Selecta: (Springer Collected Works in Mathematics)

About the Book

This volume contains research papers and survey articles written by Beno Eckmann from 1941 to 1986. The aim of the compilation is to provide a general view of the breadth of Eckmann's mathematical work. His influence was particularly strong in the development of many subfields of topology and algebra, where he repeatedly pointed out close, and often surprising, connections between them and other areas. The surveys are exemplary in terms of how they make difficult mathematical ideas easily comprehensible and accessible even to non-specialists. The topics treated here can be classified into the following, not entirely unrelated areas: algebraic topology (homotopy and homology theory), algebra, group theory and differential geometry. Beno Eckmann was Professor of Mathematics at the University of Lausanne, 1942-48, and Principal of the Institute for Mathematical Research at the ETH Zurich, 1964-84, where he was therefore an emeritus professor.

Table of Contents:
Zur Homotopietheorie gefaserter Räume.- Über die Homotopiegruppen von Gruppenräumen.- Systeme von Richtungsfeldern in Sphären und stetige Lösungen komplexer linearer Gleichungen.- L’idée de dimension.- Stetige Lösungen linearer Gleichungssysteme.- Gruppentheoretischer Beweis des Satzes von Hurwitz-Radon über die Komposition quadratischer Formen.- Topologie und Algebra.- Über monothetische Gruppen.- Harmonische Funktionen und Randwertaufgaben in einem Komplex.- Der Cohomologie-Ring einer beliebigen Gruppe.- Coverings and Betti numbers.- Formes différentielles et métrique hermitienne sans torsion. I. Structure complexe, formes pures.- Formes différentielles et métrique hermitienne sans torsion. II. Formes de classe k; formes analytiques.- Sur les variétés closes à métrique hermitienne sans torsion.- Quelques propriétés globales des variétés kählériennes.- Espaces fibrés et homotopie.- Sur l’intégrabilité des structures presque complexes.- Complex-analytic manifolds.- On complexes with operators.- Über injektive Moduln.- Cohomology of groups and transfer.- A class of compact, complex manifolds which are not algebraic.- Structures complexes et transformations infinitésimales.- Räume mit Mittelbildungen.- Zur Cohomologietheorie von Räumen und Gruppen.- Homotopie et dualité.- Groupes d’homotopie et dualité. Groupes absolus.- Groupes d’homotopie et dualité. Suites exactes.- Groupes d’homotopie et dualité. Coefficients.- Transgression homotopique et cohomologique.- Décomposition homologique d’un polyèdre simplement connexe.- On the homology and homotopy decomposition of continuous maps.- Groupes d’homotopie et dualité.- Structure maps in group theory.- Group-like structures in general categories I. Multiplications andcomultiplications.- Homotopie und Homologie.- Generalized means.- Homotopy and cohomology theory.- Unions and intersections in homotopy theory.- Composition functors and spectral sequences.- Commuting limits with colimits.- Continuous solutions of linear equations — some exceptional dimensions in topology.- Le groupe des types simples d’homotopie sur un polyèdre.- Simple homotopy type and categories of fractions.- On central group extensions and homology.- On the homology theory of central group extensions: I — The commutator map and stem extensions.- On the Schur multiplicator of a central quotient of a direct product of groups.- Groups with homological duality generalizing Poincaré duality.- Finiteness properties of duality groups.- Aspherical manifolds and higher-dimensional knots.- Cobordism for Poincaré duality groups.- Rational representations of finite groups and their Euler class.- Two-dimensional Poincaré duality groups and pairs.- On the Euler class of representations of finite groups over real fields.- Some recent developments in the homology theory of groups (Groups of finite and virtually finite dimension).- Poincaré duality groups of dimension two.- Chern classes of group representations over a number field.- Plane motion groups and virtual Poincaré duality of dimension two.- Poincaré duality groups of dimension two, II.- Profinite Chern classes for group representations.- The p-periodicity of the groups GL (n, Os(K)) and SL(n, Os(K)).- Galois action on algebraic matrix groups, Chern classes and the Euler class.- Poincaré duality groups of dimension two are surface groups.- Cyclic homology of groups and the Bass conjecture.- Notes.- I. Homotopy groups and fiber spaces.- II. Continuous solutions of linear equations.- III. Cohomology ofgroups.- IV. Homological algebra, transfer.- V. Duality in homotopy theory.- VI. Duality groups, Poincaré duality.- Bibliography of the publications of B. Eckmann.- List of Ph.D. Theses written under the supervision of B. Eckmann.

About the Author :
Prof. Beno Eckmann (1917-2008) was a distinguished mathematician who also exerted great influence on the promotion and development of Mathematics all over the globe. At Ben Gurion University he initiated the establishment of the Center for Advanced Studies in Mathematics, and served actively on its advisory committee until his passing away. Prof. Eckmann studied at ETH, where his PhD advisor was the famous Heinz Hopf. His brilliant Ph.D. work won him the Kern Prize and silver medal. He was a member of the Institute for Advanced Study in Princeton in 1947, 1951 and 1952, where he worked with Albert Einstein and John von Neumann. Prof. Eckmann is considered one of the founding fathers of Homological Algebra and Category Theory, with special emphasis on Topology and Cohomology of Groups. A close cooperation between Eckmann and Peter Hilton, who was a frequent guest at ETH since the 1950's, led to numerous papers (over 25) on these subjects. Eckmann had over 60 Ph.D. students, many of whom also became prominent mathematicians. In parallel with his scientific endeavors, Prof. Eckmann contributed significantly to the promotion and development of Mathematics, often through his ability to convince others of the importance of the field. For example, in 1964 he established the famous Institute for Advanced Studies at ETH, which he headed for 20 years until his retirement. Later he interested the Springer family, through Julius Springer, to establish the prestigious and influential series "Lecture Notes in Mathematics". He was the chief editor of this series until near his passing away. He was awarded many honorary degrees, one of them from Ben Gurion University. He was president of the Swiss Mathematical Society and secretary of the International Mathematical Union for 5 years. He also served as an Honorary President of the International Congress of Mathematicians in Zurich in 1994.