Singular Homology Theory (Graduate Texts in Mathematics)

The main purpose of this book is to give a systematic treatment of singular homology and cohomology theory. It is in some sense a sequel to the author's previous book in this Springer-Verlag series entitled Algebraic Topology: An Introduction. This earlier book is definitely not a logical prerequisite for the present volume. However, it would certainly be advantageous for a prospective reader to have an acquaintance with some of the topics treated in that earlier volume, such as 2-dimensional manifolds and the funda­ mental group. Singular homology and cohomology theory has been the subject of a number of textbooks in the last couple of decades, so the basic outline of the theory is fairly well established. Therefore, from the point of view of the mathematics involved, there can be little that is new or original in a book such as this. On the other hand, there is still room for a great deal of variety and originality in the details of the exposition. In this volume the author has tried to give a straightforward treatment of the subject matter, stripped of all unnecessary definitions, terminology, and technical machinery. He has also tried, wherever feasible, to emphasize the geometric motivation behind the various concepts. This textbook on homology and cohomology theory is geared towards the beginning graduate student. Singular homology theory is developed systematically, avoiding all unnecessary definitions, terminology, and technical machinery. Wherever possible, the geometric motivation behind various algebraic concepts is emphasized. The only formal prerequisites are knowledge of the basic facts of abelian groups and point set topology. Singular Homology Theory is a continuation of the author's earlier book, Algebraic Topology: An Introduction which presents such important supplementary material as the theory of the fundamental group and a thorough discussion of 2-dimensional manifolds. However, this earlier book is not a prerequisite to understanding Singular Homology Theory . This textbook on homology and cohomology theory is geared towards the beginning graduate student. Singular homology theory is developed systematically, avoiding all unnecessary definitions, terminology, and technical machinery. Wherever possible, the geometric motivation behind various algebraic concepts is emphasized.The only formal prerequisites are knowledge of the basic facts of abelian groups and point set topology. Singular Homology Theory is a continuation of t he author's earlier book, Algebraic Topology: An Introduction, which presents such important supplementary material as the theory of the fundamental group and a thorough discussion of 2-dimensional manifolds. However, this earlier book is not a prerequisite for understanding Singular Homology Theory. Front Matter....Pages i-xii Background and Motivation for Homology Theory....Pages 1-10 Definitions and Basic Properties of Homology Theory....Pages 11-37 Determination of the Homology Groups of Certain Spaces : Applications and Further Properties of Homology Theory....Pages 38-75 Homology of CW-complexes....Pages 76-104 Homology with Arbitrary Coefficient Groups....Pages 105-128 The Homology of Product Spaces....Pages 129-153 Cohomology Theory....Pages 154-171 Products in Homology and Cohomology....Pages 172-198 Duality Theorems for the Homology of Manifolds....Pages 199-238 Cup Products in Projective Spaces and Applications of Cup Products....Pages 239-250 Back Matter....Pages 251-267