# Exotic spheres

An exotic sphere is an n-dimensional differentiable manifold which is homeomorphic but not diffeomorphic to the standard n-sphere Sn.

• 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)