Charles B. Thomas - Böcker

Elliptic cohomology is an extremely beautiful theory with both geometric and arithmetic aspects. The former is explained by the fact that the theory is a quotient of oriented cobordism localised away from 2, the latter by the fact that the coefficients coincide with a ring of modular forms. The aim of this work is to construct this cohomology theory, and evaluate it on classifying spaces BG of finite groups G. This class of spaces is important, since (using ideas borrowed from `Monstrous Moonshine') it is possible to give a bundle-theoretic definition of EU-(BG). Concluding chapters also discuss variants, generalizations and potential applications.

J. Frank Adams was one of the world's leading topologists. He solved a number of celebrated problems in algebraic topology, a subject in which he initiated many of the most active areas of research. He wrote a large number of papers during the period 1955-1988, and they are characterised by elegant writing and depth of thought. Few of them have been superseded by later work. This selection, in two volumes, brings together all his major research contributions. They are organised by subject matter rather than in strict chronological order. The first contains papers on: the cobar construction, the Adams spectral sequence, higher-order cohomology operations, and the Hopf invariant one problem; applications of K-theory; generalised homology and cohomology theories. The second volume is mainly concerned with Adams' contributions to: characteristic classes and calculations in K-theory; modules over the Steenrod algebra and their Ext groups; finite H-spaces and compact Lie groups; maps between classifying spaces of compact groups. Every serious student or practitioner of algebraic topology will want to own a copy of these two volumes both as a historical record and as a source of continued reference.

J. Frank Adams was one of the world's leading topologists. He solved a number of celebrated problems in algebraic topology, a subject in which he initiated many of the most active areas of research. He wrote a large number of papers during the period 1955-1988, and they are characterised by elegant writing and depth of thought. Few of them have been superseded by later work. This selection, in two volumes, brings together all his major research contributions. They are organised by subject matter rather than in strict chronological order. The first contains papers on: the cobar construction, the Adams spectral sequence, higher-order cohomology operations, and the Hopf invariant one problem; applications of K-theory; generalised homology and cohomology theories. The second volume is mainly concerned with Adams' contributions to: characteristic classes and calculations in K-theory; modules over the Steenrod algebra and their Ext groups; finite H-spaces and compact Lie groups; maps between classifying spaces of compact groups. Every serious student or practitioner of algebraic topology will want to own a copy of these two volumes both as a historical record and as a source of continued reference.

Elliptic cohomology is an extremely beautiful theory with both geometric and arithmetic aspects. The former is explained by the fact that the theory is a quotient of oriented cobordism localised away from 2, the latter by the fact that the coefficients coincide with a ring of modular forms. The aim of the book is to construct this cohomology theory, and evaluate it on classifying spaces BG of finite groups G. This class of spaces is important, since (using ideas borrowed from `Monstrous Moonshine') it is possible to give a bundle-theoretic definition of EU-(BG). Concluding chapters also discuss variants, generalisations and potential applications.

This invaluable book, based on the many years of teaching experience of both authors, introduces the reader to the basic ideas in differential topology. Among the topics covered are smooth manifolds and maps, the structure of the tangent bundle and its associates, the calculation of real cohomology groups using differential forms (de Rham theory), and applications such as the Poincaré-Hopf theorem relating the Euler number of a manifold and the index of a vector field. Each chapter contains exercises of varying difficulty for which solutions are provided. Special features include examples drawn from geometric manifolds in dimension 3 and Brieskorn varieties in dimensions 5 and 7, as well as detailed calculations for the cohomology groups of spheres and tori.

This book provides an introduction to representations of both finite and compact groups. The proofs of the basic results are given for the finite case, but are so phrased as to hold without change for compact topological groups with an invariant integral replacing the sum over the group elements as an averaging tool. Among the topics covered are the relation between representations and characters, the construction of irreducible representations, induced representations and Frobenius reciprocity. Special emphasis is given to exterior powers, with the symmetric group Sn as an illustrative example. The book concludes with a chapter comparing the representations of the finite group SL2(p) and the non-compact Lie group SL2(P).

This book provides an introduction to representations of both finite and compact groups. The proofs of the basic results are given for the finite case, but are so phrased as to hold without change for compact topological groups with an invariant integral replacing the sum over the group elements as an averaging tool. Among the topics covered are the relation between representations and characters, the construction of irreducible representations, induced representations and Frobenius reciprocity. Special emphasis is given to exterior powers, with the symmetric group Sn as an illustrative example. The book concludes with a chapter comparing the representations of the finite group SL2(p) and the non-compact Lie group SL2(P).