*Heinz Hopf Collected papers* Preface

The

*Collected papers of Heinz Hopf*were published in 2001 with a Preface written by**Beno Eckmann**. The 1271 page book, published by Springer-Verlag, Berlin, also contains appendices by P J Hilton, P Aleksandrov, B Eckmann and H Hopf. The following is an extract from Eckmann's Preface:-28 years have passed since Hopf died, over 70 years since he wrote his first articles. The original publications are often difficult to find, and have been partly forgotten.

*The Selecta Heinz Hopf*appeared in 1964 and have been out of print for a long time; they contain a rather small part of his work. Only as a whole can his publications, made available here, provide a true picture of his research personality and the penetrating force of his ideas. It is fascinating to go thus to the sources: the reader will look at these collected papers with admiration and growing amazement.

It is indeed astonishing to realise that this oeuvre of a whole scientific life consists of only about 70 writings,

*Comptes Rendus Notes*and survey articles included, and of course the book

*Topology I*written jointly with Paul Aleksandrov. Astonishing also the transparent and clear style, the concreteness of the problems, and how abstract and far-reaching the methods Hopf invented to master them - abstract, but without unnecessary generalities. Astonishing too to see that through this whole oeuvre there is one major line of thought with each part contributing to the developments mentioned above.

In the sense of a very condensed survey that major line is as follows. At the beginning there is global differential geometry, namely the relation between Curvatura Integra and the Euler characteristic of a closed Riemannian manifold. This leads to the mapping degree due to Brouwer, to singularities of vector fields, and to fixed-point theorems. At the same time there is the algebraisation of the topological concepts as suggested by Emmy Noether and her circle; it immensely clarified and simplified the arguments (Euler-Poincaré formula, Lefschetz fixed-point theorem), and also led to the algebra of mappings between manifolds, not only of the same but also of different dimensions (Umkehrhomomorphismus, i.e. in modern terminology the Poincaré dual of the cohomology- homomorphism). Similar ideas lead to the surprising fact that there are essential maps of the 3-sphere to the 2-sphere, and between other spheres (Hopf invariant). On the other hand, the path is open to the topological investigation of Lie groups and of group-like spaces, to the discovery of Hopf algebras: The multiplication map yields in the cohomology algebra a comultiplication, and from that combination Hopf gets remarkable structure results. The Pontryagin multiplication in Lie group homology suggests the spanning of surfaces by two loops, and leads to the relation between the fundamental group and second homology group of spaces. This is the beginning of homological algebra: Hopf comes into the realm of the Hurewicz homotopy theory and finds algebraic descriptions for the influence of the fundamental group on the homology of aspherical spaces, in particular the decisive concept of a free resolution of a module over a group algebra. He also finds the relation between groups and ends of spaces in a similar context. He then returns to global differential geometry, this time concerned with complex and almost-complex manifolds.

The original papers are followed by several appendices. The first is the obituary written by Peter Hilton. As most of the papers are in German, it seems appropriate to include this English obituary which contains short descriptions of most of Hopf's publications. The second consists of personal recollections of Paul Aleksandrov. They bear witness to a deep friendship and provide an impression of the personality of Heinz Hopf. The third is an article Beno Eckmann wrote for the

*Neue Zürcher Zeitung*(reprinted in

*L'Enseignement Mathématique)*immediately after Hopf died. The fourth consists of the many remarks Heinz Hopf himself wrote concerning the articles which were included in the

*Selecta*1964.