Reviews of Georges de Rham's books
1. Harmonic Integrals (1950), by Georges de Rham and Kunihiko Kadaira.
These lectures, delivered during the spring term, 1950, at the Institute for Advanced Study, consist of five chapters, the first four, by the first author, giving an account of the most modern developments in the general theory of harmonic integrals, and the fifth, by Kodaira, applying the theory to certain problems concerning Kähler manifolds. The first part demonstrates the remarkable advances that have been made in the theory of harmonic integrals since the idea was first developed. Many of the details and restrictions which originally encumbered the theory have been eliminated, and the theory now takes on a stream-lined character and at the same time acquires much greater generality and width of application.
This book, distinguished by its richness and clarity, contains a development of basic theory for differential forms on differentiable manifolds and harmonic forms on Riemannian manifolds. The treatment is based on the concept of current which is closely related to that of distribution in the sense of Laurent Schwartz. A current can in fact be regarded as a differential form whose coefficients are distributions. Chains and (ordinary) differential forms can be identified as currents in the manner indicated below. This is the guiding principle by which the author shows how the homology properties of a manifold are revealed simultaneously through its chains and its differential forms.
These notes come from seminars in Lausanne, given in 1963/64, except for one given in 1960. ... Exposé 2, by de Rham, is on torsion. The treatment is more or less classical, following de Rham: it works up to Milnor's counter-example to the Hauptvermuting.
These notes partly cover a course of lectures given at the Tata Institute in 1966. They are suitable for students with an elementary knowledge of general topology and group theory. The fundamental group of a space X is defined and some properties are discussed. Related topics include homotopy equivalence and the fundamental groups of product spaces. A group-theoretic section (free products and their quotients) is followed by results useful in calculating fundamental groups, including "van Kampen's theorem". Such calculations are variously illustrated. A section on "the group of a tame link [a finite set of pairwise disjoint knots] given by a good plane projection" is followed by one on Antoine's necklace and horned sphere, then one on elementary ideals and Alexander polynomials. The concluding topics, interestingly, are "construction of 3-manifolds" (Whitehead manifold and Dehn's homology 3-sphere), followed by "involutions of S^{4}".
JOC/EFR January 2015
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