Well-posedness of parabolic difference equations IA.

No abstract

A graduate student today has a good choice of text books on the fundamentals of topology. Usually such a book is dedicated to one of the main areas in topology like homology theory, homotopy theory, differential topology or the theory of bundles and frequently the later chapters are devoted to some topics of special interest. In the volume under review the authors have chosen to package within some 500 pages the elements of homotopy theory, differential topology and bundle theory under the unifying concept of 'geometric structure' and have limited themselves to the basics. By the end of the book the reader will know for example how to classify compact differentiable surfaces via Morse theory; how to embed C r -manifolds in euclidean space; how to calculate certain homotopy groups of spheres and Lie groups using the classical fibrations and exact homotopy sequences. En route he is given a synopsis of point set topology as far as the metrization theorem and is then led very carefully through the constructions of topology which are relevant to homotopy theory. After this follows a chapter on cellular structure including simplicial complexes and the basic results on homotopy theory of maps between cellular spaces. The next chapter presents a very detailed treatment of smooth manifolds including fundamental results on C r -approximation, tubular neighbourhoods, morse theory and surgery. The fourth chapter sorts out the numerous versions of the concept of 'bundle'. Starting with bundles without group structure, the reader is introduced to locally trivial bundles, Serre bundles, Steenrod bundles, Ehresmann-Feldbau bundles, Milnor bundles, vector bundles and smooth bundles. The role of group actions in bundle theory is carefully explained. The last chapter sets up the homotopy exact sequence of a pair and of a fibration and contains a few sections on covering spaces. There are several very useful sections on the fundamental group of cellular spaces, the homotopy groups of surfaces and the construction of spaces with prescribed homotopy groups. Particular attention is paid to the lower end of the homotopy exact sequence and the action of the fundamental group on higher homotopy groups.From this volume alone the reader will learn nothing about homology theory. The authors promise a second volume covering the algebraic aspects of topology. If this second volume maintains the same standard of precision and attention to detail as the present one then the two together will form an invaluable asset for any student of topology. As a personal preference however I would have welcomed just one short chapter on singular homology theory towards the beginning of the book if only to settle, as early as possible, the question of the topological invariance of dimension. As things stand the authors have to tread very carefully when discussing the notion of topological manifolds with boundary and dimension of cells. In fact the reader does not learn officially until page 370 that U m and U n are homeomorphic only when m = n.The translation...

No abstract

No abstract