
10073:
6826:
6323:
3980:
3182:, and have been computed in numerous cases, although the general pattern is still elusive. For
1649:
1644:
50:
10727:
10622:
10320:
9428:
6087:)-fold loop space by the Hurewicz theorem. This reduces the calculation of homotopy groups of
502:. Several important patterns have been established, yet much remains unknown and unexplained.
11238:
11233:
11023:
10537:
10227:
6909:
6172:
6096:
5816:
4851:. The 2-components are hardest to calculate, and in several ways behave differently from the
4840:
3630:
1671:
is also credited with the introduction of homotopy groups in his 1935 paper and also for the
1289:
106:
11271:
10828:
6182:
is a more powerful version of the Adams spectral sequence replacing ordinary cohomology mod
5690:-homomorphism is considered to be the "hard" part of the stable homotopy groups of spheres (
11202:
11166:
11123:
11054:
11014:
10979:
10945:
10916:
10879:
10848:
10806:
10755:
10675:
10644:
10605:
10566:
10489:
10449:
10408:
10373:
10293:
10256:
10210:
10174:
10136:
10095:
6927:) of the cyclic groups of those orders. Powers indicate repeated products. (Note that when
6924:
6572:
6568:
5741:
3495:
1309:
11327:
6245:
has a unique smooth structure, and it is the last odd dimensional one β the only ones are
4785:
to an isomorphism expressing other groups of cobordism classes (e.g. of all manifolds) as
8:
10108:
6472:
5470:
1685:
1449:
1313:
1221:
1110:
777:
736:
2963:
The second and third rows of the table are the same starting in the third column (i.e.,
11255:
11101:
11081:
11058:
11032:
10904:
10705:
10554:
10506:
10343:
10281:
10244:
10222:
10218:
10104:
10014:
6790:
6464:
6457:
5827:
5474:
4843:
with a finite abelian group. In particular the homotopy groups are determined by their
1847:
1693:
1197:
1114:
511:
94:
10302:
10062:(1984), "Relations amongst Toda brackets and the Kervaire invariant in dimension 62",
9628:
833:
This construction, though simple, is of great theoretical importance. Take the circle
11283:
11259:
11188:
11152:
11142:
11105:
11062:
11000:
10965:
10933:
10865:
10834:
10794:
10741:
10630:
10591:
10475:
10394:
10361:
10307:
10198:
10186:
10162:
10148:
10049:
10028:
6920:
6818:
6814:
6813:
says that the stable homotopy groups of the spheres can be expressed in terms of the
6527:. The geometry near a critical point of such a map can be described by an element of
6187:
6031:
5831:
4889:
4802:
1689:
1653:
1324:
1320:
1089:
1029:
997:
499:
495:
254:
233:
114:
10887:
Serre, Jean-Pierre (1951), "Homologie singulière des espaces fibrés. Applications",
10444:, Pure and Applied Mathematics, vol. 8, New York & London: Academic Press,
5457:, and it is reflected in the stable homotopy groups of spheres via the image of the
4586:
one problem, because such a fibration would imply that the failed relation is true.
1639:
1440:, with only one element, the identity element, and so it can be identified with the
249:. This summary does not distinguish between two mappings if one can be continuously
102:
11397:
11247:
11091:
11072:
11042:
10992:
10896:
10784:
10774:
10733:
10583:
10546:
10516:
10465:
10386:
10385:, Undergraduate Texts in Mathematics, Springer-Verlag, New York, pp. 134β136,
10353:
10297:
10273:
10236:
10122:
10083:
10044:
10023:
8758:
6610:
6154:
6047:
5529:
5454:
4805:
showed that homotopy groups of spheres are all finite except for those of the form
3077:
1872:
1864:
1677:
1672:
1664:
1107:
1022:
519:
110:
10381:
Fine, Benjamin; Rosenberger, Gerhard (1997), "8.1 Winding Number and Proof Five",
10127:
1524:
which is contractible (it has the homotopy type of a point). In addition, because
1252:
1200:
between the associated homotopy groups. In particular, if the map is a continuous
980:
11198:
11162:
11119:
11096:
11050:
11010:
10975:
10941:
10912:
10875:
10844:
10824:
10802:
10751:
10671:
10640:
10601:
10579:
10562:
10528:
10485:
10445:
10404:
10369:
10289:
10252:
10206:
10170:
10132:
10091:
6822:
6391:
6001:
is any finite simplicial complex with finite fundamental group, in particular if
5066:
4709:
3640:
1668:
1521:
1373:
1018:
596:
349:
191:
11046:
10987:
Walschap, Gerard (2004), "Chapter 3: Homotopy groups and bundles over spheres",
6838:
Tables of homotopy groups of spheres are most conveniently organized by showing
6194:. The initial term is again quite hard to calculate; to do this one can use the
4510:{\displaystyle \pi _{30}(S^{16})\neq \pi _{30}(S^{31})\oplus \pi _{29}(S^{15}).}
10813:
10765:(1973), "The nilpotency of elements of the stable homotopy groups of spheres",
10520:
10087:
8762:
6750:
6576:
6424:
6404:
6080:)-fold repeated loop space, which is equal to the first homology group of the (
6038:"The method of killing homotopy groups", due to Cartan and Serre (
5982:
5376:
5345:
4786:
4727:
4599:
4583:
3910:
3206:
2915:
1868:
1635:
1607:
1569:
1305:
1045:
1009:
523:
385:
133:
122:
118:
11217:
10996:
10390:
10151:(1952a), "Espaces fibrΓ©s et groupes d'homotopie. I. Constructions gΓ©nΓ©rales",
1871:, which are generally easier to calculate; in particular, it shows that for a
1660:
1357:
81:
11386:
11323:
11308:
11138:
10937:
10820:
American
Mathematical Society Translations, Ser. 2, Vol. 11, pp. 1β114 (1959)
10798:
10651:
10618:
10365:
10202:
10166:
10059:
6936:
6618:
6331:
5626:
The stable homotopy groups of spheres are the direct sum of the image of the
4653:
2016:
1815:
1743:
1437:
1205:
1190:
1169:
964:
958:
Some theory requires selecting a fixed point on the sphere, calling the pair
326:
258:
11178:
11118:, The Univalent Foundations Program and Institute for Advanced Study, 2013,
10779:
6301:, which allows one to deduce motivic Adams differentials for the cofiber of
5547:
This last case accounts for the elements of unusually large finite order in
1850:, and any mapping to such a space can be deformed into a one-point mapping.
1688:
showed that there is a metastable range for the homotopy groups of spheres.
11304:
11267:
11174:
10991:, Graduate Texts in Mathematics, vol. 224, Springer-Verlag, New York,
10953:
10762:
10415:
10311:
10182:
10144:
6893:
6017:
5977:
5956:
3841:
The Hopf fibration may be constructed as follows: pairs of complex numbers
3210:
2024:
1739:
1731:
803:
10668:
Proceedings of the
International Congress of Mathematicians (Berlin, 1998)
4782:
1727:
988:
400:
in a non-trivial fashion, and so is not equivalent to a one-point mapping.
10701:
10532:
10437:
10009:
6595:
6334:
6065:
of a well behaved space shifts all the homotopy groups down by 1, so the
5819:
4348:{\displaystyle \pi _{i}(S^{8})=\pi _{i}(S^{15})\oplus \pi _{i-1}(S^{7}).}
1735:
1272:
90:
10470:
10357:
4235:{\displaystyle \pi _{i}(S^{4})=\pi _{i}(S^{7})\oplus \pi _{i-1}(S^{3}),}
3777:{\displaystyle \pi _{i}(S^{2})=\pi _{i}(S^{3})\oplus \pi _{i-1}(S^{1}).}
599:
found exactly one unit away from the origin. It is called the 2-sphere,
11251:
11234:"Γber die Abbildungen der dreidimensionalen SphΓ€re auf die KugelflΓ€che"
11229:
11113:
10908:
10789:
10670:, Documenta Mathematica, Extra Volume, vol. II, pp. 465β472,
10587:
10558:
10285:
10248:
6955:
6579:
6468:
6427:
6062:
5986:
4098:
1837:
1591:
1293:
1213:
811:
10737:
6005:
is a sphere of dimension at least 2, then its homotopy groups are all
4925:. The case of 2-dimensional spheres is slightly different: the first
4708:, computed by the algebraic sum of their points, corresponding to the
10348:
6280:
method is so far the most efficient method at the prime 2. The class
6116:
5823:
4595:
1755:
1367:
1201:
700:
606:
468:
up to 90. The stable homotopy groups form the coefficient ring of an
10900:
10550:
10277:
10240:
10189:(1952b), "Espaces fibrΓ©s et groupes d'homotopie. II. Applications",
6723:
is the cyclic subgroup represented by homotopy spheres that bound a
2893:
The first row of this table is straightforward. The homotopy groups
384:. The first such surprise was the discovery of a mapping called the
11279:
11086:
11037:
10511:
9358:
4102:
1696:
to show that most of these groups are finite, the exceptions being
1587:
1441:
1013:
1001:
654:
542:
515:
250:
153:
9681:
545:
defines a sphere rigidly, as a shape. Here are some alternatives.
10336:
Proceedings of the Japan
Academy, Series A, Mathematical Sciences
9577:
9565:
6885:
6585:
Stable homotopy groups of spheres are used to describe the group
6202:
6024:
5805:{\displaystyle \pi _{\ast }^{S}=\bigoplus _{k\geq 0}\pi _{k}^{S}}
4991:. Furthermore, the stable range can be extended in this case: if
1564:
1301:
840:
773:
708:
is the region contained by a circle, described by the inequality
367:
128:
9936:
5276:
For stable homotopy groups there are more precise results about
487:) are more erratic; nevertheless, they have been tabulated for
10924:
Serre, Jean-Pierre (1952), "Sur la suspension de
Freudenthal",
10576:
Stable homotopy groups of spheres. A computer-assisted approach
10264:
Cohen, Joel M. (1968), "The decomposition of stable homotopy",
4702:
is the cobordism group of framed 0-dimensional submanifolds of
4572:{\displaystyle S^{15}\hookrightarrow S^{31}\rightarrow S^{16},}
3979:, where the bundle projection is a double covering), there are
3064:
These patterns follow from many different theoretical results.
1863:
has also been noted already, and is an easy consequence of the
1431:
A homotopy from a circle around a sphere down to a single point
674:
538:
190:-sphere may be defined geometrically as the set of points in a
177:
167:
140:
11222:
9888:
6009:. To compute these groups, they are often factored into their
2940:, which has the same higher homotopy groups, is contractible.
6365:
4764:
which corresponds to the framed 1-dimensional submanifold of
3476:
can be deformed to a point inside the higher-dimensional one
1208:), so that the two spaces have the same topology, then their
1085:
498:, a technique first applied to homotopy groups of spheres by
9828:
8757:
and 0 otherwise. The mod 8 behavior of the table comes from
6218:
0, and therefore by the work of
Kervaire-Milnor, the sphere
4087:{\displaystyle S^{7}\hookrightarrow S^{15}\rightarrow S^{8}}
3265:{\displaystyle S^{1}\hookrightarrow S^{3}\rightarrow S^{2}.}
699:
This construction moves from geometry to pure topology. The
16:
How spheres of various dimensions can wrap around each other
9900:
9336:
9334:
9332:
8301:
7011:
6330:
identify these homotopy groups as certain quotients of the
4404:) and beyond. Although generalizations of the relations to
4031:{\displaystyle S^{3}\hookrightarrow S^{7}\rightarrow S^{4}}
3972:{\displaystyle S^{0}\hookrightarrow S^{1}\rightarrow S^{1}}
2075:). Extended tables of homotopy groups of spheres are given
10818:
Smooth manifolds and their applications in homotopy theory
9924:
9876:
1427:
1196:
A continuous map between two topological spaces induces a
1012:
is a function between spaces that preserves continuity. A
11115:
Homotopy type theoryβunivalent foundations of mathematics
9519:
9517:
9406:
6549:
sphere around the critical point maps into a topological
6343:. Under this correspondence, every nontrivial element in
1663:
in 1932. (His first paper was withdrawn on the advice of
605:, for reasons given below. The same idea applies for any
11131:
10578:, Lecture Notes in Mathematics, vol. 1423, Berlin:
9816:
9329:
7900:
5584:
have a cyclic subgroup of order 504, the denominator of
10830:
Complex cobordism and stable homotopy groups of spheres
6700:{\displaystyle \Theta _{n}/bP_{n+1}\to \pi _{n}^{S}/J,}
6463:
The stable homotopy groups of spheres are important in
3080:, which implies that the suspension homomorphism from
1052:
thus begins with continuous maps from a pointed circle
374:
The most interesting and surprising results occur when
11183:, Chicago lectures in mathematics (revised ed.),
10217:
9687:
9659:
9657:
9655:
9616:
9553:
9514:
9502:
9462:
4602:
established an isomorphism between the homotopy group
2994:). This isomorphism is induced by the Hopf fibration
1366:(There is no requirement for the continuous map to be
11365:
11344:
9912:
9717:
9583:
9571:
9529:
9370:
9346:
6633:
6510:
6486:
5752:
5641:
5082:
4529:
4419:
4251:
4138:
4047:
3991:
3932:
3680:
3507:
3302:
3222:
2924:
11112:
10057:
9942:
9804:
9693:
9453:
9434:
9382:
6919:
Where entry is a product, the homotopy group is the
6467:, which studies the structure of singular points of
6167:, and converges to something closely related to the
5487:
a cyclic group of order equal to the denominator of
4858:
In the same paper, Serre found the first place that
1900:, is isomorphic to the first nonzero homology group
1168:) β the homotopy groups of spheres β the groups are
1000:
is its continuity structure, formalized in terms of
992:
Addition of two circle maps keeping base point fixed
984:
Homotopy of two circle maps keeping base point fixed
9948:
9852:
9840:
9792:
9770:
9768:
9753:
9652:
9640:
9606:
9604:
9589:
9394:
5694:). (Adams also introduced certain order 2 elements
810:-dimensional disk is glued to a point, producing a
125:are surprisingly complex and difficult to compute.
10926:Comptes Rendus de l'AcadΓ©mie des Sciences, SΓ©rie I
10191:Comptes Rendus de l'AcadΓ©mie des Sciences, SΓ©rie I
10154:Comptes Rendus de l'AcadΓ©mie des Sciences, SΓ©rie I
10102:
9894:
9864:
9491:
8703:
6825:, leading to an identification of K-theory of the
6699:
6518:
6494:
6327:
5804:
5659:
5250:
4594:Homotopy groups of spheres are closely related to
4571:
4509:
4410:are often true, they sometimes fail; for example,
4347:
4234:
4086:
4030:
3971:
3776:
3612:
3422:For this specific bundle, each group homomorphism
3411:
3264:
2943:Beyond the first row, the higher homotopy groups (
2932:
1818:. The reason is that a continuous mapping from an
117:, forgetting about their precise geometry. Unlike
10958:Composition methods in homotopy groups of spheres
9780:
9741:
9705:
9669:
1106:are called null homotopic. The classes become an
105:can wrap around each other. They are examples of
11384:
9765:
9729:
9601:
9541:
6908:Where the entry is β, the homotopy group is the
5634:-invariant, a homomorphism from these groups to
5303:-primary component of the stable homotopy group
1590:to the ordinary 2-sphere, and was discovered by
201:located at a unit distance from the origin. The
10960:, Annals of Mathematics Studies, vol. 49,
10616:
10610:
10380:
9906:
6292:as the algebraic Novikov spectral sequence for
667:)-dimensional space. For example, the 1-sphere
278:falls into three regimes, depending on whether
11220:(1932), "HΓΆherdimensionale Homotopiegruppen",
11069:
10527:
9930:
9882:
6186:with a generalized cohomology theory, such as
5815:of the stable homotopy groups of spheres is a
1780:up to 90, and, as of 2023, unknown for larger
11310:This week's finds in mathematical physics 102
10706:"Differential topology forty-six years later"
10496:
10109:"Configurations, braids, and homotopy groups"
9987:
9340:
6859:The following table shows many of the groups
6806:for which the question was open at the time.)
6309:
5073:in terms of that of odd-dimensional spheres,
3926:Similarly (in addition to the Hopf fibration
3067:
2076:
1659:Higher homotopy groups were first defined by
1479:. This can be shown as a consequence of the
1323:) of the homotopy group with the integers is
514:provides the larger context, itself built on
404:The question of computing the homotopy group
121:, which are also topological invariants, the
10767:Journal of the Mathematical Society of Japan
10714:Notices of the American Mathematical Society
10458:Memoirs of the American Mathematical Society
10181:
10143:
10114:Journal of the American Mathematical Society
6621:). More precisely, there is an injective map
6043:
6039:
5043:, and an epimorphism if equality holds. The
4652:which are "framed", i.e. have a trivialized
3470:to zero, since the lower-dimensional sphere
1180:all maps are null homotopic, then the group
1092:based on homotopy (keeping the "base point"
11379:in MacTutor History of Mathematics archive.
10456:Isaksen, Daniel C. (2019), "Stable Stems",
8767:
6833:
6764:, in which case the image has index 1 or 2.
5630:-homomorphism, and the kernel of the Adams
5484:is congruent to 2, 4, 5, or 6 modulo 8; and
3494:. Thus the long exact sequence breaks into
1920:-sphere, this immediately implies that for
1228:
1189:consists of one element, and is called the
595:This is the set of points in 3-dimensional
222:summarizes the different ways in which the
11366:O'Connor, J. J.; Robertson, E. F. (2001),
11345:O'Connor, J. J.; Robertson, E. F. (1996),
10989:Metric structures in differential geometry
10065:Journal of the London Mathematical Society
6542:, by considering the way in which a small
6316:The computation of the homotopy groups of
4862:-torsion occurs in the homotopy groups of
4670:is homotopic to a differentiable map with
1867:: this theorem links homotopy groups with
1840:. Consequently, its image is contained in
1726:. Others who worked in this area included
1076:, where maps from one pair to another map
113:terms, the structure of spheres viewed as
11095:
11085:
11036:
10788:
10778:
10535:(1963), "Groups of homotopy spheres: I",
10510:
10469:
10347:
10301:
10126:
10077:
10048:
10027:
6745:th stable homotopy group of spheres, and
6512:
6488:
5660:{\displaystyle \mathbb {Q} /\mathbb {Z} }
5653:
5643:
4796:
2926:
2044:of such groups (written, for example, as
1836:can always be deformed so that it is not
69:Learn how and when to remove this message
10986:
10681:
10650:
10058:Barratt, Michael G.; Jones, John D. S.;
10034:
9810:
9559:
8734:-component that is accounted for by the
6881:The entry "β
" denotes the trivial group.
6201:
5992:
4839:), when the group is the product of the
1594:, who constructed a nontrivial map from
1563:
1426:
1356:
1251:
987:
979:
282:is less than, equal to, or greater than
139:
127:
80:
32:This article includes a list of general
11360:MacTutor History of Mathematics archive
11137:
10855:
10823:
10761:
10725:
10573:
10455:
10318:
9991:
9954:
9918:
9858:
9846:
9834:
9822:
9798:
9759:
9723:
9699:
9663:
9646:
9634:
9622:
9535:
9523:
9508:
9440:
9412:
9400:
9388:
9376:
9364:
9352:
6423:, which states that every non-constant
6190:or, more usually, a piece of it called
5877:is nonzero and 12 times a generator of
4995:is odd then the double suspension from
4688:-dimensional submanifold. For example,
4105:instead of complex numbers. Here, too,
3482:. This corresponds to the vanishing of
11385:
11180:A Concise Course in Algebraic Topology
11020:
10700:
10627:The Δech centennial (Boston, MA, 1993)
9870:
9637:, Stable homotopy groups, pp. 385β393.
6789:, about the existence of manifolds of
5332:, in which case it is cyclic of order
1384:These two results generalize: for all
1084:. These maps (or equivalently, closed
325:. Therefore the homotopy group is the
11272:"Stable Algebraic Topology 1945β1966"
11132:General algebraic topology references
10923:
10886:
10729:Algebraic curves and Riemann surfaces
10263:
10008:
9786:
9747:
9711:
9675:
5691:
5668:. Roughly speaking, the image of the
5339:
4903:, and has a unique subgroup of order
4866:dimensional spheres, by showing that
4358:The three fibrations have base space
3806:at least 3, the first row shows that
3175:: they are finite abelian groups for
464:and have been computed for values of
11303:
11266:
11228:
11216:
11210:
11173:
10952:
10623:"On the computation of stable stems"
10499:Publications mathΓ©matiques de l'IHΓS
10414:
9983:
9971:
9774:
9735:
9688:Cohen, Moore & Neisendorfer 1979
9610:
9595:
9547:
6073:is the first homotopy group of its (
5453:. This period 8 pattern is known as
3275:The general theory of fiber bundles
476:. The unstable homotopy groups (for
160:-sphere for brevity, and denoted as
18:
11322:
10656:"Toward a global understanding of Ο
9986:. The 2-components can be found in
9968:table of homotopy groups of spheres
6452:that every continuous map from the
6374:that is not Brunnian over the disk
5047:-torsion of the intermediate group
4589:
3195:unstable homotopy groups of spheres
1879:, the first nonzero homotopy group
1406:
13:
10436:
10383:The Fundamental Theorem of Algebra
9943:Barratt, Jones & Mahowald 1984
9424:
6802:. (This was the smallest value of
6635:
4770:defined by the standard embedding
4582:the first non-trivial case of the
4391:) as mentioned above, but not for
3923:sends any such pair to its ratio.
3881:form a 3-sphere, and their ratios
3838:is at least 3, as observed above.
3200:
1576:The first nontrivial example with
1225:make certain computations easier.
38:it lacks sufficient corresponding
14:
11409:
11329:Stable homotopy groups of spheres
11297:
10420:"Spheres, homotopy groups of the"
10107:; Wong, Yan Loi; Wu, Jie (2006),
10012:(1966), "On the groups J(X) IV",
9990:, and the 3- and 5-components in
8738:-homomorphism is cyclic of order
6896:of that order (generally written
6364:may be represented by a Brunnian
6007:finitely generated abelian groups
5735:
5569:. For example, the stable groups
3159:stable homotopy groups of spheres
1787:
975:
462:stable homotopy groups of spheres
6756:. This is an isomorphism unless
6407:(corresponding to an integer of
6046:) involves repeatedly using the
6023:, and calculating each of these
5069:gives the torsion at odd primes
2907:of the 1-sphere are trivial for
1846:with a point removed; this is a
996:The distinguishing feature of a
101:describe how spheres of various
23:
10321:"Remarks on zeta functions and
10035:Adams, J (1968), "Correction",
9977:
9960:
9446:
8704:Table of stable homotopy groups
6397:
6180:AdamsβNovikov spectral sequence
4520:Thus there can be no fibration
494:. Most modern computations use
470:extraordinary cohomology theory
9966:These tables are based on the
9486:
9473:
8723:number of cyclic groups.) For
6668:
6475:. Such singularities arise as
6421:fundamental theorem of algebra
6224:has a unique smooth structure.
5245:
5239:
5236:
5214:
5189:
5183:
5180:
5158:
5127:
5121:
5118:
5102:
4598:classes of manifolds. In 1938
4553:
4540:
4501:
4488:
4472:
4459:
4443:
4430:
4339:
4326:
4304:
4291:
4275:
4262:
4226:
4213:
4191:
4178:
4162:
4149:
4071:
4058:
4015:
4002:
3956:
3943:
3768:
3755:
3733:
3720:
3704:
3691:
3604:
3601:
3588:
3569:
3566:
3553:
3540:
3537:
3524:
3511:
3400:
3397:
3391:
3372:
3369:
3363:
3350:
3347:
3341:
3328:
3325:
3319:
3306:
3246:
3233:
1648:where the related concepts of
1481:cellular approximation theorem
1:
11369:Marie Ennemond Camille Jordan
10862:American Mathematical Society
10858:The wild world of 4-manifolds
10833:(2nd ed.), AMS Chelsea,
10609:Also see the corrections in (
10128:10.1090/S0894-0347-05-00507-2
9988:Isaksen, Wang & Xu (2023)
9584:O'Connor & Robertson 1996
9572:O'Connor & Robertson 2001
9317:
8765:, whose image is underlined.
6598:classes of oriented homotopy
6310:Isaksen, Wang & Xu (2023)
6069:th homotopy group of a space
5465:a cyclic group of order 2 if
4378:. A fibration does exist for
2019:0, the infinite cyclic group
1758:. The stable homotopy groups
1549:
1486:
1412:
1342:
1237:
1028:The first homotopy group, or
505:
11097:10.4310/ACTA.2021.v226.n2.a2
10574:Kochman, Stanley O. (1990),
10050:10.1016/0040-9383(68)90010-4
10029:10.1016/0040-9383(66)90004-8
9883:Gheorghe, Wang & Xu 2021
9492:{\textstyle \pi _{1}(S^{1})}
6892:, the homotopy group is the
6829:with stable homotopy groups.
6519:{\displaystyle \mathbb {R} }
6495:{\displaystyle \mathbb {R} }
6460:to itself has a fixed point.
6380:. For example, the Hopf map
4855:-components for odd primes.
3193:, the groups are called the
2933:{\displaystyle \mathbb {R} }
1665:Pavel Sergeyevich Alexandrov
1530:is simply connected, by the
1088:) are grouped together into
745:, described by the equality
7:
11185:University of Chicago Press
11047:10.4007/annals.2017.186.2.3
10856:Scorpan, Alexandru (2005),
10689:Encyclopedia of Mathematics
10611:Kochman & Mahowald 1995
10425:Encyclopedia of Mathematics
9907:Fine & Rosenberger 1997
9341:Isaksen, Wang & Xu 2023
8708:The stable homotopy groups
6450:Brouwer fixed point theorem
6312:computes up to the 90-stem.
6196:chromatic spectral sequence
5895:is zero because the group
5834:implies Nishida's theorem.
5280:-torsion. For example, if
4847:-components for all primes
4097:constructed using pairs of
3981:generalized Hopf fibrations
3913:, a 2-sphere. The Hopf map
3911:complex plane plus infinity
3446:, induced by the inclusion
1586:concerns mappings from the
529:
388:, which wraps the 3-sphere
264:The problem of determining
166:β generalizes the familiar
10:
11414:
11149:Cambridge University Press
10962:Princeton University Press
10521:10.1007/s10240-023-00139-1
10001:
9931:Kervaire & Milnor 1963
6787:Kervaire invariant problem
6324:combinatorial group theory
5343:
4739:represents a generator of
3068:Stable and unstable groups
1642:in his 1895 set of papers
1629:
660:as a geometric object in (
99:homotopy groups of spheres
10997:10.1007/978-0-387-21826-7
10391:10.1007/978-1-4612-1928-6
9454:Homotopy type theory 2013
6780:piecewise linear manifold
6419:can be used to prove the
6192:BrownβPeterson cohomology
5859:is nonzero and generates
5350:An important subgroup of
2077:at the end of the article
1634:In the late 19th century
1520:has the real line as its
822:: written in topology as
685:: written in topology as
253:to the other; thus, only
11274:, in I. M. James (ed.),
10682:Mahowald, Mark (2001) ,
10088:10.1112/jlms/s2-30.3.533
9322:
6834:Table of homotopy groups
5461:-homomorphism which is:
5419:special orthogonal group
5062:can be strictly larger.
4726:. The projection of the
4639:of cobordism classes of
3834:are isomorphic whenever
3038:do not vanish. However,
2914:, because the universal
2010:
1229:Low-dimensional examples
1212:-th homotopy groups are
1138:begins with the pointed
474:stable cohomotopy theory
394:around the usual sphere
10684:"EHP spectral sequence"
10319:Deitmar, Anton (2006),
6725:parallelizable manifold
6609:, this is the group of
6274:The motivic cofiber of
6178:At the odd primes, the
6104:Adams spectral sequence
6059:Serre spectral sequence
6052:EilenbergβMacLane space
5432:, the homotopy groups
4955:, then elements of the
3645:suspension homomorphism
1792:As noted already, when
683:Disk with collapsed rim
460:. These are called the
53:more precise citations.
10726:Miranda, Rick (1995),
9493:
7005:, which is denoted by
6884:Where the entry is an
6827:field with one element
6811:BarrattβPriddy theorem
6701:
6520:
6496:
6322:has been reduced to a
6207:
5806:
5722:≡ 1 or 2 (mod 8)
5661:
5421:. In the stable range
5375:, is the image of the
5252:
4797:Finiteness and torsion
4573:
4511:
4349:
4236:
4088:
4032:
3973:
3778:
3671:, giving isomorphisms
3639:, these sequences are
3614:
3413:
3289:shows that there is a
3266:
3117:is an isomorphism for
2934:
1656:were also introduced.
1573:
1432:
1362:
1319:The identification (a
1268:
993:
985:
236:continuously into the
145:
137:
107:topological invariants
86:
11348:A history of Topology
11239:Mathematische Annalen
11024:Annals of Mathematics
10889:Annals of Mathematics
10780:10.2969/jmsj/02540707
10617:Kochman, Stanley O.;
10538:Annals of Mathematics
10266:Annals of Mathematics
10228:Annals of Mathematics
9494:
6910:infinite cyclic group
6702:
6521:
6497:
6328:Berrick et al. (2006)
6205:
6173:May spectral sequence
6097:EHP spectral sequence
5993:Computational methods
5807:
5662:
5253:
5024:is an isomorphism of
4841:infinite cyclic group
4574:
4512:
4350:
4237:
4089:
4033:
3974:
3779:
3615:
3496:short exact sequences
3414:
3267:
2935:
1567:
1430:
1360:
1327:as an equality: thus
1308:of a loop around the
1290:infinite cyclic group
1255:
1064:to the pointed space
991:
983:
941:, and the suspension
820:Suspension of equator
319:to a single point of
143:
131:
84:
11282:, pp. 665β723,
10932:, Paris: 1340β1342,
9460:
9367:, Example 0.3, p. 6.
6749:is the image of the
6631:
6575:of a compact smooth
6508:
6484:
6479:of smooth maps from
5841:is the generator of
5750:
5686:by the image of the
5639:
5293:− 1) − 2
5080:
4944:is an odd prime and
4929:-torsion occurs for
4527:
4417:
4249:
4136:
4045:
3989:
3930:
3678:
3505:
3300:
3220:
3024:the homotopy groups
2922:
1957:The homology groups
1450:space-filling curves
1220:. However, the real
1124:-th homotopy group,
1120:More generally, the
1048:) topological space
892:has as equator the (
526:as a basic example.
226:-dimensional sphere
109:, which reflect, in
11276:History of Topology
10825:Ravenel, Douglas C.
10529:Kervaire, Michel A.
10358:10.3792/pjaa.82.141
10219:Cohen, Frederick R.
10105:Cohen, Frederick R.
9895:Berrick et al. 2006
9415:, pp. 123β125.
6685:
6582:is divisible by 16.
6473:algebraic varieties
6390:corresponds to the
5853:(of order 2), then
5801:
5767:
5565:for such values of
4981:have order at most
3293:of homotopy groups
3291:long exact sequence
1686:George W. Whitehead
1614:the homotopy group
1606:, now known as the
1090:equivalence classes
735:, and its rim (or "
445:are independent of
301:, any mapping from
255:equivalence classes
176:) and the ordinary
11252:10.1007/BF01457962
11144:Algebraic Topology
10588:10.1007/BFb0083795
10197:, Paris: 393β395,
10187:Serre, Jean-Pierre
10161:, Paris: 288β290,
10149:Serre, Jean-Pierre
9871:Wang & Xu 2017
9489:
8730:, the part of the
6791:Kervaire invariant
6697:
6671:
6556:sphere around the
6516:
6492:
6465:singularity theory
6208:
6115:term given by the
6032:spectral sequences
5828:nilpotence theorem
5802:
5787:
5786:
5753:
5657:
5340:The J-homomorphism
5248:
4569:
4507:
4345:
4232:
4084:
4028:
3969:
3774:
3610:
3409:
3262:
3161:, and are denoted
3074:suspension theorem
2930:
1848:contractible space
1694:spectral sequences
1682:suspension theorem
1574:
1514:. This is because
1433:
1363:
1269:
1198:group homomorphism
1174:finitely generated
1115:bouquet of spheres
1108:abstract algebraic
994:
986:
512:Algebraic topology
496:spectral sequences
366:with the group of
152:-dimensional unit
146:
138:
115:topological spaces
95:algebraic topology
87:
11307:(21 April 1997),
11289:978-0-444-82375-5
11211:Historical papers
11194:978-0-226-51183-2
11158:978-0-521-79540-1
10971:978-0-691-09586-8
10891:, Second Series,
10871:978-0-8218-3749-8
10840:978-0-8218-2967-7
10636:978-0-8218-0296-0
10619:Mahowald, Mark E.
10597:978-3-540-52468-7
10481:978-1-4704-3788-6
10471:10.1090/memo/1269
10268:, Second Series,
10231:, Second Series,
10060:Mahowald, Mark E.
9825:, pp. 67β74.
9315:
9314:
8701:
8700:
8300:
8299:
6921:cartesian product
6819:classifying space
6815:plus construction
6611:smooth structures
6569:Rokhlin's theorem
6188:complex cobordism
5832:complex cobordism
5771:
4803:Jean-Pierre Serre
4679:(1, 0, ..., 0) β
4646:-submanifolds of
2891:
2890:
1690:Jean-Pierre Serre
1654:fundamental group
1532:lifting criterion
1321:group isomorphism
1030:fundamental group
998:topological space
970:(1, 0, 0, ..., 0)
739:") is the circle
500:Jean-Pierre Serre
342:, every map from
79:
78:
71:
11405:
11378:
11377:
11376:
11357:
11356:
11355:
11341:
11340:
11339:
11334:
11319:
11318:
11317:
11292:
11280:Elsevier Science
11262:
11224:
11205:
11169:
11126:
11108:
11099:
11089:
11073:Acta Mathematica
11065:
11040:
11017:
10982:
10948:
10919:
10882:
10851:
10809:
10792:
10782:
10758:
10722:
10710:
10696:
10678:
10647:
10608:
10569:
10523:
10514:
10492:
10473:
10452:
10432:
10411:
10376:
10351:
10314:
10305:
10259:
10213:
10177:
10139:
10130:
10103:Berrick, A. J.;
10098:
10081:
10053:
10052:
10032:
10031:
9995:
9981:
9975:
9964:
9958:
9952:
9946:
9940:
9934:
9928:
9922:
9916:
9910:
9904:
9898:
9892:
9886:
9880:
9874:
9868:
9862:
9856:
9850:
9844:
9838:
9832:
9826:
9820:
9814:
9808:
9802:
9796:
9790:
9784:
9778:
9772:
9763:
9757:
9751:
9745:
9739:
9733:
9727:
9721:
9715:
9709:
9703:
9697:
9691:
9685:
9679:
9673:
9667:
9661:
9650:
9644:
9638:
9632:
9626:
9620:
9614:
9608:
9599:
9593:
9587:
9581:
9575:
9569:
9563:
9557:
9551:
9545:
9539:
9533:
9527:
9521:
9512:
9506:
9500:
9498:
9496:
9495:
9490:
9485:
9484:
9472:
9471:
9456:, Section 8.1, "
9450:
9444:
9438:
9432:
9422:
9416:
9410:
9404:
9398:
9392:
9386:
9380:
9374:
9368:
9362:
9356:
9350:
9344:
9338:
8768:
8759:Bott periodicity
8756:
8749:
8741:
8737:
8733:
8729:
8721:
8720:
8719:
8302:
8217:
8160:
7993:
7758:
7012:
7008:
7004:
6966:
6953:
6934:
6930:
6915:
6904:
6891:
6876:
6855:
6805:
6801:
6800:2 − 2 = 62
6796:
6793:1 in dimensions
6777:
6763:
6759:
6753:
6748:
6744:
6740:
6739:
6738:
6722:
6706:
6704:
6703:
6698:
6690:
6684:
6679:
6667:
6666:
6648:
6643:
6642:
6616:
6608:
6601:
6593:
6565:Vladimir Rokhlin
6555:
6548:
6541:
6526:
6525:
6523:
6522:
6517:
6515:
6502:
6501:
6499:
6498:
6493:
6491:
6455:
6447:
6418:
6389:
6379:
6373:
6363:
6356:
6342:
6321:
6306:
6300:
6291:
6285:
6279:
6268:
6262:
6256:
6250:
6244:
6237:
6223:
6185:
6170:
6166:
6155:Steenrod algebra
6153:
6149:
6135:
6134:
6114:
6090:
6086:
6079:
6072:
6068:
6048:Hurewicz theorem
6027:
6022:
6013:
6004:
6000:
5975:
5974:
5954:
5943:
5933:
5932:
5931:
5922:are elements of
5921:
5917:
5913:
5906:
5905:
5904:
5894:
5888:
5887:
5886:
5876:
5870:
5869:
5868:
5858:
5852:
5851:
5850:
5840:
5817:supercommutative
5811:
5809:
5808:
5803:
5800:
5795:
5785:
5766:
5761:
5731:
5723:
5716:
5715:
5714:
5702:
5689:
5685:
5684:
5683:
5671:
5667:
5666:
5664:
5663:
5658:
5656:
5651:
5646:
5633:
5629:
5622:
5621:
5619:
5618:
5615:
5612:
5605:
5603:
5602:
5599:
5596:
5583:
5568:
5564:
5542:
5530:Bernoulli number
5527:
5515:
5514:
5512:
5511:
5505:
5502:
5483:
5468:
5460:
5455:Bott periodicity
5452:
5445:
5431:
5416:
5408:
5374:
5367:
5335:
5331:
5324:is divisible by
5323:
5317:vanishes unless
5316:
5315:
5314:
5302:
5298:
5294:
5279:
5272:
5268:
5257:
5255:
5254:
5249:
5235:
5234:
5213:
5212:
5179:
5178:
5157:
5156:
5117:
5116:
5101:
5100:
5072:
5061:
5046:
5042:
5027:
5023:
5008:
4994:
4990:
4986:
4980:
4958:
4954:
4943:
4939:
4928:
4924:
4913:
4906:
4902:
4887:
4883:
4865:
4861:
4854:
4850:
4846:
4838:
4834:
4818:
4779:
4769:
4763:
4757:
4756:
4738:
4725:
4707:
4701:
4687:
4683:
4669:
4651:
4645:
4638:
4632:
4631:
4619:
4590:Framed cobordism
4578:
4576:
4575:
4570:
4565:
4564:
4552:
4551:
4539:
4538:
4516:
4514:
4513:
4508:
4500:
4499:
4487:
4486:
4471:
4470:
4458:
4457:
4442:
4441:
4429:
4428:
4409:
4403:
4396:
4390:
4383:
4377:
4370:
4363:
4354:
4352:
4351:
4346:
4338:
4337:
4325:
4324:
4303:
4302:
4290:
4289:
4274:
4273:
4261:
4260:
4241:
4239:
4238:
4233:
4225:
4224:
4212:
4211:
4190:
4189:
4177:
4176:
4161:
4160:
4148:
4147:
4128:
4116:
4093:
4091:
4090:
4085:
4083:
4082:
4070:
4069:
4057:
4056:
4037:
4035:
4034:
4029:
4027:
4026:
4014:
4013:
4001:
4000:
3978:
3976:
3975:
3970:
3968:
3967:
3955:
3954:
3942:
3941:
3922:
3908:
3907:
3905:
3904:
3896:
3893:
3880:
3878:
3869:
3858:
3837:
3833:
3819:
3805:
3801:
3783:
3781:
3780:
3775:
3767:
3766:
3754:
3753:
3732:
3731:
3719:
3718:
3703:
3702:
3690:
3689:
3670:
3638:
3628:
3619:
3617:
3616:
3611:
3600:
3599:
3587:
3586:
3565:
3564:
3552:
3551:
3536:
3535:
3523:
3522:
3493:
3481:
3475:
3469:
3455:
3445:
3418:
3416:
3415:
3410:
3390:
3389:
3362:
3361:
3340:
3339:
3318:
3317:
3288:
3271:
3269:
3268:
3263:
3258:
3257:
3245:
3244:
3232:
3231:
3192:
3181:
3174:
3173:
3172:
3156:
3145:
3127:
3116:
3097:
3078:Hans Freudenthal
3059:
3052:
3037:
3023:
3013:
3003:
2993:
2986:
2952:
2939:
2937:
2936:
2931:
2929:
2913:
2906:
2820:
2819:
2745:
2744:
2653:
2652:
2572:
2571:
2557:
2556:
2540:
2539:
2507:
2506:
2496:
2495:
2448:
2447:
2437:
2436:
2412:
2411:
2337:
2336:
2326:
2325:
2301:
2300:
2082:
2081:
2074:
2065:
2064:
2054:
2039:
2030:
2023:, b) the finite
2022:
2006:
1996:
1982:
1972:
1953:
1926:
1919:
1915:
1899:
1892:
1873:simply-connected
1865:Hurewicz theorem
1862:
1845:
1835:
1825:
1821:
1813:
1799:
1795:
1783:
1779:
1775:
1725:
1709:
1678:Hans Freudenthal
1673:Hurewicz theorem
1625:
1605:
1599:
1585:
1560:
1545:
1539:
1529:
1519:
1513:
1497:
1478:
1464:
1447:
1423:
1404:
1390:
1380:
1353:
1338:
1299:
1288:is therefore an
1287:
1267:
1248:
1219:
1211:
1188:
1179:
1167:
1163:
1157:
1153:
1141:
1137:
1123:
1105:
1095:
1083:
1079:
1075:
1063:
1051:
1043:
1023:complex analysis
971:
961:
953:
947:
940:
938:
937:
923:
922:
911:
910:
898:
891:
889:
888:
875:
874:
863:
862:
850:
846:
838:
828:
809:
801:
795:
789:
771:
769:
768:
757:
756:
744:
734:
732:
731:
720:
719:
707:
694:
672:
666:
657:
652:
650:
649:
636:
635:
624:
623:
611:
604:
590:
588:
587:
576:
575:
564:
563:
550:Implicit surface
532:
520:abstract algebra
493:
486:
467:
459:
448:
444:
425:
421:
399:
393:
383:
365:
348:to itself has a
347:
341:
324:
318:
312:
306:
300:
285:
281:
277:
248:
242:
240:
231:
225:
221:
204:
200:
189:
185:
175:
165:
159:
151:
74:
67:
63:
60:
54:
49:this article by
40:inline citations
27:
26:
19:
11413:
11412:
11408:
11407:
11406:
11404:
11403:
11402:
11393:Homotopy theory
11383:
11382:
11374:
11372:
11353:
11351:
11337:
11335:
11332:
11315:
11313:
11300:
11290:
11213:
11195:
11159:
11134:
11129:
11007:
10972:
10901:10.2307/1969485
10872:
10841:
10814:Pontrjagin, Lev
10748:
10738:10.1090/gsm/005
10708:
10702:Milnor, John W.
10659:
10637:
10598:
10580:Springer-Verlag
10551:10.2307/1970128
10533:Milnor, John W.
10482:
10442:Homotopy theory
10416:Fuks, Dmitry B.
10401:
10331:
10278:10.2307/1970586
10241:10.2307/1971238
10079:10.1.1.212.1163
10010:Adams, J. Frank
10004:
9999:
9998:
9982:
9978:
9965:
9961:
9953:
9949:
9941:
9937:
9929:
9925:
9917:
9913:
9905:
9901:
9893:
9889:
9881:
9877:
9869:
9865:
9857:
9853:
9845:
9841:
9833:
9829:
9821:
9817:
9809:
9805:
9797:
9793:
9785:
9781:
9773:
9766:
9758:
9754:
9746:
9742:
9734:
9730:
9722:
9718:
9710:
9706:
9698:
9694:
9686:
9682:
9674:
9670:
9662:
9653:
9645:
9641:
9633:
9629:
9621:
9617:
9609:
9602:
9594:
9590:
9582:
9578:
9570:
9566:
9558:
9554:
9546:
9542:
9534:
9530:
9522:
9515:
9507:
9503:
9480:
9476:
9467:
9463:
9461:
9458:
9457:
9451:
9447:
9439:
9435:
9423:
9419:
9411:
9407:
9399:
9395:
9387:
9383:
9375:
9371:
9363:
9359:
9351:
9347:
9339:
9330:
9325:
9320:
9269:
9257:27β
5β
7β
13β
19β
37
9216:
9164:
9111:
9058:
9005:
8956:
8899:
8850:
8809:
8751:
8743:
8739:
8735:
8731:
8724:
8718:
8713:
8712:
8711:
8709:
8706:
8665:
8622:
8579:
8536:
8493:
8450:
8407:
8364:
8252:
8215:
8195:
8158:
8138:
8083:
8028:
7991:
7971:
7916:
7903:
7854:
7796:
7756:
7736:
7678:
7620:
7562:
7504:
7446:
7388:
7330:
7272:
7214:
7156:
7099:
7006:
7003:
6999:
6995:
6987:
6979:
6975:
6965:
6959:
6952:
6946:
6940:
6932:
6928:
6923:(equivalently,
6913:
6903:
6897:
6889:
6870:
6860:
6849:
6839:
6836:
6823:symmetric group
6817:applied to the
6803:
6799:
6794:
6776:
6770:
6761:
6760:is of the form
6757:
6751:
6746:
6742:
6737:
6732:
6731:
6730:
6728:
6721:
6712:
6686:
6680:
6675:
6656:
6652:
6644:
6638:
6634:
6632:
6629:
6628:
6614:
6603:
6599:
6592:
6586:
6550:
6543:
6535:
6528:
6511:
6509:
6506:
6505:
6504:
6487:
6485:
6482:
6481:
6480:
6477:critical points
6453:
6441:
6434:
6412:
6408:
6400:
6392:Borromean rings
6381:
6375:
6369:
6358:
6350:
6344:
6338:
6317:
6302:
6299:
6293:
6287:
6281:
6275:
6264:
6258:
6252:
6246:
6240:
6236:
6230:
6219:
6206:Borromean rings
6183:
6168:
6157:
6151:
6147:
6141:
6133:
6123:
6122:
6121:
6119:
6113:
6107:
6088:
6081:
6074:
6070:
6066:
6025:
6020:
6011:
6002:
5998:
5995:
5983:Massey products
5960:
5959:
5945:
5935:
5930:
5927:
5926:
5925:
5923:
5919:
5915:
5911:
5903:
5900:
5899:
5898:
5896:
5890:
5885:
5882:
5881:
5880:
5878:
5872:
5867:
5864:
5863:
5862:
5860:
5854:
5849:
5846:
5845:
5844:
5842:
5838:
5796:
5791:
5775:
5762:
5757:
5751:
5748:
5747:
5738:
5732:to save space.
5725:
5718:
5713:
5708:
5707:
5706:
5704:
5701:
5695:
5687:
5682:
5677:
5676:
5675:
5673:
5669:
5652:
5647:
5642:
5640:
5637:
5636:
5635:
5631:
5627:
5616:
5613:
5610:
5609:
5607:
5600:
5597:
5595:
5589:
5588:
5586:
5585:
5577:
5570:
5566:
5558:
5548:
5541:β 1 β‘ 3 (mod 4)
5533:
5526:
5517:
5506:
5503:
5501:
5492:
5491:
5489:
5488:
5481:
5466:
5458:
5447:
5446:only depend on
5439:
5433:
5422:
5410:
5402:
5388:
5379:
5369:
5361:
5351:
5348:
5342:
5333:
5325:
5318:
5313:
5308:
5307:
5306:
5304:
5300:
5296:
5281:
5277:
5270:
5269:means take the
5262:
5221:
5217:
5199:
5195:
5165:
5161:
5137:
5133:
5109:
5105:
5087:
5083:
5081:
5078:
5077:
5070:
5067:James fibration
5055:
5048:
5044:
5029:
5028:-components if
5025:
5017:
5010:
5002:
4996:
4992:
4988:
4982:
4974:
4964:
4956:
4945:
4941:
4930:
4926:
4915:
4908:
4904:
4893:
4885:
4877:
4867:
4863:
4859:
4852:
4848:
4844:
4836:
4828:
4820:
4812:
4806:
4799:
4787:homotopy groups
4771:
4765:
4755:
4752:
4751:
4750:
4744:
4740:
4730:
4713:
4703:
4695:
4689:
4685:
4671:
4657:
4647:
4643:
4630:
4625:
4624:
4623:
4621:
4613:
4603:
4592:
4560:
4556:
4547:
4543:
4534:
4530:
4528:
4525:
4524:
4495:
4491:
4482:
4478:
4466:
4462:
4453:
4449:
4437:
4433:
4424:
4420:
4418:
4415:
4414:
4405:
4398:
4392:
4385:
4379:
4372:
4365:
4359:
4333:
4329:
4314:
4310:
4298:
4294:
4285:
4281:
4269:
4265:
4256:
4252:
4250:
4247:
4246:
4220:
4216:
4201:
4197:
4185:
4181:
4172:
4168:
4156:
4152:
4143:
4139:
4137:
4134:
4133:
4122:
4118:
4110:
4106:
4078:
4074:
4065:
4061:
4052:
4048:
4046:
4043:
4042:
4022:
4018:
4009:
4005:
3996:
3992:
3990:
3987:
3986:
3963:
3959:
3950:
3946:
3937:
3933:
3931:
3928:
3927:
3914:
3903:
3897:
3894:
3892:
3886:
3885:
3883:
3882:
3877:
3871:
3870:| + |
3868:
3862:
3860:
3856:
3849:
3842:
3835:
3827:
3821:
3813:
3807:
3803:
3795:
3788:
3762:
3758:
3743:
3739:
3727:
3723:
3714:
3710:
3698:
3694:
3685:
3681:
3679:
3676:
3675:
3664:
3654:
3647:
3634:
3624:
3595:
3591:
3576:
3572:
3560:
3556:
3547:
3543:
3531:
3527:
3518:
3514:
3506:
3503:
3502:
3487:
3483:
3477:
3471:
3463:
3457:
3447:
3439:
3429:
3423:
3379:
3375:
3357:
3353:
3335:
3331:
3313:
3309:
3301:
3298:
3297:
3276:
3253:
3249:
3240:
3236:
3227:
3223:
3221:
3218:
3217:
3203:
3201:Hopf fibrations
3183:
3176:
3171:
3166:
3165:
3164:
3162:
3157:are called the
3147:
3139:
3129:
3118:
3110:
3099:
3091:
3081:
3070:
3054:
3046:
3039:
3031:
3025:
3015:
3008:
2995:
2988:
2980:
2970:
2964:
2944:
2925:
2923:
2920:
2919:
2908:
2900:
2894:
2887:
2881:
2869:
2863:
2857:
2818:
2815:
2814:
2813:
2809:
2803:
2791:
2785:
2779:
2743:
2740:
2739:
2738:
2734:
2730:
2724:
2718:
2706:
2700:
2694:
2663:
2659:
2651:
2648:
2647:
2646:
2642:
2636:
2630:
2624:
2618:
2612:
2606:
2600:
2570:
2567:
2566:
2565:
2563:
2555:
2554:
2550:
2545:
2544:
2543:
2538:
2535:
2534:
2533:
2529:
2523:
2517:
2513:
2505:
2502:
2501:
2500:
2494:
2491:
2490:
2489:
2485:
2479:
2473:
2446:
2443:
2442:
2441:
2435:
2432:
2431:
2430:
2428:
2422:
2418:
2410:
2407:
2406:
2405:
2401:
2395:
2389:
2383:
2377:
2371:
2365:
2359:
2335:
2332:
2331:
2330:
2324:
2321:
2320:
2319:
2317:
2311:
2307:
2299:
2296:
2295:
2294:
2290:
2284:
2278:
2272:
2266:
2260:
2254:
2248:
2174:
2168:
2162:
2156:
2150:
2144:
2138:
2132:
2126:
2120:
2114:
2108:
2102:
2096:
2090:
2073:
2069:
2063:
2060:
2059:
2058:
2056:
2053:
2049:
2045:
2042:direct products
2038:
2032:
2028:
2020:
2013:
1998:
1990:
1984:
1974:
1966:
1958:
1947:
1934:
1928:
1921:
1917:
1909:
1901:
1894:
1886:
1880:
1869:homology groups
1854:
1841:
1827:
1823:
1819:
1807:
1801:
1797:
1793:
1790:
1781:
1777:
1769:
1759:
1719:
1711:
1703:
1697:
1669:Witold Hurewicz
1632:
1619:
1615:
1601:
1595:
1577:
1562:
1554:
1550:
1541:
1535:
1534:, any map from
1525:
1522:universal cover
1515:
1507:
1503:
1499:
1491:
1487:
1472:
1466:
1456:
1445:
1436:is therefore a
1425:
1417:
1413:
1398:
1392:
1385:
1378:
1355:
1347:
1343:
1332:
1328:
1297:
1281:
1277:
1261:
1257:
1250:
1242:
1238:
1231:
1217:
1209:
1187:
1181:
1177:
1165:
1159:
1155:
1143:
1139:
1131:
1125:
1121:
1097:
1093:
1081:
1077:
1065:
1053:
1049:
1037:
1033:
1019:residue theorem
978:
969:
960:(sphere, point)
959:
949:
942:
936:
930:
929:
928:
921:
918:
917:
916:
909:
906:
905:
904:
900:
893:
887:
882:
881:
880:
873:
870:
869:
868:
861:
858:
857:
856:
852:
848:
844:
834:
823:
807:
797:
796:. For example,
791:
781:
767:
764:
763:
762:
755:
752:
751:
750:
746:
740:
730:
727:
726:
725:
718:
715:
714:
713:
709:
703:
686:
668:
661:
655:
648:
643:
642:
641:
634:
631:
630:
629:
622:
619:
618:
617:
613:
612:; the equation
609:
600:
597:Euclidean space
586:
583:
582:
581:
574:
571:
570:
569:
562:
559:
558:
557:
553:
535:
530:
524:homotopy groups
508:
488:
477:
465:
450:
446:
438:
428:
423:
415:
405:
395:
389:
375:
359:
353:
343:
333:
320:
314:
308:
302:
291:
283:
279:
271:
265:
244:
238:
237:
227:
223:
215:
209:
202:
195:
192:Euclidean space
187:
181:
171:
161:
157:
149:
123:homotopy groups
119:homology groups
75:
64:
58:
55:
45:Please help to
44:
28:
24:
17:
12:
11:
5:
11411:
11401:
11400:
11395:
11381:
11380:
11363:
11342:
11324:Hatcher, Allen
11320:
11299:
11298:External links
11296:
11295:
11294:
11288:
11264:
11246:(1): 637β665,
11226:
11212:
11209:
11208:
11207:
11193:
11171:
11157:
11139:Hatcher, Allen
11133:
11130:
11128:
11127:
11110:
11080:(2): 319β407,
11067:
11031:(2): 501β580,
11018:
11005:
10984:
10970:
10950:
10921:
10895:(3): 425β505,
10884:
10870:
10853:
10839:
10821:
10811:
10773:(4): 707β732,
10759:
10746:
10723:
10698:
10679:
10657:
10652:Mahowald, Mark
10648:
10635:
10614:
10596:
10571:
10545:(3): 504β537,
10525:
10494:
10480:
10453:
10434:
10412:
10399:
10378:
10342:(8): 141β146,
10329:
10316:
10272:(2): 305β320,
10261:
10235:(3): 549β565,
10223:Moore, John C.
10215:
10179:
10141:
10121:(2): 265β326,
10100:
10072:(3): 533β550,
10055:
10005:
10003:
10000:
9997:
9996:
9992:Ravenel (2003)
9976:
9959:
9947:
9935:
9923:
9911:
9899:
9887:
9875:
9863:
9851:
9839:
9827:
9815:
9803:
9791:
9779:
9764:
9752:
9740:
9728:
9716:
9704:
9692:
9680:
9668:
9651:
9639:
9627:
9625:, p. 342.
9615:
9600:
9598:, p. 203.
9588:
9576:
9564:
9552:
9540:
9528:
9526:, p. 349.
9513:
9511:, p. 348.
9501:
9488:
9483:
9479:
9475:
9470:
9466:
9445:
9433:
9417:
9405:
9393:
9381:
9379:, p. 129.
9369:
9357:
9355:, p. xii.
9345:
9327:
9326:
9324:
9321:
9319:
9316:
9313:
9312:
9303:
9300:
9297:
9294:
9285:
9282:
9276:
9270:
9264:
9260:
9259:
9250:
9247:
9244:
9241:
9232:
9229:
9223:
9217:
9211:
9207:
9206:
9197:
9194:
9191:
9188:
9179:
9176:
9170:
9165:
9159:
9155:
9154:
9145:
9142:
9139:
9136:
9127:
9124:
9118:
9112:
9106:
9102:
9101:
9088:
9085:
9082:
9079:
9074:
9071:
9065:
9059:
9053:
9049:
9048:
9039:
9036:
9033:
9030:
9021:
9018:
9012:
9006:
9000:
8996:
8995:
8986:
8983:
8980:
8977:
8972:
8969:
8963:
8957:
8951:
8947:
8946:
8933:
8930:
8927:
8924:
8915:
8912:
8906:
8900:
8894:
8890:
8889:
8880:
8877:
8874:
8871:
8866:
8863:
8857:
8851:
8845:
8841:
8840:
8835:
8832:
8829:
8826:
8821:
8818:
8813:
8810:
8804:
8800:
8799:
8796:
8793:
8790:
8787:
8784:
8781:
8778:
8775:
8763:J-homomorphism
8714:
8705:
8702:
8699:
8698:
8695:
8692:
8689:
8686:
8683:
8680:
8677:
8674:
8671:
8660:
8656:
8655:
8652:
8649:
8646:
8643:
8640:
8637:
8634:
8631:
8628:
8617:
8613:
8612:
8609:
8606:
8603:
8600:
8597:
8594:
8591:
8588:
8585:
8574:
8570:
8569:
8566:
8563:
8560:
8557:
8554:
8551:
8548:
8545:
8542:
8531:
8527:
8526:
8523:
8520:
8517:
8514:
8511:
8508:
8505:
8502:
8499:
8488:
8484:
8483:
8480:
8477:
8474:
8471:
8468:
8465:
8462:
8459:
8456:
8445:
8441:
8440:
8437:
8434:
8431:
8428:
8425:
8422:
8419:
8416:
8413:
8402:
8398:
8397:
8394:
8391:
8388:
8385:
8382:
8379:
8376:
8373:
8370:
8359:
8355:
8354:
8349:
8344:
8339:
8334:
8329:
8324:
8319:
8314:
8309:
8298:
8297:
8294:
8291:
8288:
8285:
8282:
8279:
8276:
8273:
8270:
8267:
8264:
8261:
8258:
8247:
8243:
8242:
8239:
8236:
8233:
8230:
8227:
8224:
8221:
8218:
8213:
8210:
8207:
8204:
8201:
8190:
8186:
8185:
8182:
8179:
8176:
8173:
8170:
8167:
8164:
8161:
8156:
8153:
8150:
8147:
8144:
8133:
8129:
8128:
8125:
8122:
8119:
8116:
8113:
8110:
8107:
8104:
8101:
8098:
8095:
8092:
8089:
8078:
8074:
8073:
8070:
8067:
8064:
8061:
8058:
8055:
8052:
8049:
8046:
8043:
8040:
8037:
8034:
8023:
8019:
8018:
8015:
8012:
8009:
8006:
8003:
8000:
7997:
7994:
7989:
7986:
7983:
7980:
7977:
7966:
7962:
7961:
7958:
7955:
7952:
7949:
7946:
7943:
7940:
7937:
7934:
7931:
7928:
7925:
7922:
7911:
7907:
7906:
7899:
7896:
7893:
7890:
7887:
7884:
7881:
7878:
7875:
7872:
7869:
7866:
7863:
7860:
7849:
7845:
7844:
7841:
7838:
7835:
7832:
7829:
7826:
7823:
7820:
7817:
7814:
7811:
7808:
7805:
7802:
7791:
7787:
7786:
7783:
7780:
7777:
7774:
7771:
7768:
7765:
7762:
7759:
7754:
7751:
7748:
7745:
7742:
7731:
7727:
7726:
7723:
7720:
7717:
7714:
7711:
7708:
7705:
7702:
7699:
7696:
7693:
7690:
7687:
7684:
7673:
7669:
7668:
7665:
7662:
7659:
7656:
7653:
7650:
7647:
7644:
7641:
7638:
7635:
7632:
7629:
7626:
7615:
7611:
7610:
7607:
7604:
7601:
7598:
7595:
7592:
7589:
7586:
7583:
7580:
7577:
7574:
7571:
7568:
7557:
7553:
7552:
7549:
7546:
7543:
7540:
7537:
7534:
7531:
7528:
7525:
7522:
7519:
7516:
7513:
7510:
7499:
7495:
7494:
7491:
7488:
7485:
7482:
7479:
7476:
7473:
7470:
7467:
7464:
7461:
7458:
7455:
7452:
7441:
7437:
7436:
7433:
7430:
7427:
7424:
7421:
7418:
7415:
7412:
7409:
7406:
7403:
7400:
7397:
7394:
7383:
7379:
7378:
7375:
7372:
7369:
7366:
7363:
7360:
7357:
7354:
7351:
7348:
7345:
7342:
7339:
7336:
7325:
7321:
7320:
7317:
7314:
7311:
7308:
7305:
7302:
7299:
7296:
7293:
7290:
7287:
7284:
7281:
7278:
7267:
7263:
7262:
7259:
7256:
7253:
7250:
7247:
7244:
7241:
7238:
7235:
7232:
7229:
7226:
7223:
7220:
7209:
7205:
7204:
7201:
7198:
7195:
7192:
7189:
7186:
7183:
7180:
7177:
7174:
7171:
7168:
7165:
7162:
7151:
7147:
7146:
7143:
7140:
7137:
7134:
7131:
7128:
7125:
7122:
7119:
7116:
7113:
7110:
7107:
7105:
7094:
7090:
7089:
7084:
7079:
7074:
7069:
7064:
7059:
7054:
7049:
7044:
7039:
7034:
7029:
7024:
7019:
7009:in the table.
7001:
6997:
6993:
6985:
6977:
6969:
6968:
6961:
6948:
6942:
6917:
6906:
6899:
6882:
6862:
6841:
6835:
6832:
6831:
6830:
6807:
6783:
6772:
6766:
6765:
6733:
6716:
6709:
6708:
6707:
6696:
6693:
6689:
6683:
6678:
6674:
6670:
6665:
6662:
6659:
6655:
6651:
6647:
6641:
6637:
6623:
6622:
6619:exotic spheres
6602:-spheres (for
6588:
6583:
6561:
6558:critical value
6530:
6514:
6490:
6461:
6436:
6433:The fact that
6431:
6410:
6405:winding number
6399:
6396:
6346:
6314:
6313:
6297:
6271:
6270:
6234:
6226:
6225:
6214:
6213:
6200:
6199:
6176:
6143:
6137:
6124:
6111:
6102:The classical
6100:
6093:
6055:
5994:
5991:
5928:
5901:
5883:
5865:
5847:
5813:
5812:
5799:
5794:
5790:
5784:
5781:
5778:
5774:
5770:
5765:
5760:
5756:
5737:
5736:Ring structure
5734:
5709:
5697:
5678:
5655:
5650:
5645:
5593:
5572:
5550:
5545:
5544:
5521:
5496:
5485:
5478:
5435:
5394:
5384:
5377:J-homomorphism
5353:
5346:J-homomorphism
5344:Main article:
5341:
5338:
5309:
5259:
5258:
5247:
5244:
5241:
5238:
5233:
5230:
5227:
5224:
5220:
5216:
5211:
5208:
5205:
5202:
5198:
5194:
5191:
5188:
5185:
5182:
5177:
5174:
5171:
5168:
5164:
5160:
5155:
5152:
5149:
5146:
5143:
5140:
5136:
5132:
5129:
5126:
5123:
5120:
5115:
5112:
5108:
5104:
5099:
5096:
5093:
5090:
5086:
5050:
5012:
4998:
4966:
4869:
4835:(for positive
4822:
4808:
4798:
4795:
4789:of spaces and
4753:
4742:
4728:Hopf fibration
4691:
4641:differentiable
4626:
4620:and the group
4605:
4600:Lev Pontryagin
4591:
4588:
4584:Hopf invariant
4580:
4579:
4568:
4563:
4559:
4555:
4550:
4546:
4542:
4537:
4533:
4518:
4517:
4506:
4503:
4498:
4494:
4490:
4485:
4481:
4477:
4474:
4469:
4465:
4461:
4456:
4452:
4448:
4445:
4440:
4436:
4432:
4427:
4423:
4356:
4355:
4344:
4341:
4336:
4332:
4328:
4323:
4320:
4317:
4313:
4309:
4306:
4301:
4297:
4293:
4288:
4284:
4280:
4277:
4272:
4268:
4264:
4259:
4255:
4243:
4242:
4231:
4228:
4223:
4219:
4215:
4210:
4207:
4204:
4200:
4196:
4193:
4188:
4184:
4180:
4175:
4171:
4167:
4164:
4159:
4155:
4151:
4146:
4142:
4120:
4108:
4095:
4094:
4081:
4077:
4073:
4068:
4064:
4060:
4055:
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3485:
3459:
3456:, maps all of
3435:
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3273:
3272:
3261:
3256:
3252:
3248:
3243:
3239:
3235:
3230:
3226:
3207:Hopf fibration
3205:The classical
3202:
3199:
3167:
3131:
3128:. The groups
3101:
3083:
3069:
3066:
3062:
3061:
3041:
3027:
3005:
2976:
2966:
2961:
2958:
2928:
2916:covering space
2896:
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2034:
2012:
2009:
1986:
1962:
1943:
1930:
1905:
1882:
1822:-sphere to an
1803:
1789:
1788:General theory
1786:
1776:are known for
1761:
1748:Daniel Isaksen
1713:
1699:
1645:Analysis situs
1640:Henri PoincarΓ©
1636:Camille Jordan
1631:
1628:
1617:
1608:Hopf fibration
1570:Hopf fibration
1561:
1552:
1548:
1505:
1498:
1489:
1485:
1468:
1424:
1415:
1411:
1394:
1354:
1345:
1341:
1330:
1306:winding number
1279:
1259:
1249:
1240:
1236:
1230:
1227:
1183:
1176:. If for some
1164:(for positive
1127:
1046:path connected
1035:
1010:continuous map
977:
976:Homotopy group
974:
965:pointed sphere
956:
955:
931:
919:
907:
883:
871:
859:
830:
829:
816:
815:
765:
753:
728:
716:
696:
695:
679:
678:
644:
632:
620:
592:
591:
584:
572:
560:
534:
528:
507:
504:
430:
407:
402:
401:
386:Hopf fibration
372:
355:
330:
267:
211:
207:homotopy group
134:Hopf fibration
77:
76:
59:September 2022
31:
29:
22:
15:
9:
6:
4:
3:
2:
11410:
11399:
11396:
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11312:
11311:
11306:
11302:
11301:
11291:
11285:
11281:
11277:
11273:
11269:
11268:May, J. Peter
11265:
11261:
11257:
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11249:
11245:
11241:
11240:
11235:
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11227:
11223:
11219:
11215:
11214:
11204:
11200:
11196:
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11182:
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11176:
11175:May, J. Peter
11172:
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11052:
11048:
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11039:
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11030:
11026:
11025:
11019:
11016:
11012:
11008:
11006:0-387-20430-X
11002:
10998:
10994:
10990:
10985:
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10973:
10967:
10963:
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10819:
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10812:
10808:
10804:
10800:
10796:
10791:
10786:
10781:
10776:
10772:
10768:
10764:
10763:Nishida, Goro
10760:
10757:
10753:
10749:
10747:0-8218-0268-2
10743:
10739:
10735:
10731:
10730:
10724:
10720:
10716:
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10703:
10699:
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10439:
10435:
10431:
10427:
10426:
10421:
10417:
10413:
10410:
10406:
10402:
10400:0-387-94657-8
10396:
10392:
10388:
10384:
10379:
10375:
10371:
10367:
10363:
10359:
10355:
10350:
10345:
10341:
10337:
10333:
10328:
10325:-theory over
10324:
10317:
10313:
10309:
10304:
10299:
10295:
10291:
10287:
10283:
10279:
10275:
10271:
10267:
10262:
10258:
10254:
10250:
10246:
10242:
10238:
10234:
10230:
10229:
10224:
10220:
10216:
10212:
10208:
10204:
10200:
10196:
10192:
10188:
10184:
10183:Cartan, Henri
10180:
10176:
10172:
10168:
10164:
10160:
10156:
10155:
10150:
10146:
10145:Cartan, Henri
10142:
10138:
10134:
10129:
10124:
10120:
10116:
10115:
10110:
10106:
10101:
10097:
10093:
10089:
10085:
10080:
10075:
10071:
10067:
10066:
10061:
10056:
10051:
10046:
10042:
10038:
10030:
10025:
10021:
10017:
10016:
10011:
10007:
10006:
9993:
9989:
9985:
9980:
9973:
9969:
9963:
9956:
9951:
9944:
9939:
9932:
9927:
9921:, p. 32.
9920:
9915:
9908:
9903:
9896:
9891:
9884:
9879:
9872:
9867:
9860:
9855:
9848:
9843:
9836:
9831:
9824:
9819:
9812:
9811:Mahowald 2001
9807:
9800:
9795:
9788:
9783:
9776:
9771:
9769:
9761:
9756:
9749:
9744:
9737:
9732:
9726:, p. 25.
9725:
9720:
9713:
9708:
9701:
9696:
9689:
9684:
9677:
9672:
9665:
9660:
9658:
9656:
9648:
9643:
9636:
9631:
9624:
9619:
9612:
9607:
9605:
9597:
9592:
9585:
9580:
9573:
9568:
9562:, p. 90.
9561:
9560:Walschap 2004
9556:
9549:
9544:
9538:, p. 61.
9537:
9532:
9525:
9520:
9518:
9510:
9505:
9481:
9477:
9468:
9464:
9455:
9449:
9443:, p. 29.
9442:
9437:
9430:
9426:
9421:
9414:
9409:
9402:
9397:
9391:, p. 28.
9390:
9385:
9378:
9373:
9366:
9361:
9354:
9349:
9342:
9337:
9335:
9333:
9328:
9311:
9307:
9304:
9301:
9298:
9295:
9293:
9289:
9286:
9283:
9280:
9277:
9274:
9271:
9268:
9262:
9261:
9258:
9254:
9251:
9248:
9245:
9242:
9240:
9236:
9233:
9230:
9227:
9224:
9221:
9218:
9215:
9209:
9208:
9205:
9201:
9198:
9195:
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9180:
9177:
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9137:
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9125:
9122:
9119:
9116:
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9031:
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9022:
9019:
9016:
9013:
9010:
9007:
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8998:
8997:
8994:
8990:
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8967:
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8961:
8958:
8955:
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8907:
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8891:
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8773:
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7257:
7254:
7251:
7248:
7245:
7242:
7239:
7236:
7233:
7230:
7227:
7224:
7221:
7218:
7213:
7207:
7206:
7202:
7199:
7196:
7193:
7190:
7187:
7184:
7181:
7178:
7175:
7172:
7169:
7166:
7163:
7160:
7155:
7149:
7148:
7144:
7141:
7138:
7135:
7132:
7129:
7126:
7123:
7120:
7117:
7114:
7111:
7108:
7106:
7103:
7098:
7092:
7091:
7088:
7085:
7083:
7080:
7078:
7075:
7073:
7070:
7068:
7065:
7063:
7060:
7058:
7055:
7053:
7050:
7048:
7045:
7043:
7040:
7038:
7035:
7033:
7030:
7028:
7025:
7023:
7020:
7017:
7014:
7013:
7010:
6991:
6983:
6973:
6964:
6957:
6951:
6945:
6938:
6937:common factor
6926:
6922:
6918:
6911:
6907:
6902:
6895:
6887:
6883:
6880:
6879:
6878:
6874:
6869:
6865:
6857:
6853:
6848:
6844:
6828:
6824:
6820:
6816:
6812:
6808:
6792:
6788:
6784:
6781:
6775:
6768:
6767:
6755:
6754:-homomorphism
6736:
6726:
6719:
6715:
6710:
6694:
6691:
6687:
6681:
6676:
6672:
6663:
6660:
6657:
6653:
6649:
6645:
6639:
6627:
6626:
6625:
6624:
6620:
6612:
6606:
6597:
6591:
6584:
6581:
6578:
6574:
6570:
6566:
6562:
6559:
6553:
6546:
6539:
6533:
6478:
6474:
6470:
6466:
6462:
6459:
6456:-dimensional
6451:
6445:
6439:
6432:
6429:
6426:
6422:
6416:
6406:
6402:
6401:
6395:
6393:
6388:
6384:
6378:
6372:
6367:
6361:
6354:
6349:
6341:
6336:
6333:
6329:
6325:
6320:
6311:
6305:
6296:
6290:
6284:
6278:
6273:
6272:
6267:
6261:
6255:
6249:
6243:
6233:
6228:
6227:
6222:
6216:
6215:
6210:
6209:
6204:
6197:
6193:
6189:
6181:
6177:
6174:
6164:
6160:
6156:
6150:over the mod
6146:
6140:
6131:
6127:
6118:
6110:
6105:
6101:
6098:
6094:
6084:
6077:
6064:
6060:
6056:
6053:
6049:
6045:
6041:
6037:
6036:
6035:
6033:
6029:
6019:
6015:
6008:
5990:
5988:
5984:
5979:
5972:
5968:
5964:
5958:
5955:, there is a
5952:
5948:
5941:
5938:
5908:
5893:
5875:
5857:
5835:
5833:
5829:
5825:
5821:
5818:
5797:
5792:
5788:
5782:
5779:
5776:
5772:
5768:
5763:
5758:
5754:
5746:
5745:
5744:
5743:
5733:
5729:
5721:
5712:
5700:
5693:
5681:
5648:
5624:
5592:
5581:
5575:
5562:
5557:
5553:
5540:
5536:
5531:
5525:
5520:
5510:
5500:
5495:
5486:
5479:
5476:
5472:
5464:
5463:
5462:
5456:
5450:
5443:
5438:
5429:
5425:
5420:
5414:
5406:
5401:
5397:
5392:
5387:
5382:
5378:
5372:
5365:
5360:
5356:
5347:
5337:
5329:
5321:
5312:
5292:
5288:
5284:
5274:
5266:
5242:
5231:
5228:
5225:
5222:
5218:
5209:
5206:
5203:
5200:
5196:
5192:
5186:
5175:
5172:
5169:
5166:
5162:
5153:
5150:
5147:
5144:
5141:
5138:
5134:
5130:
5124:
5113:
5110:
5106:
5097:
5094:
5091:
5088:
5084:
5076:
5075:
5074:
5068:
5063:
5059:
5053:
5040:
5036:
5032:
5021:
5015:
5006:
5001:
4985:
4978:
4973:
4969:
4962:
4952:
4948:
4937:
4933:
4922:
4918:
4911:
4900:
4896:
4891:
4881:
4876:
4872:
4856:
4842:
4832:
4826:
4816:
4811:
4804:
4794:
4792:
4788:
4784:
4778:
4774:
4768:
4761:
4748:
4737:
4733:
4729:
4724:
4720:
4716:
4711:
4706:
4699:
4694:
4682:
4678:
4674:
4668:
4664:
4660:
4656:. Every map
4655:
4654:normal bundle
4650:
4642:
4636:
4629:
4617:
4612:
4608:
4601:
4597:
4587:
4585:
4566:
4561:
4557:
4548:
4544:
4535:
4531:
4523:
4522:
4521:
4504:
4496:
4492:
4483:
4479:
4475:
4467:
4463:
4454:
4450:
4446:
4438:
4434:
4425:
4421:
4413:
4412:
4411:
4408:
4401:
4395:
4388:
4382:
4375:
4368:
4362:
4342:
4334:
4330:
4321:
4318:
4315:
4311:
4307:
4299:
4295:
4286:
4282:
4278:
4270:
4266:
4257:
4253:
4245:
4244:
4229:
4221:
4217:
4208:
4205:
4202:
4198:
4194:
4186:
4182:
4173:
4169:
4165:
4157:
4153:
4144:
4140:
4132:
4131:
4130:
4126:
4114:
4104:
4100:
4079:
4075:
4066:
4062:
4053:
4049:
4041:
4040:
4023:
4019:
4010:
4006:
3997:
3993:
3985:
3984:
3983:
3982:
3964:
3960:
3951:
3947:
3938:
3934:
3924:
3921:
3917:
3912:
3900:
3889:
3874:
3865:
3853:
3846:
3839:
3831:
3826:
3817:
3812:
3802:vanishes for
3799:
3793:
3771:
3763:
3759:
3750:
3747:
3744:
3740:
3736:
3728:
3724:
3715:
3711:
3707:
3699:
3695:
3686:
3682:
3674:
3673:
3672:
3668:
3663:
3658:
3652:
3646:
3642:
3637:
3632:
3627:
3607:
3596:
3592:
3583:
3580:
3577:
3573:
3561:
3557:
3548:
3544:
3532:
3528:
3519:
3515:
3508:
3501:
3500:
3499:
3497:
3491:
3480:
3474:
3467:
3462:
3454:
3450:
3443:
3438:
3433:
3428:
3406:
3403:
3394:
3386:
3383:
3380:
3376:
3366:
3358:
3354:
3344:
3336:
3332:
3322:
3314:
3310:
3303:
3296:
3295:
3294:
3292:
3287:
3283:
3279:
3259:
3254:
3250:
3241:
3237:
3228:
3224:
3216:
3215:
3214:
3212:
3208:
3198:
3196:
3190:
3186:
3179:
3170:
3160:
3154:
3150:
3143:
3138:
3134:
3125:
3121:
3114:
3108:
3104:
3095:
3090:
3086:
3079:
3075:
3065:
3057:
3050:
3044:
3035:
3030:
3022:
3018:
3011:
3006:
3002:
2998:
2991:
2984:
2979:
2974:
2969:
2962:
2959:
2956:
2955:
2954:
2951:
2947:
2941:
2917:
2911:
2904:
2899:
2883:
2877:
2874:
2871:
2865:
2859:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2827:
2824:
2823:
2811:
2805:
2799:
2796:
2793:
2787:
2781:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2752:
2749:
2748:
2736:
2726:
2720:
2714:
2711:
2708:
2702:
2696:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2670:
2667:
2666:
2655:
2644:
2638:
2632:
2626:
2620:
2614:
2608:
2602:
2596:
2593:
2590:
2587:
2584:
2581:
2579:
2576:
2575:
2559:
2542:
2531:
2525:
2519:
2509:
2498:
2487:
2481:
2475:
2469:
2466:
2463:
2460:
2457:
2455:
2452:
2451:
2439:
2424:
2414:
2403:
2397:
2391:
2385:
2379:
2373:
2367:
2361:
2355:
2352:
2349:
2346:
2344:
2341:
2340:
2328:
2313:
2303:
2292:
2286:
2280:
2274:
2268:
2262:
2256:
2250:
2244:
2241:
2238:
2235:
2233:
2230:
2229:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2181:
2178:
2177:
2170:
2164:
2158:
2152:
2146:
2140:
2134:
2128:
2122:
2116:
2110:
2104:
2098:
2092:
2086:
2084:
2083:
2080:
2078:
2043:
2040:), or c) the
2037:
2026:
2025:cyclic groups
2018:
2017:trivial group
2008:
2005:
2001:
1994:
1989:
1981:
1977:
1970:
1965:
1961:
1955:
1951:
1946:
1942:
1938:
1933:
1924:
1913:
1908:
1904:
1897:
1890:
1885:
1878:
1874:
1870:
1866:
1861:
1857:
1851:
1849:
1844:
1839:
1834:
1830:
1826:-sphere with
1817:
1816:trivial group
1811:
1806:
1796:is less than
1785:
1773:
1768:
1764:
1757:
1753:
1749:
1745:
1744:Mark Mahowald
1741:
1737:
1733:
1729:
1723:
1717:
1707:
1702:
1695:
1691:
1687:
1683:
1679:
1674:
1670:
1666:
1662:
1657:
1655:
1651:
1647:
1646:
1641:
1637:
1627:
1623:
1613:
1609:
1604:
1598:
1593:
1589:
1584:
1580:
1571:
1566:
1558:
1547:
1544:
1538:
1533:
1528:
1523:
1518:
1511:
1495:
1484:
1482:
1476:
1471:
1463:
1459:
1453:
1451:
1443:
1439:
1438:trivial group
1429:
1421:
1410:
1408:
1402:
1397:
1388:
1382:
1376:
1375:
1369:
1359:
1351:
1340:
1336:
1326:
1325:often written
1322:
1317:
1315:
1311:
1307:
1303:
1296:to the group
1295:
1291:
1285:
1274:
1265:
1254:
1246:
1235:
1226:
1223:
1215:
1207:
1206:homeomorphism
1203:
1199:
1194:
1192:
1191:trivial group
1186:
1175:
1171:
1162:
1151:
1147:
1135:
1130:
1118:
1116:
1112:
1109:
1104:
1100:
1091:
1087:
1073:
1069:
1061:
1057:
1047:
1041:
1031:
1026:
1024:
1020:
1015:
1011:
1007:
1006:neighborhoods
1003:
999:
990:
982:
973:
967:
966:
952:
946:
934:
927:
915:
903:
896:
886:
879:
867:
855:
842:
837:
832:
831:
827:
821:
818:
817:
813:
805:
800:
794:
788:
784:
779:
775:
761:
749:
743:
738:
724:
712:
706:
702:
698:
697:
693:
689:
684:
681:
680:
676:
671:
664:
659:
653:produces the
647:
640:
628:
616:
608:
603:
598:
594:
593:
580:
568:
556:
551:
548:
547:
546:
544:
540:
527:
525:
521:
517:
513:
503:
501:
497:
491:
484:
480:
475:
471:
463:
457:
453:
442:
437:
433:
422:for positive
419:
414:
410:
398:
392:
387:
382:
378:
373:
369:
363:
358:
351:
346:
340:
336:
331:
328:
327:trivial group
323:
317:
311:
305:
299:
295:
289:
288:
287:
275:
270:
262:
260:
259:abelian group
256:
252:
247:
235:
230:
219:
214:
208:
198:
194:of dimension
193:
184:
179:
174:
169:
164:
156:β called the
155:
142:
135:
130:
126:
124:
120:
116:
112:
108:
104:
100:
96:
92:
83:
73:
70:
62:
52:
48:
42:
41:
35:
30:
21:
20:
11373:, retrieved
11368:
11352:, retrieved
11347:
11336:, retrieved
11328:
11314:, retrieved
11309:
11275:
11243:
11237:
11221:
11218:Δech, Eduard
11179:
11143:
11114:
11077:
11071:
11028:
11022:
10988:
10957:
10954:Toda, Hirosi
10929:
10925:
10892:
10888:
10857:
10829:
10817:
10770:
10766:
10728:
10721:(6): 804β809
10718:
10712:
10687:
10667:
10661:
10626:
10575:
10542:
10536:
10502:
10498:
10461:
10457:
10441:
10438:Hu, Sze-tsen
10423:
10382:
10349:math/0605429
10339:
10335:
10326:
10322:
10269:
10265:
10232:
10226:
10194:
10190:
10158:
10152:
10118:
10112:
10069:
10063:
10040:
10036:
10022:(1): 21β71,
10019:
10013:
9979:
9962:
9955:Deitmar 2006
9950:
9938:
9926:
9919:Hatcher 2002
9914:
9902:
9890:
9878:
9866:
9859:Isaksen 2019
9854:
9847:Kochman 1990
9842:
9837:, Chapter 5.
9835:Ravenel 2003
9830:
9823:Ravenel 2003
9818:
9806:
9799:Ravenel 2003
9794:
9782:
9760:Nishida 1973
9755:
9743:
9731:
9724:Ravenel 2003
9719:
9707:
9702:, p. 4.
9700:Ravenel 2003
9695:
9683:
9671:
9664:Scorpan 2005
9647:Hatcher 2002
9642:
9635:Hatcher 2002
9630:
9623:Hatcher 2002
9618:
9591:
9579:
9567:
9555:
9543:
9536:Hatcher 2002
9531:
9524:Hatcher 2002
9509:Hatcher 2002
9504:
9448:
9441:Hatcher 2002
9436:
9420:
9413:Miranda 1995
9408:
9403:, p. 3.
9401:Hatcher 2002
9396:
9389:Hatcher 2002
9384:
9377:Hatcher 2002
9372:
9365:Hatcher 2002
9360:
9353:Hatcher 2002
9348:
9309:
9305:
9291:
9287:
9278:
9272:
9266:
9256:
9252:
9238:
9234:
9225:
9219:
9213:
9203:
9199:
9185:
9181:
9172:
9167:
9161:
9151:
9147:
9133:
9129:
9120:
9114:
9108:
9098:
9094:
9090:
9076:
9067:
9061:
9055:
9045:
9041:
9027:
9023:
9014:
9008:
9002:
8992:
8988:
8974:
8965:
8959:
8953:
8943:
8939:
8935:
8921:
8917:
8908:
8902:
8896:
8886:
8882:
8868:
8859:
8853:
8847:
8837:
8823:
8815:
8806:
8771:
8752:
8745:
8725:
8715:
8707:
8667:
8662:
8624:
8619:
8581:
8576:
8538:
8533:
8495:
8490:
8452:
8447:
8409:
8404:
8366:
8361:
8351:
8346:
8341:
8336:
8331:
8326:
8321:
8316:
8311:
8305:
8254:
8249:
8197:
8192:
8140:
8135:
8085:
8080:
8030:
8025:
7973:
7968:
7918:
7913:
7901:
7856:
7851:
7798:
7793:
7738:
7733:
7680:
7675:
7622:
7617:
7564:
7559:
7506:
7501:
7448:
7443:
7390:
7385:
7332:
7327:
7274:
7269:
7216:
7211:
7158:
7153:
7101:
7096:
7086:
7081:
7076:
7071:
7066:
7061:
7056:
7051:
7046:
7041:
7036:
7031:
7026:
7021:
7015:
6989:
6981:
6971:
6970:
6962:
6949:
6943:
6900:
6894:cyclic group
6872:
6867:
6863:
6858:
6851:
6846:
6842:
6837:
6773:
6734:
6717:
6713:
6604:
6589:
6551:
6544:
6537:
6531:
6448:implies the
6443:
6437:
6414:
6398:Applications
6386:
6382:
6376:
6370:
6359:
6352:
6347:
6339:
6335:braid groups
6318:
6315:
6303:
6294:
6288:
6282:
6276:
6265:
6259:
6253:
6247:
6241:
6231:
6220:
6162:
6158:
6144:
6138:
6129:
6125:
6108:
6082:
6075:
5996:
5978:Hiroshi Toda
5970:
5966:
5962:
5957:Toda bracket
5950:
5946:
5939:
5936:
5909:
5907:is trivial.
5891:
5873:
5855:
5837:Example: If
5836:
5814:
5739:
5727:
5719:
5710:
5698:
5679:
5625:
5590:
5579:
5573:
5560:
5555:
5551:
5546:
5538:
5534:
5523:
5518:
5508:
5498:
5493:
5448:
5441:
5436:
5427:
5423:
5417:denotes the
5412:
5404:
5399:
5395:
5390:
5385:
5380:
5370:
5363:
5358:
5354:
5349:
5327:
5319:
5310:
5295:for a prime
5290:
5286:
5282:
5275:
5264:
5260:
5064:
5057:
5051:
5038:
5034:
5030:
5019:
5013:
5004:
4999:
4983:
4976:
4971:
4967:
4950:
4946:
4935:
4931:
4920:
4916:
4909:
4898:
4894:
4879:
4874:
4870:
4857:
4830:
4824:
4814:
4809:
4800:
4776:
4772:
4766:
4759:
4746:
4735:
4731:
4722:
4718:
4714:
4704:
4697:
4692:
4680:
4676:
4672:
4666:
4662:
4658:
4648:
4634:
4627:
4615:
4610:
4606:
4593:
4581:
4519:
4406:
4399:
4393:
4386:
4380:
4373:
4366:
4360:
4357:
4124:
4112:
4096:
3925:
3919:
3915:
3898:
3887:
3872:
3863:
3851:
3844:
3840:
3829:
3824:
3815:
3810:
3797:
3791:
3786:
3666:
3661:
3656:
3650:
3635:
3625:
3622:
3489:
3478:
3472:
3465:
3460:
3452:
3448:
3441:
3436:
3431:
3426:
3421:
3285:
3281:
3277:
3274:
3211:fiber bundle
3204:
3194:
3188:
3184:
3177:
3168:
3158:
3152:
3148:
3141:
3136:
3132:
3123:
3119:
3112:
3106:
3102:
3093:
3088:
3084:
3071:
3063:
3055:
3048:
3042:
3033:
3028:
3020:
3016:
3012:= 2, 3, 4, 5
3009:
3000:
2996:
2989:
2982:
2977:
2972:
2967:
2949:
2945:
2942:
2909:
2902:
2897:
2892:
2825:
2750:
2668:
2577:
2453:
2342:
2231:
2179:
2035:
2031:(written as
2014:
2003:
1999:
1992:
1987:
1979:
1975:
1968:
1963:
1959:
1956:
1949:
1944:
1940:
1936:
1931:
1922:
1911:
1906:
1902:
1895:
1888:
1883:
1876:
1859:
1855:
1852:
1842:
1832:
1828:
1809:
1804:
1791:
1771:
1766:
1762:
1752:Guozhen Wang
1740:J. Peter May
1732:Hiroshi Toda
1721:
1715:
1705:
1700:
1658:
1643:
1633:
1621:
1602:
1596:
1582:
1578:
1575:
1556:
1542:
1536:
1526:
1516:
1509:
1500:
1493:
1474:
1469:
1461:
1457:
1454:
1434:
1419:
1400:
1395:
1386:
1383:
1372:
1364:
1349:
1334:
1318:
1283:
1270:
1263:
1256:Elements of
1244:
1232:
1195:
1184:
1160:
1149:
1145:
1133:
1128:
1119:
1102:
1098:
1071:
1067:
1059:
1055:
1039:
1027:
995:
963:
957:
950:
944:
932:
925:
913:
901:
894:
884:
877:
865:
853:
835:
825:
819:
804:line segment
798:
792:
786:
782:
759:
747:
741:
722:
710:
704:
691:
687:
682:
669:
662:
645:
638:
626:
614:
601:
578:
566:
554:
549:
537:An ordinary
536:
509:
489:
482:
478:
461:
455:
451:
440:
435:
431:
417:
412:
408:
403:
396:
390:
380:
376:
361:
356:
344:
338:
334:
321:
315:
309:
303:
297:
293:
273:
268:
263:
245:
241:-dimensional
228:
217:
212:
206:
196:
182:
172:
162:
147:
98:
91:mathematical
88:
65:
56:
37:
11230:Hopf, Heinz
10790:2433/220059
10505:: 107β243,
10033:. See also
9972:Toda (1962)
9452:See, e.g.,
6795:2 − 2
6769:The groups
6762:2 − 2
6596:h-cobordism
6469:smooth maps
6430:has a zero.
6014:-components
5820:graded ring
5480:trivial if
4099:quaternions
1736:Frank Adams
1661:Eduard Δech
1610:. This map
1273:rubber band
51:introducing
11387:Categories
11375:2007-11-14
11354:2007-11-14
11338:2007-10-20
11316:2007-10-09
11305:Baez, John
11087:1809.09290
11038:1601.02184
10512:2001.04511
10043:(3): 331,
9787:Cohen 1968
9748:Adams 1966
9712:Serre 1952
9676:Serre 1951
9427:, p.
9318:References
9310:3β
25β
11β
41
8748:− 1)
6956:isomorphic
6925:direct sum
6580:4-manifold
6567:, implies
6428:polynomial
6326:question.
6117:Ext groups
6063:loop space
5987:cohomology
5742:direct sum
5692:Adams 1966
5473:to 0 or 1
5330:− 1)
3909:cover the
3879:| = 1
3631:suspension
1916:. For the
1838:surjective
1592:Heinz Hopf
1294:isomorphic
1214:isomorphic
839:to be the
812:CW complex
506:Background
103:dimensions
34:references
11270:(1999a),
11260:123533891
11177:(1999b),
11106:119303902
11063:119147703
10938:0764-4442
10799:0025-5645
10694:EMS Press
10430:EMS Press
10418:(2001) ,
10366:0386-2194
10203:0764-4442
10167:0764-4442
10074:CiteSeerX
9984:Fuks 2001
9775:Toda 1962
9736:Fuks 2001
9611:May 1999a
9596:Δech 1932
9548:Hopf 1931
9465:π
9186:9β
7β
11β
31
9084:16β
2β
9β
5
8229:504β
24β
2
8159:24β
12β
4β
2
8005:240β
24β
4
6673:π
6669:→
6636:Θ
6573:signature
6571:that the
6554:− 1
6547:− 1
6016:for each
5824:nilpotent
5789:π
5780:≥
5773:⨁
5759:∗
5755:π
5471:congruent
5383: : Ο
5299:then the
5229:−
5197:π
5193:⊕
5173:−
5151:−
5135:π
5085:π
4961:component
4801:In 1951,
4783:RenΓ© Thom
4684:a framed
4596:cobordism
4554:→
4541:↪
4480:π
4476:⊕
4451:π
4447:≠
4422:π
4376:= 1, 2, 3
4319:−
4312:π
4308:⊕
4283:π
4254:π
4206:−
4199:π
4195:⊕
4170:π
4141:π
4103:octonions
4072:→
4059:↪
4016:→
4003:↪
3957:→
3944:↪
3748:−
3741:π
3737:⊕
3712:π
3683:π
3605:→
3581:−
3574:π
3570:→
3545:π
3541:→
3516:π
3512:→
3404:⋯
3401:→
3384:−
3377:π
3373:→
3355:π
3351:→
3333:π
3329:→
3311:π
3307:→
3304:⋯
3247:→
3234:↪
2027:of order
1853:The case
1756:Zhouli Xu
1728:JosΓ© Adem
1612:generates
1368:injective
1292:, and is
1202:bijection
1158:equal to
1002:open sets
948:produces
899:)-sphere
897:− 1
790:produces
607:dimension
472:, called
111:algebraic
93:field of
11232:(1931),
11141:(2002),
10956:(1962),
10827:(2003),
10704:(2011),
10654:(1998),
10621:(1995),
10464:(1269),
10440:(1959),
10312:16591550
10037:Topology
10015:Topology
9038:4β
2β
3β
5
8761:via the
8750:divides
8694:ββ
264β
2
8676:264β
4β
2
8510:ββ
480β
2
8241:480β
4β
2
8216:120β
12β
2
7992:2520β
6β
2
7757:120β
12β
2
6935:have no
6534:−1
6440:−1
6332:Brunnian
5973:⟩
5961:⟨
5889:, while
5516:, where
5409:, where
5041:+ 1) β 3
4717: :
4712:of maps
4661: :
1652:and the
1650:homology
1588:3-sphere
1465:, then
1442:subgroup
1302:integers
1216:for all
1142:-sphere
1014:homotopy
851:-sphere
737:boundary
543:Geometry
516:topology
371:mapping.
368:integers
251:deformed
186:). The
11398:Spheres
11203:1702278
11167:1867354
11124:3204653
11055:3702672
11015:2045823
10980:0143217
10946:0046048
10917:0045386
10909:1969485
10880:2136212
10849:0860042
10807:0341485
10756:1326604
10676:1648096
10645:1320997
10606:1052407
10567:0148075
10559:1970128
10490:4046815
10450:0106454
10409:1454356
10374:2279281
10294:0231377
10286:1970586
10257:0554384
10249:1971238
10211:0046046
10175:0046045
10137:2188127
10096:0810962
10002:Sources
9425:Hu 1959
9255:β
8β
4β
2β
9228:β
4β
2β
3
9064:β
4β
2β
3
9046:3β
25β
11
9028:27β
7β
19
8639:24β
8β
2
8278:1056β
8
8223:24β
6β
2
8017:48β
4β
2
7936:24β
6β
2
6992:) = ZΓZ
6972:Example
6886:integer
6821:of the
6741:is the
6425:complex
6028:-groups
5620:
5608:
5604:
5587:
5513:
5490:
5451:(mod 8)
5261:(where
4938:β 3 + 1
4890:torsion
4884:has no
4791:spectra
3906:
3884:
3643:by the
1973:, with
1893:, with
1630:History
1312:in the
1170:abelian
841:equator
774:balloon
772:. If a
658:-sphere
533:-sphere
522:, with
492:< 20
292:0 <
243:sphere
232:can be
89:In the
47:improve
11286:
11258:
11201:
11191:
11165:
11155:
11122:
11104:
11061:
11053:
11013:
11003:
10978:
10968:
10944:
10936:
10915:
10907:
10878:
10868:
10847:
10837:
10805:
10797:
10754:
10744:
10674:
10643:
10633:
10604:
10594:
10565:
10557:
10488:
10478:
10448:
10407:
10397:
10372:
10364:
10310:
10303:224450
10300:
10292:
10284:
10255:
10247:
10209:
10201:
10173:
10165:
10135:
10094:
10076:
9302:4β
2β
3
9296:4β
2β
5
9284:4β
2β
3
9204:3β
5β
17
9152:3β
5β
29
9099:5β
7β
13
9077:8β
3β
23
9073:8β
2β
3
8993:3β
5β
17
8944:5β
7β
13
8838:16β
3β
5
8728:> 5
8697:264β
2
8691:264β
2
8688:264β
2
8685:264β
2
8682:264β
2
8679:264β
2
8673:264β
2
8645:8β
4β
2
8525:480β
2
8522:480β
2
8519:480β
2
8516:480β
2
8513:480β
2
8507:480β
2
8504:480β
2
8501:480β
2
8296:264β
2
8293:264β
2
8290:264β
6
8287:264β
2
8284:264β
2
8281:264β
2
8275:264β
2
8272:132β
2
8269:132β
2
8238:8β
4β
2
8121:240β
2
8109:504β
2
8072:240β
2
8069:240β
2
8066:240β
2
8063:240β
2
8060:120β
2
8057:120β
2
7840:ββ
504
7831:504β
2
7828:504β
2
7825:504β
2
7822:504β
4
7819:504β
2
7594:ββ
120
6711:where
6417:) = Z)
6362:> 2
6263:, and
5871:, and
5826:; the
5475:modulo
5393:)) β Ο
5368:, for
5285:< 2
4897:< 2
4710:degree
4371:, for
3861:|
3787:Since
3623:Since
2912:> 1
1997:, for
1898:> 0
1875:space
1814:, the
1754:, and
1389:> 0
1374:degree
1310:origin
1086:curves
1044:of a (
924:+ β― +
876:+ β― +
847:, the
675:circle
637:+ β― +
539:sphere
350:degree
234:mapped
178:sphere
168:circle
154:sphere
97:, the
36:, but
11333:(PDF)
11256:S2CID
11102:S2CID
11082:arXiv
11059:S2CID
11033:arXiv
10905:JSTOR
10709:(PDF)
10555:JSTOR
10507:arXiv
10344:arXiv
10282:JSTOR
10245:JSTOR
9323:Notes
9308:β
4β
2β
9290:β
2β
9β
9275:β
2β
3
9237:β
4β
2β
9222:β
4β
2
9202:β
4β
2β
9132:β
8β
2β
9123:β
2β
3
9117:β
4β
2
9093:β
4β
2β
9044:β
2β
3β
8938:β
8β
2β
8869:8β
9β
7
8547:24β
2
8458:16β
2
8266:12β
2
8235:24β
2
8232:24β
2
8226:24β
2
8220:24β
2
8212:12β
2
8209:12β
2
8155:12β
2
8054:60β
6
8051:30β
2
8014:16β
2
8011:16β
2
8008:16β
4
8002:24β
4
7999:12β
2
7904:below
7868:84β
2
7816:84β
2
7813:84β
2
7810:12β
2
7776:12β
2
7773:24β
2
7770:24β
2
7767:24β
2
7764:72β
2
7761:72β
2
7753:12β
2
7646:24β
2
7524:24β
3
7350:ββ
12
6984:) = Ο
6446:) = Z
6368:over
6366:braid
6212:find.
6044:1952b
6040:1952a
6018:prime
5934:with
5532:, if
5528:is a
5033:<
4762:) = Z
4749:) = Ξ©
4700:) = Z
4364:with
3859:with
3659:) β Ο
3641:split
3629:is a
3434:) β Ο
3209:is a
3151:>
3146:with
3122:>
3051:) = 0
2975:) = Ο
2948:>
2011:Table
2002:>
1978:>
1952:) = Z
1831:<
1812:) = 0
1692:used
1624:) = Z
1581:>
1559:) = Z
1512:) = 0
1496:) = 0
1477:) = 0
1460:<
1422:) = 0
1407:below
1405:(see
1403:) = Z
1352:) = Z
1337:) = Z
1314:plane
1247:) = Z
1222:plane
1111:group
1080:into
802:is a
778:slash
673:is a
481:<
379:>
332:When
296:<
11284:ISBN
11189:ISBN
11153:ISBN
11001:ISBN
10966:ISBN
10934:ISSN
10866:ISBN
10835:ISBN
10795:ISSN
10742:ISBN
10631:ISBN
10592:ISBN
10476:ISBN
10395:ISBN
10362:ISSN
10308:PMID
10199:ISSN
10163:ISSN
9299:4β
2
9249:4β
2
9243:2β
3
9231:8β
2
9196:2β
3
9144:4β
2
9138:2β
3
9126:2β
3
9087:2β
3
9035:2β
3
9032:2β
3
9020:4β
2
8985:2β
3
8971:2β
3
8932:2β
2
8926:8β
3
8922:3β
11
8914:8β
2
8865:2β
3
8654:8β
2
8651:8β
2
8648:8β
2
8642:8β
2
8636:8β
2
8633:8β
2
8630:8β
2
8602:ββ
2
8464:4β
2
8461:8β
2
8418:ββ
3
8172:6β
2
8163:4β
2
8152:6β
2
8103:6β
2
8100:6β
2
7996:6β
2
7960:6β
2
7957:6β
2
7948:6β
2
7939:6β
2
7880:240
7843:504
7837:504
7834:504
7779:6β
2
7716:ββ
2
7609:240
7606:240
7603:240
7600:240
7597:240
7591:120
7095:<
6986:9+10
6931:and
6809:The
6785:The
6577:spin
6458:ball
6403:The
6357:for
6106:has
6095:The
6057:The
5944:and
5918:and
5914:and
5740:The
5717:for
5440:(SO(
5389:(SO(
4914:and
4117:and
3820:and
3098:to
3053:for
3014:and
3007:For
2987:for
1939:) =
1710:and
1568:The
1172:and
1008:. A
701:disk
518:and
449:for
290:For
205:-th
148:The
132:The
11358:in
11248:doi
11244:104
11092:doi
11078:226
11043:doi
11029:186
10993:doi
10930:234
10897:doi
10785:hdl
10775:doi
10734:doi
10584:doi
10547:doi
10517:doi
10503:137
10466:doi
10462:262
10387:doi
10354:doi
10298:PMC
10274:doi
10237:doi
10233:110
10195:234
10159:234
10123:doi
10084:doi
10045:doi
10024:doi
9970:in
9429:107
9281:β
2
9265:72+
9212:64+
9200:128
9184:β
2β
9175:β
2
9160:56+
9150:β
3β
9107:48+
9097:β
3β
9070:β
2
9054:40+
9026:β
2β
9017:β
2
9011:β
2
9001:32+
8991:β
2β
8975:8β
3
8968:β
2
8962:β
2
8952:24+
8942:β
3β
8920:β
2β
8911:β
2
8905:β
2
8895:16+
8887:3β
5
8885:β
2β
8862:β
2
8856:β
2
8824:8β
3
8755:+ 1
8742:if
8661:19+
8618:18+
8575:17+
8532:16+
8489:15+
8446:14+
8403:13+
8360:12+
8248:19+
8191:18+
8134:17+
8097:30
8079:16+
8048:30
8045:30
8042:30
8024:15+
7988:30
7967:14+
7912:13+
7902:See
7892:12
7850:12+
7792:11+
7732:10+
7634:15
7588:60
7585:30
7582:15
7579:15
7402:12
7377:24
7374:24
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