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Collected Papers — Gesammelte Abhandlungen

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All seventy-one of Hopf’s papers are here. And the collection is supplemented by appendices, or rather obituaries, by Peter Hilton, Paul Alexandroff, and Beno Eckmann, now also deceased, Hopf himself having passed in 1971. Let me quote from Hilton’s obituary, published in the Bulletin of the London Mathematical Society in 1972:

Hopf wrote with limpid clarity and elegant simplicity, always explaining to the reader the underlying motive for the direction his argument was taking. Moreover, it is a revelation, on reading Hopf’s papers, to discover how many basic ideas of algebraic topology and homological algebra stem from Hopf’s genius … In 1925, Hopf came under the influence of Emmy Noether in Göttingen, and was perhaps the first topologist to appreciate the significance of her point of view that the proper objects to describe homology relations in a simplicial complex were not numbers (Betti numbers, torsion coefficients) but algebraic structures … in particular abelian groups. This appreciation may … be regarded as the precursor of the functorial approach which now imbues the whole of algebraic topology…

Indeed, is it even possible to regard algebraic topology otherwise at this stage in history, almost a century later, a century during which the subject grew into one of the major subdisciplines of mathematics with connections across the whole field, from number theory and algebraic geometry to differential geometry and physics?

Thus, in these Gesammelte Abhandlungen we encounter something of a record of the evolution of a critical part of modern mathematics in the form of publications of an acknowledged grandmaster of the subject, and reading his writings can only serve to amplify our appreciation of the subject in question. Additionally, as Hilton takes pains to make explicit, Hopf’s style is both eminently accessible and pedagogical in the best sense, i.e. in the Abelian sense: we should read the masters, not their pupils. Or, putting it perhaps somewhat less unforgivingly and much more practically, we should not neglect to read the masters, even if we have learnt from their pupils.

Clearly no one should come to the vast majority of Hopf’s papers without a background in topology already in place. But this is not as exclusive a prerequisite as it might seem. Consider, for example, the first paper in the collection written in English, “A New Proof of the Lefschetz Formula on Invariant Points,” no. 9 of the 71, and dating to 1928. The reader is treated to a crystalline argument in four pages, using, beyond some very elementary topological notions in the old-fashioned simplicial context, no more than linear algebra at an elementary level. This paper can easily be used as a source for undergraduate level independent study. And, yes, the prose, even in English, is limpidly clear and elegantly simple, to paraphrase Hilton.

But the reader should certainly not come to this book without a decent grounding in German, as this is the language of the majority of the articles. There are gems to be mined, though, such as no’s. 11 and 14, being parts one and two of a work whose titles translate to “On the topology of presentations of manifolds, Part One: a new presentation of the theory … for topological manifolds” and “Part Two: class invariants for [these] presentations.” Or consider no’s. 42–45, which concern relations between the fundamental group and the second Betti group (with no. 43 in fact being in English and carrying this title).

It is interesting to note, too, that Hopf’s publications are not restricted to algebraic topology or even, more broadly, general topology; there are a number of entries in other areas, e.g., no. 15, Über die Verteilung quadratischer Reste, or “On the distribution of quadratic residues,” and no. 46, Maximale Toroide und Singuläre Elemente in Geschlossenen Lieschen Gruppen, or “Maximal tori and singular elements in closed Lie groups.”

Nonetheless, topology is surely in the forefront, and it would be criminal not to mention papers no. 68 (dating to 1966), whose title translates to “Some personal recollections of the prehistory of contemporary topology” and no. 69, also dating to 1966, whose title translates to “A snapshot of the development of topology.” In the latter article, Hopf starts off by saying (in loose translation) that “it is not my goal to serve as a historian — such a person would have to be ‘objective’ and ‘complete’ in his coverage; I prefer to present myself as more of a reporter who presents what he himself has witnessed so that the remarks that follow are necessarily subjective and incomplete.” And then follow some fabulous reminiscences involving such historical players as Erhard Schmidt, L. E. J. Brouwer, Paul Alexandroff, O. Veblen, S. Lefshetz, J. W. Alexander, E. Stiefel, and L. Pontryagin — and I’ve surely missed a few. Hopf presents us with a master’s account of the development of what are today mainstays of topology, with the main players represented from first hand knowledge, he himself of course prominently among them.

Now that I’ve dropped these names, let me finish by mentioning paper no. 30 in the Gesammelte Abhandlungen, written in 1937 together with Pontryangin and Hopf’s dear friend Alexandroff: Über den Brouwerschen Dimensionsbegriff: it doesn’t get more fundamental than that! 

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

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Thursday, January 30, 2014
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Heinz Hopf
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