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**Commentary on “On the parallelizability of the spheres” by R. Bott and J. Milnor and “On the nonexistence of elements of Hopf invariant one” by J. F. Adams.**
*(English)*
Zbl 1277.57002

From the text: The first of my two selections is the 1958 announcement by Raoul Bott and John Milnor [Bull. Am. Math. Soc. 64, 87–89 (1958; Zbl 0082.16602)] that the following conditions on \(n\geq 1\) are equivalent:

1. There exists an \(n\)-dimensional division algebra;

2. There exists an \(n\)-dimensional vector bundle over \(S^n\) with \(w_n\neq 0\);

3. \(S^{n-1}\) is parallelizable;

4. \(n = 1, 2, 4\), or \(8\).

The full account was published in [J. Milnor, Ann. Math. (2) 68, 444–449 (1958; Zbl 0085.17301)]. M. A. Kervaire [Proc. Natl. Acad. Sci. USA 44, 280–283 (1958; Zbl 0093.37303)] proved the nonparallelizability of \(S^{n-1}\) for \(n > 8\) independently. Both proofs made use of the Bott periodicity theorem, which was still quite new then. Incidentally, it followed from the theorem that the \(K\)-groups of all real vector bundles over spheres are generated by the trivial bundles and the Hopf bundles.

The second of my two selections is the 1958 announcement by Frank Adams [Bull. Am. Math. Soc. 64, 279–282 (1958; Zbl 0178.26106)], that the following conditions on \(n\geq 2\) are equivalent:

1. \(S^{n-1}\) is an \(H\)-space, i.e., equipped with a continuous multiplication map \(S^{n-1}\times S^{n-1}\to S^{n-1}\) with an identity;

2. There exists a map \(f : S^{2n-1} \to S^n\) with Hopf invariant \(H(f) = 1\in\mathbb Z\);

3. n = 2, 4, or 8.

The proof made use of secondary cohomology operations; the full account was published in [F. Adams, Ann. Math. (2) 72, 20–104 (1960; Zbl 0096.17404)]. The more elegant \(K\)-theoretic proof obtained a few years later by Adams and Atiyah [Q. J. Math., Oxf. II. Ser. 17, 31–38 (1966; Zbl 0136.43903)] did not diminish the achievement of Adams’ 1958 paper.

F. Hirzebruch [in: Numbers. Graduate Texts in Mathematics, 123. New York etc.: Springer-Verlag (1991; Zbl 0705.00001), pp. 281–302)] has written a very readable account of the connections between division algebras and topology.

The two announcements are classic applications of algebraic topology to both algebra and topology, which changed the scenery.”

The comment ends with hints to recent results of M. A. Hill, M. J. Hopkins and D. C. Ravenel [On the non-existence of elements of Kervaire invariant one, http://arxiv.org/abs/0908.3724] (see also Current developments in mathematics, 2010. Somerville, MA: International Press, 1–43 (2011; Zbl 1249.55005) and a 1969 result of W. Browder on “The Kervaire invariant of framed manifolds” [Ann. Math. (2) 90, 157–186 (1969; Zbl 0198.28501)].

The two texts are reprinted in their entirety immediately following the commentary.

1. There exists an \(n\)-dimensional division algebra;

2. There exists an \(n\)-dimensional vector bundle over \(S^n\) with \(w_n\neq 0\);

3. \(S^{n-1}\) is parallelizable;

4. \(n = 1, 2, 4\), or \(8\).

The full account was published in [J. Milnor, Ann. Math. (2) 68, 444–449 (1958; Zbl 0085.17301)]. M. A. Kervaire [Proc. Natl. Acad. Sci. USA 44, 280–283 (1958; Zbl 0093.37303)] proved the nonparallelizability of \(S^{n-1}\) for \(n > 8\) independently. Both proofs made use of the Bott periodicity theorem, which was still quite new then. Incidentally, it followed from the theorem that the \(K\)-groups of all real vector bundles over spheres are generated by the trivial bundles and the Hopf bundles.

The second of my two selections is the 1958 announcement by Frank Adams [Bull. Am. Math. Soc. 64, 279–282 (1958; Zbl 0178.26106)], that the following conditions on \(n\geq 2\) are equivalent:

1. \(S^{n-1}\) is an \(H\)-space, i.e., equipped with a continuous multiplication map \(S^{n-1}\times S^{n-1}\to S^{n-1}\) with an identity;

2. There exists a map \(f : S^{2n-1} \to S^n\) with Hopf invariant \(H(f) = 1\in\mathbb Z\);

3. n = 2, 4, or 8.

The proof made use of secondary cohomology operations; the full account was published in [F. Adams, Ann. Math. (2) 72, 20–104 (1960; Zbl 0096.17404)]. The more elegant \(K\)-theoretic proof obtained a few years later by Adams and Atiyah [Q. J. Math., Oxf. II. Ser. 17, 31–38 (1966; Zbl 0136.43903)] did not diminish the achievement of Adams’ 1958 paper.

F. Hirzebruch [in: Numbers. Graduate Texts in Mathematics, 123. New York etc.: Springer-Verlag (1991; Zbl 0705.00001), pp. 281–302)] has written a very readable account of the connections between division algebras and topology.

The two announcements are classic applications of algebraic topology to both algebra and topology, which changed the scenery.”

The comment ends with hints to recent results of M. A. Hill, M. J. Hopkins and D. C. Ravenel [On the non-existence of elements of Kervaire invariant one, http://arxiv.org/abs/0908.3724] (see also Current developments in mathematics, 2010. Somerville, MA: International Press, 1–43 (2011; Zbl 1249.55005) and a 1969 result of W. Browder on “The Kervaire invariant of framed manifolds” [Ann. Math. (2) 90, 157–186 (1969; Zbl 0198.28501)].

The two texts are reprinted in their entirety immediately following the commentary.

### MSC:

57-03 | History of manifolds and cell complexes |

57R22 | Topology of vector bundles and fiber bundles |

01A60 | History of mathematics in the 20th century |

### Keywords:

reprints of classics### Citations:

Zbl 0082.16602; Zbl 0085.17301; Zbl 0093.37303; Zbl 0178.26106; Zbl 0096.17404; Zbl 0136.43903; Zbl 0705.00001; Zbl 1249.55005; Zbl 0198.28501
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XMLCite

\textit{A. Ranicki}, Bull. Am. Math. Soc., New Ser. 48, No. 4, 509--511 (2011; Zbl 1277.57002)

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### References:

[1] | J. F. Adams, On the nonexistence of elements of Hopf invariant one, Bull. Amer. Math. Soc. 64 (1958), 279 – 282. · Zbl 0178.26106 |

[2] | J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20 – 104. · Zbl 0096.17404 · doi:10.2307/1970147 |

[3] | J. F. Adams and M. F. Atiyah, \?-theory and the Hopf invariant, Quart. J. Math. Oxford Ser. (2) 17 (1966), 31 – 38. · Zbl 0136.43903 · doi:10.1093/qmath/17.1.31 |

[4] | R. Bott and J. Milnor, On the parallelizability of the spheres, Bull. Amer. Math. Soc. 64 (1958), 87 – 89. · Zbl 0082.16602 |

[5] | William Browder, The Kervaire invariant of framed manifolds and its generalization, Ann. of Math. (2) 90 (1969), 157 – 186. · Zbl 0198.28501 · doi:10.2307/1970686 |

[6] | M. Hill, M. Hopkins and D. Ravenel, On the non-existence of elements of Kervaire invariant one, arXiv:0908.3724. · Zbl 1366.55007 |

[7] | F. Hirzebruch, “Division algebras and topology”, in Numbers, Graduate Texts in Mathematics 123, Springer-Verlag, New York, 1991, pp. 281-302. |

[8] | H. Hopf, Über die Abbildungen von Sphären auf Sphären niedriger Dimension, Fund. Math. 25 (1935), 427-440. · Zbl 0012.31902 |

[9] | M. Kervaire, Non-parallelizability of the \( n\)-sphere, \( n>7\), Proc. Nat. Acad. U.S.A. 44 (1958), 280-283. · Zbl 0093.37303 |

[10] | John Milnor, Some consequences of a theorem of Bott, Ann. of Math. (2) 68 (1958), 444 – 449. · Zbl 0085.17301 · doi:10.2307/1970255 |

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