Commentary on the Kervaire-Milnor correspondence 1958-1961

Authors:
Andrew Ranicki and Claude Weber

Journal:
Bull. Amer. Math. Soc. **52** (2015), 603-609

Published electronically:
July 1, 2015

MathSciNet review:
3393348

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Abstract | References | Additional Information

Abstract: The extant letters exchanged between Kervaire and Milnor during their collaboration from 1958-1961 concerned their work on the classification of exotic spheres, culminating in their 1963 *Annals of Mathematics* paper. Michel Kervaire died in 2007; for an account of his life, see the obituary by Shalom Eliahou, Pierre de la Harpe, Jean-Claude Hausmann, and Claude Weber in the September 2008 issue of the *Notices of the American Mathematical Society*. The letters were made public at the 2009 Kervaire Memorial Conference in Geneva. Their publication in this issue of the *Bulletin of the American Mathematical Society* is preceded by our commentary on these letters, providing some historical background.

**[1]**Alfred Aeppli,*Modifikation von reellen und komplexen Mannigfaltigkeiten*, Comment. Math. Helv.**31**(1957), 219–301 (German). MR**0116363****[2]**Michel A. Kervaire,*Non-parallelizability of the 𝑛-sphere for 𝑛>7*, Proc. Natl. Acad. Sci. USA**44**(1958), no. 3, 280–283. MR**3075371**, 10.1073/pnas.44.3.280**[3]**Michel A. Kervaire,*A manifold which does not admit any differentiable structure*, Comment. Math. Helv.**34**(1960), 257–270. MR**0139172****[4]**Michel A. Kervaire and John W. Milnor,*Groups of homotopy spheres. I*, Ann. of Math. (2)**77**(1963), 504–537. MR**0148075****[5]**J. P. Levine,*Lectures on groups of homotopy spheres*, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 62–95. MR**802786**, 10.1007/BFb0074439**[6]**John Milnor,*On manifolds homeomorphic to the 7-sphere*, Ann. of Math. (2)**64**(1956), 399–405. MR**0082103**- [7] John Milnor: ``Differentiable manifolds which are homotopy spheres'' Mimeographed Notes Princeton. Dated January 23rd 1959. Reprinted in Milnor's Collected Works III.
**[8]**John Milnor,*Differentiable structures on spheres*, Amer. J. Math.**81**(1959), 962–972. MR**0110107****[9]**John Milnor,*A procedure for killing homotopy groups of differentiable manifolds.*, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I, 1961, pp. 39–55. MR**0130696****[10]**John W. Milnor and Michel A. Kervaire,*Bernoulli numbers, homotopy groups, and a theorem of Rohlin*, Proc. Internat. Congress Math. 1958, Cambridge Univ. Press, New York, 1960, pp. 454–458. MR**0121801****[11]**Stephen Smale,*Generalized Poincaré’s conjecture in dimensions greater than four*, Ann. of Math. (2)**74**(1961), 391–406. MR**0137124****[12]**C. T. C. Wall,*Killing the middle homotopy groups of odd dimensional manifolds*, Trans. Amer. Math. Soc.**103**(1962), 421–433. MR**0139185**, 10.1090/S0002-9947-1962-0139185-4**[13]**Andrew H. Wallace,*Modifications and cobounding manifolds*, Canad. J. Math.**12**(1960), 503–528. MR**0125588**

Additional Information

**Andrew Ranicki**

Affiliation:
School of Mathematics, University of Edinburgh, Scotland, United Kingdom

Email:
a.ranicki@ed.ac.uk

**Claude Weber**

Affiliation:
Section de Mathématiques, Université de Genève, Switzerland

Email:
claude.weber@unige.ch

DOI:
http://dx.doi.org/10.1090/bull/1508

Published electronically:
July 1, 2015

Additional Notes:
We are grateful to Mme. Aimée Kervaire and John Milnor for permitting us to publish the correspondence.

Article copyright:
© Copyright 2015
American Mathematical Society