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:
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;
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)];
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;
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.
The volume starts with a biographical notice and a comprehensive bibliography of Thom’s publication. The editors have appended to the volume an overview of the history of the Morse complex, a summary of Thom’s work in algebraic topology, a note on Thom’s proof of the vanishing of the group of oriented cobordism in dimension \(3\), together with a bunch of notes and reviews on Thom’s articles in singularity theory and on the classification of immersions. This volume also includes two unpublished documents about the Lefschetz hyperplane section theorem and the homology of Stein manifolds, as well as some excerpts of his correspondence with Henri Cartan (concerning his first publications and his thesis) and some excerpts of personal letters sent during his stay in Princeton in 1952.
This is a welcome publication which will delight the many admirers of René Thom’s works in mathematics.


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

Biographic References:

Thom, René