# Exotic spheres

** An exotic sphere is an ***n*-dimensional differentiable manifold which is
homeomorphic but not diffeomorphic to the standard *n*-sphere
*S*^{n}.

The articles on exotic spheres on the
Wikipedia and the
Manifold Atlas Project.

Bulletin of the AMS Volume 52 Number 4

On manifolds homeomorphic to the 7-sphere,
by J.Milnor, Ann. of Math. (2) 64, 399--405 (1956)

Hedrick Lectures on Differential Topology by J. Milnor (1965)

Classification of (n-1)-connected
2n-dimensional manifolds and the discovery of the exotic spheres

by J.Milnor, *Surveys in Surgery Theory Theory*, Vol. 1,
Ann. of Math. Studies 145, 25--30 (2000)

Fifty years ago: topology in the
50's and 60's by J.Milnor (2006)

Spheres Abel Prize Lecture by J. Milnor (2011)

Differential topology 46 years later by J.Milnor, Notices A.M.S. 58, 804--809 (2011)

The tabulation of exotic spheres in the On-Line Encyclopedia of Integer Sequences.

The number
of differential structures on the $n$-sphere, from
the Online Encyclopedia of Integer Sequences

Differentiable manifolds which are homotopy spheres, by J.Milnor,
Princeton notes (1959)

Differentiable structures on spheres, by J.Milnor,
Am. J. Math. 81, 962--972 (1959)

Bernoulli numbers, homotopy groups, and a theorem of Rohlin
by J.Milnor and M.Kervaire, Proc. 1958 Edinburgh ICM, 454--458 (1959)

The Kervaire-Milnor correspondence
(1958--1961). Transcription.

A manifold which does not admit any differentiable structure,
by M.Kervaire, Comm. Math. Helv. 34, 257--270 (1960)

A procedure for killing homotopy groups of differentiable manifolds,
by J.Milnor, Proc. Sympos. Pure Math., Vol. III, 39--55, A.M.S. (1961)

Differentiable structures on the 15-sphere
and Pontrjagin classes of certain manifolds, by N.Shimada,
Nagoya Math. J. 12, 59--69 (1957)

8-manifolds admitting no differentiable structure,
by I.Tamura, J. Math. Soc. Japan 13, 377--382 (1961)

Remarks on differentiable structures on
spheres, by I.Tamura, J. Math. Soc. Japan 13, 383--386 (1961)

Classification of (n-1)-connected 2n-manifolds
by C.T.C. Wall, Ann. of Math. (2) 75, 163--189 (1962)

An invariant for certain smooth manifolds
by J. Eells and N. Kuiper, Ann. Mat. Pura Appl. (4) 60, 93-110 (1962)

Groups of homotopy spheres I.
by M.Kervaire and J.Milnor, Ann. of Math. (2) 77, 504--537 (1963)

Beispiele zur Differentialtopologie
von Singularitäten by E. Brieskorn, Inv. Math. 2, 1--14 (1966)

Singularities and exotic spheres
by F. Hirzebruch, Seminaire Bourbaki, Vol. 10, Exp. 314, 13-32 (1967)

O(n)-Mannigfaltigkeiten, exotische
Sphären und Singularitäten by F.Hirzebruch and K.-H.Mayer,
Springer Lecture Notes 57 (1968)

An invariant for almost-closed manifolds
by D.Frank, Bull. Amer. Math. Soc. 74, 562--567 (1968)

On the homotopy groups of BPL and PL/O
by G.Brumfiel, Ann. of Math. 88, 291--311 (1968)

Lectures on groups of homotopy spheres
by J.Levine, Springer Lecture Notes 1126, 62--95 (1969/1985)

On the homotopy groups of BPL and PL/O III.
by G.Brumfiel, Michigan Math. J. 17, 217--224 (1970)

An exotic sphere with nonnegative sectional
curvature.
by D.Gromoll and W.Meyer, Ann. of Maths. 100, 401--406 (1974)

Combinatorial Pontrjagin classes by
J.Milnor and J.Stasheff, Chapter 20 of *Characteristic Classes*,
Ann. of Math. Studies 76 (1974)

Spheres exotiques et l'espace de Whitehead
(English translation) by M.Weiss, C.R. Acad. Sci. Paris Ser. I. Math. 303,
885--888 (1986)

Cobordism theory and exotic spheres by
J.Plotkin, Harvard Senior Thesis (1999)

Differentiable structures on
manifolds by T.Lance, *Surveys in Surgery Theory Theory*, Vol. 1,
Ann. of Math. Studies 145, 73--104 (2000)

The structure set by A.Ranicki,
Chapter 13 of *Algebraic and Geometric Surgery*, Oxford (2002)

Exotic spheres and curvature
by M.Joachim and D.J.Wraith, Bull. A.M.S. 45, 595--616 (2008)

A minimal Brieskorn 5-sphere in the
Gromoll-Meyer sphere and its applications.
by C.Duran and T.Püttmann, Michigan Math. J. 56, 419--451 (2008)

On the work of Michel Kervaire in surgery
and knot theory by A.Ranicki, Slides of lecture given at Kervaire
memorial symposium, Geneva, 10-12 February, 2009.

Addendum Exotic spheres and the Kervaire invariant
(8 May 2009)

An introduction to exotic spheres and singularities
by A.Ranicki, Slides of lecture given in Edinburgh, 4 May 2012

Dusa McDuff and Jack Milnor (Somewhere in Scotland, 2011)