Selecta

Algebraic topology is surely one of the 20th century’s grandest mathematical developments, a tectonic movement par excellence, eventually crystallizing into one of the mainstays of contemporary mathematics and exerting its influence in any number of directions. Perhaps only algebraic geometry can be said to have had a comparable flowering in this time frame, and indeed the two evolutions are linked: just consider the huge role played by the French school surrounding Henri Cartan and including Jean-Pierre Serre and Alexander Grothendieck in both fields. Prior to this, one of the earlier pioneers of this algebraic topology proper was Heinz Hopf, who finished his PhD at Berlin in 1925 when topology itself was still, if not in its infancy, then certainly in its childhood: Henri Poincaré died in 1912, and by the 1920s L. E. J. Brouwer had left analysis situs for intuitionism and his own flavor of philosophy. But algebraic topology was already growing. In 1941, Beno Eckmann, whose Selecta we review here, earned his PhD at the ETH in Zürich with Heinz Hopf. Accordingly Eckmann can be taken to belong to the second generation, so to speak, on the early heroic age of algebraic topology.

It is therefore altogether fitting that Springer should re-issue, in soft-cover form, the Selecta under review. The original edition appeared in 1986, the year prior to Eckmann’s 70th birthday, as something of a survey and selection of his papers, “seek[ing] to give an impression of the breadth and depth of Eckmann’s work.” These dimensions are conveyed even by a spare cross-section of the offerings in this Selecta: in chronological order we encounter (p. 102) “l’Idée de dimension,” (p. 240) “Coverings and Betti numbers,” (p. 259) “Espaces fibrés et homotopie,” (p. 299) “Cohomology groups and transfer,” (p. 591) “On central group extensions and homology” (co-written with P. J. Hilton), (p. 655) “Aspherical manifolds and higher-dimensional knots,” and (p. 813) “Cyclic homology groups and the Bass conjecture.”

The first paper in this cross-section appeared in 1943, marvelously in Revue de Théologie et Philosophie, and the last one mentioned appeared in 1986 in Comment. Math. Helv. It is fascinating, in both a historical and a biographical sense, to meditate on the dramatic changes in style, context as well as content, and mathematical flavor of these two papers, signaling a major evolution in both the scholar and his field: from an apologia for the notion of dimension rooted in geometry but now infiltrating topology to something very modern indeed, involving the interplay of cyclic homology, linear groups, and K-theory. Wonderful scholarship.

The book is supplemented by two pages on notes by Eckmann himself, covering six areas of his broad research activity: “homotopy groups and fibre spaces, continuous solutions of linear equations, cohomology of groups, homological algebra [and] transfer, duality in homotopy theory, and duality groups [and] Poincaré duality.”

It would be ridiculous to claim that these papers would go out of style: they are still eminently readable and of high pedagogical value. Eckmann’s Selecta presents in its 800+ pages wonderful scholarship in topology by an established master. 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.