Exotic sphere

Short description: Smooth manifold that is homeomorphic but not diffeomorphic to a sphere

It has been suggested that Milnor's sphere be merged into this article. (Discuss) Proposed since March 2026.

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In an area of mathematics called 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 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. These groups are known as Kervaire–Milnor groups.

More generally, in any dimension n ≠ 4, there is a finite Abelian group whose elements are the equivalence classes of smooth structures on Sn, where two structures are considered equivalent if there is an orientation preserving diffeomorphism carrying one structure onto the other. The group operation is defined by [x] + [y] = [x \+ y], where x and y are arbitrary representatives of their equivalence classes, and x \+ y denotes the smooth structure on the smooth Sn that is the connected sum of x and y. It is necessary to show that such a definition does not depend on the choices made; indeed this can be shown.

## Contents

* 1 Introduction
* 2 Classification
* 2.1 Parallelizable manifolds
* 2.2 Map between quotients
* 2.3 Order of Θn
* 3 Explicit examples of exotic spheres
* 3.1 Milnor's construction
* 3.2 Brieskorn spheres
* 4 Twisted spheres
* 5 Applications
* 6 4-dimensional exotic spheres and Gluck twists
* 7 See also
* 8 References
* 9 External links

## Introduction

The unit n-sphere, Sn, is the set of all (n+1)-tuples (x1,x2,…,xn+1) of real numbers, such that the sum x12+x22+⋯+xn+12=1. For instance, S1 is a circle, while 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 there is a homeomorphism between them, i.e. every point in X may be assigned to exactly one point in the unit n-sphere by a continuous bijection with continuous inverse. For example, a point x on an n-sphere of radius r can be matched homeomorphically with a point on the unit n-sphere by multiplying its distance from the origin by 1/r. Similarly, an n-cube of any radius is homeomorphic to an n-sphere.

In differential topology, two smooth manifolds are considered smoothly equivalent if there exists a diffeomorphism from one to the other, which is a homeomorphism between them, with the additional condition that it be smooth — that is, it should have derivatives of all orders at all its points — and its inverse homeomorphism must also be smooth. To calculate derivatives, one needs to have local coordinate systems defined consistently in X. Mathematicians (including Milnor himself) were surprised in 1956 when Milnor showed that consistent local 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-, 12-, 56- or 61-sphere.[1] 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 Θn 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 Θn has a cyclic subgroup

bPn+1

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

Θn/bPn+1

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 bPn+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 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 126; the case n=126 is still open. The order of bP4k for k≥2 is

22k−2(22k−1−1)B,

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

Θn/bPn+1->πnS/J,

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 n=4k+3, in which case it can be large, with its order related to the Bernoulli numbers. The quotient group πnS/J is the "hard" part of the stable homotopy groups of spheres, and accordingly Θn/bPn+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.

The Kervaire invariant problem is almost completely solved, with only the case n=126 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.[2] Previous work of (Browder 1969), proved that such manifolds only existed in dimension n=2j−2, and (Hill Hopkins), 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. 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 Θn

The order of the group Θn is given in this table (sequence A001676 in the OEIS) from (Kervaire Milnor) (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

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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.

By computations of stable homotopy groups of spheres, (Wang Xu) proves that the sphere S61 has a unique smooth structure, and that it is the last odd-dimensional sphere with this property – the only ones are S1, S3, S5, and S61.

## 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)

### Milnor's construction

Main article: Milnor's sphere

One of the first examples of an exotic sphere found by (Milnor 1956) was the following. Let B4 be the unit ball in ℝ4, and let S3 be its boundary—a 3-sphere which we identify with the group of unit quaternions. Now take two copies of B4×S3, each with boundary S3×S3, and glue them together by identifying (a,b) in the first boundary with (a,a2ba−1) in the second boundary. 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.

### Brieskorn spheres

Main article: Brieskorn manifold

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

a2+b2+c2+d3+e6k−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 Γn to be the group of twisted n-spheres (under connect sum), one obtains the exact sequence

π0Diff+(Dn)->π0Diff+(Sn−1)->Γn->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 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 π0Diff+(Dn) is the trivial group provided n≥6, and so Γn≃π0Diff+(Sn−1) provided n≥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 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 group 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), Selman Akbulut (2010), (Gompf 2010), (Kim Yamada).

## See also

* Milnor's sphere
* Gromoll–Meyer sphere, special Milnor sphere
* Atlas (topology)
* Clutching construction
* Exotic R4
* Cerf theory
* Seven-dimensional space

## References

1. ↑ Behrens, M.; Hill, M.; Hopkins, M. J.; Mahowald, M. (2020). "Detecting exotic spheres in low dimensions using coker J" (in en). Journal of the London Mathematical Society 101 (3): 1173–1218. doi:10.1112/jlms.12301. ISSN 1469-7750. https://onlinelibrary.wiley.com/doi/abs/10.1112/jlms.12301.
2. ↑ "Computing differentials in the Adams spectral sequence | Math". https://www.math.princeton.edu/events/computing-differentials-adams-spectral-sequence-2024-05-30t170000.

* Akbulut, Selman (2010), "Cappell–Shaneson homotopy spheres are standard", Annals of Mathematics 171 (3): 2171–2175, doi:10.4007/annals.2010.171.2171
* 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, doi:10.1073/pnas.55.6.1395, PMID 16578636, Bibcode: 1966PNAS...55.1395B
* Brieskorn, Egbert (1966b), "Beispiele zur Differentialtopologie von Singularitäten", Invent. Math. 2 (1): 1–14, doi:10.1007/BF01403388, Bibcode: 1966InMat...2....1B
* Browder, William (1969), "The Kervaire invariant of framed manifolds and its generalization", Annals of Mathematics 90 (1): 157–186, doi:10.2307/1970686
* Cappell, Sylvain E.; Shaneson, Julius L. (1976), "Some new four-manifolds", Annals of Mathematics 104 (1): 61–72, doi:10.2307/1971056
* 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, 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, doi:10.2307/1993581
* Hughes, Mark; Kim, Seungwon; Miller, Maggie (2018), Gluck Twists Of S4 Are Diffeomorphic to S4
* Gompf, Robert E (1991), "Killing the Akbulut-Kirby 4-sphere, with relevance to the Andres-Curtis and Schoenflies problems", Topology 30: 123–136, doi:10.1016/0040-9383(91)90036-4
* Gompf, Robert E (2010), "More Cappell-Shaneson spheres are standard", Algebraic & Geometric Topology 10 (3): 1665–1681, doi:10.2140/agt.2010.10.1665
* Gordon, Cameron McA. (1976), "Knots in the 4-sphere", Commentarii Mathematici Helvetici 51: 585–596, doi:10.1007/BF02568175
* Habiro, Kazuo; Marumoto, Yoshihiko; Yamada, Yuichi (2000), "Gluck surgery and framed links in 4-manifolds", Series on Knots and Everything (World Scientific) 24: 80–93, ISBN 978-9810243401
* Hill, Michael A.; Hopkins, Michael J.; Ravenel, Douglas C. (2016). "On the non-existence of elements of Kervaire invariant one". Annals of Mathematics 184 (1): 1–262. doi:10.4007/annals.2016.184.1.1.
* Hirzebruch, Friedrich; Mayer, Karl Heinz (1968), O(n)-Mannigfaltigkeiten, exotische Sphären und Singularitäten, Lecture Notes in Mathematics, 57, Berlin-New York: Springer-Verlag, doi:10.1007/BFb0074355, ISBN 978-3-540-04227-3 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". Annals of Mathematics 77 (3): 504–537. doi:10.2307/1970128. http://www.uni-math.gwdg.de/schick/publ/Groups of homotopy spheres I.pdf. – This paper describes the structure of the group of smooth structures on an n-sphere for n > 4\. 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
* Levine, Jerome 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, doi:10.1007/BFb0074439, ISBN 978-3-540-15235-4
* Milnor, John W. (1956), "On manifolds homeomorphic to the 7-sphere", Annals of Mathematics 64 (2): 399–405, doi:10.2307/1969983
* Milnor, John W. (1959), "Sommes de variétés différentiables et structures différentiables des sphères", Bulletin de la Société Mathématique de France 87: 439–444, doi:10.24033/bsmf.1538
* Milnor, John W. (1959b), "Differentiable structures on spheres", American Journal of Mathematics 81 (4): 962–972, doi:10.2307/2372998
* Milnor, John (2000), "Classification of (n−1)-connected 2n-dimensional manifolds and the discovery of exotic spheres", in Cappell, Sylvain; Ranicki, Andrew; Rosenberg, Jonathan, Surveys on Surgery Theory: Volume 1, Annals of Mathematics Studies 145, Princeton University Press, pp. 25–30, ISBN 9780691049380 .
* Milnor, John Willard (2009), "Fifty years ago: topology of manifolds in the 50s and 60s", 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., IAS/Park City Math. Ser., 15, Providence, R.I.: American Mathematical Society, pp. 9–20, ISBN 978-0-8218-4766-4, http://www.math.sunysb.edu/~jack/PREPRINTS/pcity-lec.pdf
* Milnor, John W. (2011), "Differential topology forty-six years later", Notices of the American Mathematical Society 58 (6): 804–809, https://www.ams.org/notices/201106/rtx110600804p.pdf
* 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, https://eprints.ucm.es/17187/1/On twins in the four sphere.pdf
* Plotnick, Steven P (1984), Gordon, Cameron McA., ed., Fibered knots in S4 – twisted, spinning, rolling, surgery, and branching, American Mathematical Society, Contemporary Mathematics Volume 35, pp. 437–459, ISBN 978-0-8218-5033-6 .
* Wang, Guozhen; Xu, Zhouli (2017), "The triviality of the 61-stem in the stable homotopy groups of spheres", Annals of Mathematics 186 (2): 501–580, doi:10.4007/annals.2017.186.2.3 .
* Hazewinkel, Michiel, ed. (2001), "Milnor sphere", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=M/m063800

## 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 presentation by Niles Johnson at the Second Abel conference in honor of John Milnor.
* The Gluck construction on the Manifold Atlas

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