In differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one (hence the name "exotic").
The first exotic spheres were constructed by John Milnor (1956) in dimension as -bundles over . He showed that there are at least 7 differentiable structures on the 7-sphere. In any dimension Milnor (1959) showed that the diffeomorphism classes of oriented exotic spheres form the non-trivial elements of an abelian monoid under connected sum, which is a finite abelian group if the dimension is not 4. The classification of exotic spheres by Michel Kervaire and Milnor (1963) showed that the oriented exotic 7-spheres are the non-trivial elements of a cyclic group of order 28 under the operation of connected sum.
The unit n-sphere, , is the set of all (n+1)-tuples of real numbers, such that the sum . ( is a circle; is the surface of an ordinary ball of radius one in 3 dimensions.) Topologists consider a space, X, to be an n-sphere if every point in X can be assigned to exactly one point in the unit n-sphere in a continuous way, which means that sufficiently nearby points in X get assigned to nearby points in Sn and vice versa. For example, a point x on an n-sphere of radius r can be matched with a point on the unit n-sphere by adjusting its distance from the origin by .
In differential topology, a more stringent condition is added, that the functions matching points in X with points in should be smooth, that is they should have derivatives of all orders everywhere. To calculate derivatives, one needs to have local coordinate systems defined consistently in X. Mathematicians were surprised in 1956 when Milnor showed that consistent coordinate systems could be set up on the 7-sphere in two different ways that were equivalent in the continuous sense, but not in the differentiable sense. Milnor and others set about trying to discover how many such exotic spheres could exist in each dimension and to understand how they relate to each other. No exotic structures are possible on the 1-, 2-, 3-, 5-, 6- or 12-spheres. Some higher-dimensional spheres have only two possible differentiable structures, others have thousands. Whether exotic 4-spheres exist, and if so how many, is an unsolved problem.
The monoid of smooth structures 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 , this monoid is a group and is isomorphic to the group of h-cobordism classes of oriented 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 Gluck twists. 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.
The group has a cyclic subgroup
represented by n-spheres that bound parallelizable manifolds. The structures of and the quotient
are described separately in the paper (Kervaire & Milnor 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 here.
The group is a cyclic group, and is trivial or order 2 except in case , in which case it can be large, with its order related to the Bernoulli numbers. 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 William Browder (1969) proved that it has order 2 if n = 1 mod 4 is not of the form 2k – 3. It follows from the now almost completely resolved Kervaire invariant problem that it has order 2 for all n bigger than 125; the case n = 125 is still open. The order of bP4k for k ≥ 2 is
where B is the numerator of |4B2k/k|, and B2k 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 Θn/bPn+1 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
where πnS is the nth stable homotopy group of spheres, and J is the image of the J-homomorphism. As with bPn+1, the image of J is a cyclic group, and is trivial or order 2 except in case in which case it can be large, with its order related to the Bernoulli numbers. The quotient group is the "hard" part of the stable homotopy groups of spheres, and accordingly 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 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.
As of 2012[update], the Kervaire invariant problem is almost completely solved, with only the case n = 126 remaining open; see that article for details. This is primarily the work of Browder (1969), which proved that such manifolds only existed in dimension n = 2j − 2, and Hill, Hopkins & Ravenel (2016), which proved that there were no such manifolds for dimension 254 = 28 − 2 and above. Manifolds with Kervaire invariant 1 have been constructed in dimension 2, 6, 14, 30, and 62, but dimension 126 is open, with no manifold being either constructed or disproven.
Order of Θn
The order of the group Θn is given in this table (sequence A001676 in the OEIS) from (Kervaire & Milnor 1963) (except that the entry for n = 19 is wrong by a factor of 2 in their paper; see the correction in volume III p. 97 of Milnor's collected works).
Dim n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 order Θn 1 1 1 1 1 1 28 2 8 6 992 1 3 2 16256 2 16 16 523264 24 bPn+1 1 1 1 1 1 1 28 1 2 1 992 1 1 1 8128 1 2 1 261632 1 Θn/bPn+1 1 1 1 1 1 1 1 2 2×2 6 1 1 3 2 2 2 2×2×2 8×2 2 24 πnS/J 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 n = 4k-1, then Θn are 28 = 22(23-1), 992 = 25(25-1), 16256 = 27(27-1), and 523264 = 210(29-1). Further entries in this table can be computed from the information above together with the table of stable homotopy groups of spheres.
Explicit examples of exotic spheres
John Milnor (2009, p.12)
One of the first examples of an exotic sphere found by Milnor (1956, section 3) was the following: Take two copies of B4×S3, each with boundary S3×S3, and glue them together by identifying (a,b) in the boundary with (a, a2ba−1), (where we identify each S3 with the group of unit quaternions). The resulting manifold has a natural smooth structure and is homeomorphic to S7, but is not diffeomorphic to S7. Milnor showed that it is not the boundary of any smooth 8-manifold with vanishing 4th Betti number, and has no orientation-reversing diffeomorphism to itself; either of these properties implies that it is not a standard 7-sphere. Milnor showed that this manifold has a Morse function with just two critical points, both non-degenerate, which implies that it is topologically a sphere.
with a small sphere around the origin for k = 1, 2, ..., 28 gives all 28 possible smooth structures on the oriented 7-sphere. Similar manifolds are called Brieskorn spheres.
Given an (orientation-preserving) diffeomorphism f : Sn−1 → Sn−1, gluing the boundaries of two copies of the standard disk Dn together by f yields a manifold called a twisted sphere (with twist f). It is homotopy equivalent to the standard n-sphere because the gluing map is homotopic to the identity (being an orientation-preserving diffeomorphism, hence degree 1), but not in general diffeomorphic to the standard sphere. (Milnor 1959b) Setting to be the group of twisted n-spheres (under connect sum), one obtains the exact sequence
For n > 5, every exotic n-sphere is diffeomorphic to a twisted sphere, a result proven by Stephen Smale which can be seen as a consequence of the h-cobordism theorem. (In contrast, in the piecewise linear setting the left-most map is onto via radial extension: every piecewise-linear-twisted sphere is standard.) The group Γn of twisted spheres is always isomorphic to the group Θn. The notations are different because it was not known at first that they were the same for n = 3 or 4; for example, the case n = 3 is equivalent to the Poincaré conjecture.
In 1970 Jean Cerf proved the pseudoisotopy theorem which implies that is the trivial group provided , so provided .
If M is a piecewise linear manifold then the problem of finding the compatible smooth structures on M depends on knowledge of the groups Γk = Θk. More precisely, the obstructions to the existence of any smooth structure lie in the groups Hk+1(M, Γk) for various values of k, while if such a smooth structure exists then all such smooth structures can be classified using the groups Hk(M, Γk). In particular the groups Γk vanish if k < 7, so all PL manifolds of dimension at most 7 have a smooth structure, which is essentially unique if the manifold has dimension at most 6.
The following finite abelian groups are essentially the same:
- The group Θn of h-cobordism classes of oriented homotopy n-spheres.
- The group of h-cobordism classes of oriented n-spheres.
- The group Γn of twisted oriented n-spheres.
- The homotopy group πn(PL/DIFF)
- If n ≠ 3, the homotopy πn(TOP/DIFF) (if n = 3 this group has order 2; see Kirby–Siebenmann invariant).
- The group of smooth structures of an oriented PL n-sphere.
- If n ≠ 4, the group of smooth structures of an oriented topological n-sphere.
- If n ≠ 5, the group of components of the group of all orientation-preserving diffeomorphisms of Sn−1.
4-dimensional exotic spheres and Gluck twists
In 4 dimensions it is not known whether there are any exotic smooth structures on the 4-sphere. The statement that they do not exist is known as the "smooth Poincaré conjecture", and is discussed by Michael Freedman, Robert Gompf, and Scott Morrison et al. (2010) who say that it is believed to be false.
Some candidates proposed for exotic 4-spheres are the Cappell–Shaneson spheres (Sylvain Cappell and Julius Shaneson (1976)) and those derived by Gluck twists (Gluck 1962). Gluck twist spheres are constructed by cutting out a tubular neighborhood of a 2-sphere S in S4 and gluing it back in using a diffeomorphism of its boundary S2×S1. The result is always homeomorphic to S4. Many cases over the years were ruled out as possible counterexamples to the smooth 4 dimensional Poincaré conjecture. For example, Cameron Gordon (1976), José Montesinos (1983), Steven P. Plotnick (1984), Gompf (1991), Habiro, Marumoto & Yamada (2000), Selman Akbulut (2010), Gompf (2010), Kim & Yamada (2017).
- Akbulut, Selman (2010), "Cappell–Shaneson homotopy spheres are standard", Annals of Mathematics, 171 (3): 2171–2175, arXiv:0907.0136, doi:10.4007/annals.2010.171.2171
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- Gluck, Herman (1962), "The embedding of two-spheres in the four-sphere", Transactions of the American Mathematical Society, 104 (2): 308–333, doi:10.2307/1993581, JSTOR 1993581, MR 0146807
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- Gompf, Robert E (2010), "More Cappell-Shaneson spheres are standard", Algebraic & Geometric Topology, 10 (3): 1665–1681, arXiv:0908.1914, doi:10.2140/agt.2010.10.1665
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- Hill, Michael A.; Hopkins, Michael J.; Ravenel, Douglas C. (2016) [First published as arXiv 2009]. "On the non-existence of elements of Kervaire invariant one". Annals of Mathematics. 184 (1): 1–262. arXiv:0908.3724. doi:10.4007/annals.2016.184.1.1.
- Hirzebruch, Friedrich; Mayer, Karl Heinz (1968), O(n)-Mannigfaligkeiten, Exotische Sphären und Singularitäten, Lecture Notes in Mathematics, 57, Berlin-New York: Springer-Verlag, doi:10.1007/BFb0074355, ISBN 978-3-540-04227-3, MR 0229251 This book describes Brieskorn's work relating exotic spheres to singularities of complex manifolds.
- Kervaire, Michel A.; Milnor, John W. (1963). "Groups of homotopy spheres: I" (PDF). Annals of Mathematics. 77 (3): 504–537. doi:10.2307/1970128. JSTOR 1970128. MR 0148075. – This paper describes the structure of the group of smooth structures on an n-sphere for n > 4. Sadly, the promised paper "Groups of Homotopy Spheres: II" never appeared, but Levine's lecture notes contain the material which it might have been expected to contain.
- Kim, Min Hoon; Yamada, Shohei (2017), Ideal classes and Cappell-Shaneson homotopy 4-spheres, arXiv:1707.03860v1
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- Milnor, John W. (1956), "On manifolds homeomorphic to the 7-sphere", Annals of Mathematics, 64 (2): 399–405, doi:10.2307/1969983, JSTOR 1969983, MR 0082103
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- Milnor, John Willard (2009), "Fifty years ago: topology of manifolds in the 50's and 60's" (PDF), in Mrowka, Tomasz S.; Ozsváth, Peter S. (eds.), Low dimensional topology. Lecture notes from the 15th Park City Mathematics Institute (PCMI) Graduate Summer School held in Park City, UT, Summer 2006., IAS/Park City Math. Ser., 15, Providence, R.I.: American Mathematical Society, pp. 9–20, ISBN 978-0-8218-4766-4, MR 2503491
- Milnor, John W. (2011), "Differential topology forty-six years later" (PDF), Notices of the American Mathematical Society, 58 (6): 804–809
- Montesinos, José M. (1983), "On twins in the four-sphere I", The Quarterly Journal of Mathematics, 34 (6): 171–199, doi:10.1093/qmath/34.2.171
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- Exotic spheres on the Manifold Atlas
- Exotic sphere home page on the home page of Andrew Ranicki. Assorted source material relating to exotic spheres.