Editing Exotic sphere (section)

==Classification==
The monoid of [[smooth structure]]s on ''n''-spheres is the collection of oriented smooth ''n''-manifolds which are homeomorphic to the ''n''-sphere, taken up to orientation-preserving diffeomorphism.  The monoid operation is the [[connected sum]]. Provided <math>n\ne  4</math>,  this monoid is a group and is isomorphic to the group <math>\Theta_n</math>  of [[H-cobordism|''h''-cobordism]] classes of oriented [[homotopy sphere|homotopy ''n''-spheres]], which is finite and abelian. In dimension 4 almost nothing is known about the monoid of smooth spheres, beyond the facts that it is finite or countably infinite, and abelian, though it is suspected to be infinite; see the section on [[#4-dimensional exotic spheres and Gluck twists|Gluck twist]]s. All homotopy ''n''-spheres are homeomorphic to the ''n''-sphere by the [[generalized Poincaré conjecture]], proved by [[Stephen Smale]] in dimensions bigger than 4, [[Michael Freedman]] in dimension 4, and [[Grigori Perelman]] in dimension 3. In dimension 3, [[Edwin E. Moise]] proved that every topological manifold has an essentially unique smooth structure (see [[Moise's theorem]]), so the monoid of smooth structures on the 3-sphere is trivial.

=== Parallelizable manifolds===
The group <math>\Theta_n</math>  has a cyclic subgroup

:<math>bP_{n+1}</math>

represented by ''n''-spheres that bound [[parallelizable manifold]]s. The structures of <math>bP_{n+1}</math> and the quotient

:<math>\Theta_n/bP_{n+1}</math>

are described separately in the paper {{harvs|authorlink1=Michel Kervaire|last1=Kervaire|last2=Milnor|authorlink2=John Milnor|year=1963}}, which was influential in the development of [[surgery theory]]. In fact, these calculations can be formulated in a modern language in terms of the [[surgery exact sequence]] as indicated [[surgery exact sequence#Examples|here]].

The group <math>bP_{n+1}</math> is a cyclic group, and is trivial or order 2 except in case <math>n = 4k+3</math>, in which case it can be large, with its order related to the [[Bernoulli number]]s. It is trivial if ''n'' is even. If ''n'' is 1 mod 4 it has order 1 or 2; in particular it has order 1 if ''n'' is 1, 5,  13,  29, or 61, and {{harvs|txt|first=William|last=Browder|authorlink=William Browder (mathematician)|year=1969}} proved that it has order 2 if <math>n = 1</math> mod 4 is not of the form <math>2^k - 3</math>. It follows from the now almost completely resolved [[Kervaire invariant]] problem that it has order 2 for all ''n'' bigger than 126; the case <math>n = 126</math> is still open. The order of <math>bP_{4k}</math> for <math>k\ge 2</math> is

:<math>2^{2k-2}(2^{2k-1}-1)B,</math>

where ''B'' is the numerator of <math>4B_{2k}/k</math>, and <math>B_{2k}</math> is a [[Bernoulli number]]. (The formula in the topological literature differs slightly because topologists use a different convention for naming Bernoulli numbers; this article uses the number theorists' convention.)

===Map between quotients===
The quotient group <math>\Theta_n/bP_{n+1}</math> has a description in terms of [[stable homotopy groups of spheres]] modulo the image of the [[J-homomorphism]]; it is either equal to the quotient or index 2. More precisely there is an injective map
:<math>\Theta_n/bP_{n+1}\to \pi_n^S/J,</math>
where <math>\pi_n^S</math> is the ''n''th stable homotopy group of spheres, and ''J'' is the image of the ''J''-homomorphism. As with <math>bP_{n+1}</math>, the image of ''J'' is a cyclic group, and is trivial or order 2 except in case <math>n = 4k+3</math>, in which case it can be large, with its order related to the [[Bernoulli number]]s. The quotient group <math>\pi_n^S/J</math> is the "hard" part of the stable homotopy groups of spheres, and accordingly <math>\Theta_n/bP_{n+1}</math> is the hard part of the exotic spheres, but almost completely reduces to computing homotopy groups of spheres. The map is either an isomorphism (the image is the whole group), or an injective map with [[Index of a subgroup|index]] 2. The latter is the case if and only if there exists an ''n''-dimensional framed manifold with [[Kervaire invariant]] 1, which is known as the [[Kervaire invariant problem]]. Thus a factor of 2 in the classification of exotic spheres depends on the Kervaire invariant problem.

The Kervaire invariant problem is almost completely solved, with only the case <math>n=126</math> remaining open, although Zhouli Xu (in collaboration with Weinan Lin and Guozhen Wang), announced during a seminar at Princeton University, on May 30, 2024, that the final case of dimension 126 has been settled and that there exist manifolds of Kervaire invariant 1 in dimension 126.<ref>{{Cite web |title=Computing differentials in the Adams spectral sequence {{!}} Math |url=https://www.math.princeton.edu/events/computing-differentials-adams-spectral-sequence-2024-05-30t170000 |access-date=2025-05-04 |website=www.math.princeton.edu}}</ref> Previous work of {{harvtxt|Browder|1969}}, proved that such manifolds only existed in dimension <math>n=2^j-2</math>, and {{harvtxt|Hill|Hopkins|Ravenel|2016}}, which proved that there were no such manifolds for dimension <math>254=2^8-2</math> and above. Manifolds with Kervaire invariant 1 have been constructed in dimension 2, 6, 14, 30. While it is known that there are manifolds of Kervaire invariant 1 in dimension 62, no such manifold has yet been constructed. Similarly for dimension 126.

===Order of Θ<sub>n</sub>===
The order of the group <math>\Theta_n</math> is given in this table {{OEIS|id=A001676}} from {{harv|Kervaire|Milnor|1963}} (except that the entry for <math>n = 19</math> is wrong by a factor of 2 in their paper; see the correction in volume III p.&nbsp;97 of Milnor's collected works).

:{| class="wikitable" style="text-align:center"
|-
! Dim  n !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 20
|-
! order <math>\Theta_n</math>
| 1 || 1 || 1 || 1 || 1 || 1 || 28 || 2 || 8 || 6 || 992 || 1 || 3 || 2 || 16256 || 2 || 16 || 16 || 523264 || 24
|-
!<math>bP_{n+1}</math>
| 1 || 1 || 1 || 1 || 1 || 1 || 28 || 1 || 2 || 1 || 992 || 1 || 1 || 1 || 8128 || 1 || 2 || 1 || 261632 || 1
|-
!<math>\Theta_n/bP_{n+1}</math>
| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 2 || 2×2 || 6 || 1 || 1 || 3 || 2 || 2 || 2 || 2×2×2 || 8×2 || 2 || 24
|-
!<math>\pi_n^S/J</math> 
| 1 || 2 || 1 || 1 || 1 || 2 || 1 || 2 || 2×2 || 6 || 1 || 1 || 3 || 2×2 || 2 || 2 || 2×2×2 || 8×2 || 2 || 24
|-
!index
| – || 2 || – || – || – || 2 || – || – || – || – || – || – || – || 2 || – || – || – || – || – || –
|}

Note that for dim <math>n = 4k - 1</math>, then <math>\theta_n</math> are <math>28 = 2^2(2^3-1)</math>, <math>992 = 2^5(2^5 - 1)</math>, <math>16256 = 2^7(2^7 - 1) </math>, and <math>523264 = 2^{10}(2^9 - 1)  </math>. Further entries in this table can be computed from the information above together with the table of [[stable homotopy groups of spheres]].

By computations of stable homotopy groups of spheres, {{harvtxt|Wang|Xu|2017}} proves that the sphere {{math|''S''<sup>61</sup>}} has a unique smooth structure, and that it is the last odd-dimensional sphere with this property – the only ones are {{math|''S''<sup>1</sup>}}, {{math|''S''<sup>3</sup>}}, {{math|''S''<sup>5</sup>}}, and {{math|''S''<sup>61</sup>}}.