Exotic sphere

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 n = 7 as S³-bundles over S⁴. 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 John 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.

Introduction

The unit n-sphere, Sⁿ, is the set of all (n+1)-tuples (x₁, x₂, ... x_n+1) of real numbers, such that the sum x₁² + x₂² + ... + x_n+1² = 1. (S¹ is a circle; S² 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 1/r.

In differential topology, a more stringent condition is added, that the functions matching points in X with points in Sⁿ 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 John 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.

Classification

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 n ≠ 4, this monoid is a group and is isomorphic to the group Θ_n 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.

Parallelizable manifolds

The group Θ_n has a cyclic subgroup

bP_{n+1}\

represented by n-spheres that bound parallelizable manifolds. The structures of bP_n+1 and the quotient

\Theta_n/bP_{n+1}\

are described separately in the paper (Michel Kervaire & John 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 bP_n+1 is a cyclic group, and is trivial or order 2 except in case n = 4k+3, 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 Browder (1969) proved that it has order 2 if n = 1 mod 4 is not of the form 2^k – 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 bP_4k for k ≥ 2 is

2^{2k-2}(2^{2k-1}-1)B\,\!

where B is the numerator of |4B_2k/k|, and B_2k 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 Θ_n/bP_n+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

\Theta_n/bP_{n+1}\to \pi_n^S/J\,

where π_n^S is the nth stable homotopy group of spheres, and J is the image of the J-homomorphism. As with bP_n+1, the image of J is a cyclic group, and is trivial or order 2 except in case $n = 4k+3,$ in which case it can be large, with its order related to the Bernoulli numbers. The quotient group $\pi_n^S/J\,$ is the "hard" part of the stable homotopy groups of spheres, and accordingly $\Theta_n/bP_{n+1}$ 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, 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 = 2^j − 2, and Hill, Hopkins & Ravenel (2009), 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 Θ_n

The order of the group Θ_n 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 Θ_n	1	1	1	1	1	1	28	2	8	6	992	1	3	2	16256	2	16	16	523264	24
bP_n+1	1	1	1	1	1	1	28	1	2	1	992	1	1	1	8128	1	2	1	261632	1
Θ_n/bP_n+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
π_n^S/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 Θ_n 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

When I came upon such an example in the mid-50s, I was very puzzled and didn't know what to make of it. At first, I thought I'd found a counterexample to the generalized Poincaré conjecture in dimension seven. But careful study showed that the manifold really was homeomorphic to S⁷. Thus, there exists a differentiable structure on S⁷ not diffeomorphic to the standard one.

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.

As shown by Egbert Brieskorn (1966, 1966b) (see also (Hirzebruch & Mayer 1968)) the intersection of the complex manifold of points in C⁵ satisfying

a^2 + b^2 + c^2 + d^3 + e^{6k-1} = 0\

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.

Twisted 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 $\Gamma_n$ to be the group of twisted n-spheres (under connect sum), one obtains the exact sequence

\pi _{0}\,{\text{Diff}}^{+}(D^{n})\to \pi _{0}\,{\text{Diff}}^{+}(S^{n-1})\to \Gamma _{n}\to 0.

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 Γ_n of twisted spheres is always isomorphic to the group Θ_n. 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 $\pi_0\,\text{Diff}^+(D^n)$ is the trivial group provided $n \geq 6$ , so $\Gamma_n \simeq \pi_0\,\text{Diff}^+(S^{n-1})$ provided $n \geq 6$ .

Applications

If M is a piecewise linear manifold then the problem of finding the compatible smooth structures on M depends on knowledge of the groups Γ_k = Θ_k. More precisely, the obstructions to the existence of any smooth structure lie in the groups H_k+1(M, Γ_k) for various values of k, while if such a smooth structure exists then all such smooth structures can be classified using the groups H_k(M, Γ_k). In particular the groups Γ_k 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 Θ_n of h-cobordism classes of oriented homotopy n-spheres.
The group of h-cobordism classes of oriented n-spheres.
The group Γ_n of twisted oriented n-spheres.
The homotopy group π_n(PL/DIFF)
If n ≠ 3, the homotopy π_n(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 for exotic 4-spheres are given by Gluck twists (Gluck 1962). These 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. But in most cases it is unknown whether or not the result is diffeomorphic to S⁴. (If the 2-sphere is unknotted, or given by spinning a knot in the 3-sphere, then the Gluck twist is known to be diffeomorphic to S⁴, but there are plenty of other ways to knot a 2-sphere in S⁴.)

Akbulut (2009) showed that a certain family of candidates for 4-dimensional exotic spheres constructed by Cappell and Shaneson are in fact standard.

References

Akbulut, Selman (2009), Cappell–Shaneson homotopy spheres are standard, arXiv:0907.0136 
Brieskorn, Egbert V. (1966), "Examples of singular normal complex spaces which are topological manifolds", Proceedings of the National Academy of Sciences, 55 (6): 1395–1397, MR 0198497, PMC 224331 , PMID 16578636, doi:10.1073/pnas.55.6.1395
Brieskorn, Egbert (1966b), "Beispiele zur Differentialtopologie von Singularitäten", Invent. Math., 2 (1): 1–14, MR 0206972, doi:10.1007/BF01403388
Browder, William (1969), "The Kervaire invariant of framed manifolds and its generalization", Annals of Mathematics, 90 (1): 157–186, JSTOR 1970686, MR 0251736, doi:10.2307/1970686
Freedman, Michael; Gompf, Robert; Morrison, Scott; Walker, Kevin (2010), "Man and machine thinking about the smooth 4-dimensional Poincaré conjecture", Quantum Topology, 1 (2): 171–208, arXiv:0906.5177 , doi:10.4171/qt/5
Gluck, Herman (1962), "The embedding of two-spheres in the four-sphere", Transactions of the American Mathematical Society, 104 (2): 308–333, JSTOR 1993581, MR 0146807, doi:10.2307/1993581
Hirzebruch, Friedrich; Mayer, Karl Heinz (1968), O(n)-Mannigfaligkeiten, Exotische Sphären und Singularitäten, Lecture Notes in Mathematics, 57, Berlin-New York: Springer-Verlag, ISBN 978-3-540-04227-3, MR 0229251, doi:10.1007/BFb0074355 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. Princeton University Press. 77 (3): 504–537. JSTOR 1970128. MR 0148075. doi:10.2307/1970128. – 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.
Levine, J.P. (1985), "Lectures on groups of homotopy spheres", Algebraic and geometric topology, Lecture Notes in Mathematics, 1126, Berlin-New York: Springer-Verlag, pp. 62–95, ISBN 978-3-540-15235-4, MR 8757031, doi:10.1007/BFb0074439
Milnor, John W. (1956), "On manifolds homeomorphic to the 7-sphere", Annals of Mathematics, 64 (2): 399–405, JSTOR 1969983, MR 0082103, doi:10.2307/1969983
Milnor, John W. (1959), "Sommes de variétes différentiables et structures différentiables des sphères", Bulletin de la Société Mathématique de France, 87: 439–444, MR 0117744
Milnor, John W. (1959b), "Differentiable structures on spheres", American Journal of Mathematics, 81 (4): 962–972, JSTOR 2372998, MR 0110107, doi:10.2307/2372998
Milnor, John (2000), "Classification of (n − 1)-connected 2n-dimensional manifolds and the discovery of exotic spheres", in Cappell, Sylvain; Ranick, Andrew; Rosenberg, Jonathan, Surveys on Surgery Theory: Volume 1, Annals of Mathematics Studies 145, Princeton University Press, pp. 25–30, ISBN 9780691049380, MR 1747528 .
Milnor, John Willard (2009), "Fifty years ago: topology of manifolds in the 50's and 60's", in Mrowka, Tomasz S.; Ozsváth., Peter S., Low dimensional topology. Lecture notes from the 15th Park City Mathematics Institute (PCMI) Graduate Summer School held in Park City, UT, Summer 2006. (PDF), IAS/Park City Math. Ser., 15, Providence, R.I.: Amer. Math. Soc., 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
Rudyak, Yu.B. (2001) [1994], "Milnor sphere", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

External links

Exotic spheres on the Manifold Atlas
Exotic sphere home page on the home page of Andrew Ranicki. Assorted source material relating to exotic spheres.

An animation of exotic 7-spheres Video from a presenation by Niles Johnson at the Second Abel conference in honor of John Milnor.
The Gluck construction on the Manifold Atlas

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