A Basic Course in Algebraic Topology

ISBN-13: 9780387974309 / Angielski / Twarda / 1991 / 431 str.

W. S. Massey; William S. Massey; P. R. Halmos

Opis

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.

Spis treści

1: Two-Dimensional Manifolds .- 2: The Fundamental Group .- 3: Free Groups and Free Products of Groups.-  4: Seifert and Van Kampen Theorem on the Fundamental Group of the Union of Two Spaces. Applications .-  5: Covering Spaces .- 6: Background and Motivation for Homology Theory .- 7: Definitions and Basic Properties of Homology Theory .- 8: Determination of the Homology Groups of Certain Spaces: Applications and Further Properties of Homology Theory .- 9: Homology of CW-Complexes.-  10: Homology with Arbitrary Coefficient Groups .- 11: The Homology of Product Spaces.- 12: Cohomology Theory.- 13: Products in Homology and Cohomology.- 14: Duality Theorems for the Homology of Manifolds.- 15: Cup Products in Projective Spaces and Applications of Cup Products. Appendix A: A Proof of De Rham's Theorem..-  Appendix B: Permutation Groups or Tranformation Groups.

O autorze

William S. Massey (1920-2017) was an American mathematician known for his work in algebraic topology. The Blakers-Massey theorem and the Massey product were both named for him. His textbooks Singular Homology Theory and Algebraic Topology: An Introduction are also in the Graduate Texts in Mathematics series.

Blurb

This textbook is intended for a course in algebraic topology at the beginning graduate level. The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory. These topics are developed systematically, avoiding all unnecessary definitions, terminology, and technical machinery. The text consists of material from the first five chapters of the author's earlier book, Algebraic Topology; an Introduction (GTM 56) together with almost all of his book, Singular Homology Theory (GTM 70). The material from the two earlier books has been substantially revised, corrected, and brought up to date.