Mathematical works. Volume I.
(Œuvres mathématiques. Volume I.)

*(French, German)*Zbl 1381.01013
Documents Mathématiques 15. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-816-9/hbk). ix, 573 (2017).

This is the first volume of a commented edition of the mathematical research papers of René Thom. This volume covers the period 1949–1960 with the following highlights:

This is a welcome publication which will delight the many admirers of René Thom’s works in mathematics.

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- his first publication [“Sur une partition en cellules associée à une fonction sur une variété”, C. R. Acad. Sci., Paris 228, 973–975 (1949; Zbl 0034.20802)], where he outlines the definition of what is now called the Morse complex;
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- his famous contributions to algebraic topology, which include his work on the characteristic classes of sphere bundles [“Espaces fibrés en sphères et carrés de Steenrod”, Ann. Sci. Éc. Norm. Supér. (3) 69, 109–182 (1952; Zbl 0049.40001)], his applications of transversality theorems leading to the determination of cobordism groups through the homotopy of what is now called the Thom complex [“Quelques propriétés globales des variétés différentiables”, Comment. Math. Helv. 28, 17–86 (1954; Zbl 0057.15502)], and his collaboration with A. Dold about the homotopy of infinite symmetric products [“Quasifaserungen und unendliche symmetrische Produkte”, Ann. Math. (2) 67, 239–281 (1958; Zbl 0091.37102)];
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- his foundational article on singularity theory [“Les singularités des applications différentiables”, Ann. Inst. Fourier 6, 43–87 (1955/56; Zbl 0075.32104)], where he starts his program on the classification of generic singularities of mappings between differentiable manifolds;
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- his influential Bourbaki report on Smale’s classification of immersions [“La classification des immersions”, Séminaire Bourbaki 10e année, Textes des Conf., Exposé No. 157, 11 p. (1958; Zbl 0116.40504)], followed by his article on differential inequalities [“Remarques sur les problèmes comportant des inéquations différentielles globales”, Bull. Soc. Math. Fr. 87, 455–461 (1959; Zbl 0213.25302)], which anticipates the development of what is now called the \(h\)-principle.

This is a welcome publication which will delight the many admirers of René Thom’s works in mathematics.

Reviewer: Benoît Fresse (Villeneuve d’Ascq)

##### MSC:

01A75 | Collected or selected works; reprintings or translations of classics |

57-03 | History of manifolds and cell complexes |

58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |

55R25 | Sphere bundles and vector bundles in algebraic topology |

57R75 | \(\mathrm{O}\)- and \(\mathrm{SO}\)-cobordism |

55S15 | Symmetric products and cyclic products in algebraic topology |

57R45 | Singularities of differentiable mappings in differential topology |

58A20 | Jets in global analysis |

57R42 | Immersions in differential topology |

01A70 | Biographies, obituaries, personalia, bibliographies |