Homology Theory

An Introduction to Algebraic Topology

Author: P. J. Hilton,S. Wylie

Publisher: CUP Archive

ISBN: 9780521094221

Category: Mathematics

Page: 484

View: 2549

This account of algebraic topology is complete in itself, assuming no previous knowledge of the subject. It is used as a textbook for students in the final year of an undergraduate course or on graduate courses and as a handbook for mathematicians in other branches who want some knowledge of the subject.

Homology Theory

An Introduction to Algebraic Topology

Author: James W. Vick

Publisher: Springer Science & Business Media

ISBN: 1461208815

Category: Mathematics

Page: 245

View: 7479

This introduction to some basic ideas in algebraic topology is devoted to the foundations and applications of homology theory. After the essentials of singular homology and some important applications are given, successive topics covered include attaching spaces, finite CW complexes, cohomology products, manifolds, Poincare duality, and fixed point theory. This second edition includes a chapter on covering spaces and many new exercises.

Singular Homology Theory

Author: W.S. Massey

Publisher: Springer Science & Business Media

ISBN: 1468492314

Category: Mathematics

Page: 268

View: 6154

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.

Elements of Homology Theory

Author: Viktor Vasilʹevich Prasolov

Publisher: American Mathematical Soc.

ISBN: 0821838121

Category: Mathematics

Page: 418

View: 2562

The book is a continuation of the previous book by the author (Elements of Combinatorial and Differential Topology, Graduate Studies in Mathematics, Volume 74, American Mathematical Society, 2006). It starts with the definition of simplicial homology and cohomology, with many examples and applications. Then the Kolmogorov-Alexander multiplication in cohomology is introduced. A significant part of the book is devoted to applications of simplicial homology and cohomology to obstruction theory, in particular, to characteristic classes of vector bundles. The later chapters are concerned with singular homology and cohomology, and Cech and de Rham cohomology. The book ends with various applications of homology to the topology of manifolds, some of which might be of interest to experts in the area. The book contains many problems; almost all of them are provided with hints or complete solutions.

An Introduction to Intersection Homology Theory, Second Edition

Author: Frances Kirwan,Jonathan Woolf

Publisher: CRC Press

ISBN: 9781584881841

Category: Mathematics

Page: 248

View: 931

Now more that a quarter of a century old, intersection homology theory has proven to be a powerful tool in the study of the topology of singular spaces, with deep links to many other areas of mathematics, including combinatorics, differential equations, group representations, and number theory. Like its predecessor, An Introduction to Intersection Homology Theory, Second Edition introduces the power and beauty of intersection homology, explaining the main ideas and omitting, or merely sketching, the difficult proofs. It treats both the basics of the subject and a wide range of applications, providing lucid overviews of highly technical areas that make the subject accessible and prepare readers for more advanced work in the area. This second edition contains entirely new chapters introducing the theory of Witt spaces, perverse sheaves, and the combinatorial intersection cohomology of fans. Intersection homology is a large and growing subject that touches on many aspects of topology, geometry, and algebra. With its clear explanations of the main ideas, this book builds the confidence needed to tackle more specialist, technical texts and provides a framework within which to place them.

A Geometric Approach to Homology Theory

Author: S. Buonchristiano,C. P. Rourke,B. J. Sanderson

Publisher: Cambridge University Press

ISBN: 0521209404

Category: Mathematics

Page: 149

View: 1645

The purpose of these notes is to give a geometrical treatment of generalized homology and cohomology theories. The central idea is that of a 'mock bundle', which is the geometric cocycle of a general cobordism theory, and the main new result is that any homology theory is a generalized bordism theory. The book will interest mathematicians working in both piecewise linear and algebraic topology especially homology theory as it reaches the frontiers of current research in the topic. The book is also suitable for use as a graduate course in homology theory.

Homology Theory on Algebraic Varieties

Author: Andrew H. Wallace

Publisher: Courier Corporation

ISBN: 0486787842

Category: Mathematics

Page: 128

View: 6216

Concise and authoritative monograph, geared toward advanced undergraduate and graduate students, covers linear sections, singular and hyperplane sections, Lefschetz's first and second theorems, the Poincaré formula, and invariant and relative cycles. 1958 edition.

Homology Theory

Acyclic Space, Alexander-Spanier Cohomology, Aspherical Space, Borel-Moore Homology, Cap Product, Cech Cohomology, Cellular Homology,

Author: Source Wikipedia

Publisher: University-Press.org

ISBN: 9781230633404


Page: 98

View: 8155

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 42. Chapters: Acyclic space, Alexander-Spanier cohomology, Aspherical space, Borel-Moore homology, Cap product, ech cohomology, Cellular homology, Chern-Simons form, Cocycle class, Cohomology ring, Compactly-supported homology, Continuation map, Cup product, Cyclic category, De Rham cohomology, Eilenberg-Moore spectral sequence, Eilenberg-Steenrod axioms, Excision theorem, Floer homology, Good cover (algebraic topology), Gromov norm, Hodge conjecture, Homology (mathematics), Homology sphere, Hurewicz theorem, Khovanov homology, Kirby-Siebenmann class, Mayer-Vietoris sequence, Morse homology, Poincare complex, Poincare duality, Polar homology, Pontryagin product, Products in algebraic topology, Reduced homology, Relative contact homology, Relative homology, Simplicial homology, Singular homology, Steenrod homology, Steenrod problem, Stratifold.

Cycles, Transfers, and Motivic Homology Theories. (AM-143)

Author: Vladimir Voevodsky,Andrei Suslin,Eric M. Friedlander

Publisher: Princeton University Press

ISBN: 140083712X

Category: Mathematics

Page: 256

View: 6582

The original goal that ultimately led to this volume was the construction of "motivic cohomology theory," whose existence was conjectured by A. Beilinson and S. Lichtenbaum. This is achieved in the book's fourth paper, using results of the other papers whose additional role is to contribute to our understanding of various properties of algebraic cycles. The material presented provides the foundations for the recent proof of the celebrated "Milnor Conjecture" by Vladimir Voevodsky. The theory of sheaves of relative cycles is developed in the first paper of this volume. The theory of presheaves with transfers and more specifically homotopy invariant presheaves with transfers is the main theme of the second paper. The Friedlander-Lawson moving lemma for families of algebraic cycles appears in the third paper in which a bivariant theory called bivariant cycle cohomology is constructed. The fifth and last paper in the volume gives a proof of the fact that bivariant cycle cohomology groups are canonically isomorphic (in appropriate cases) to Bloch's higher Chow groups, thereby providing a link between the authors' theory and Bloch's original approach to motivic (co-)homology.

Foundations of Algebraic Topology

Author: Samuel Eilenberg,Norman Steenrod

Publisher: Princeton University Press

ISBN: 1400877490

Category: Mathematics

Page: 346

View: 8199

The need for an axiomatic treatment of homology and cohomology theory has long been felt by topologists. Professors Eilenberg and Steenrod present here for the first time an axiomatization of the complete transition from topology to algebra. Originally published in 1952. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

A Homology Theory for Smale Spaces

Author: Ian F. Putnam

Publisher: American Mathematical Soc.

ISBN: 1470409097

Category: Mathematics

Page: 122

View: 5640

The author develops a homology theory for Smale spaces, which include the basics sets for an Axiom A diffeomorphism. It is based on two ingredients. The first is an improved version of Bowen's result that every such system is the image of a shift of finite type under a finite-to-one factor map. The second is Krieger's dimension group invariant for shifts of finite type. He proves a Lefschetz formula which relates the number of periodic points of the system for a given period to trace data from the action of the dynamics on the homology groups. The existence of such a theory was proposed by Bowen in the 1970s.

Modern Geometry - Methods and Applications

Part 3: Introduction to Homology Theory

Author: B.A. Dubrovin,A.T. Fomenko,Sergeĭ Petrovich Novikov,S.P. Novikov,Robert G. Burns

Publisher: Springer Science & Business Media

ISBN: 9780387972718

Category: Mathematics

Page: 418

View: 7774

Over the past fifteen years, the geometrical and topological methods of the theory of manifolds have as- sumed a central role in the most advanced areas of pure and applied mathematics as well as theoretical physics. The three volumes of Modern Geometry - Methods and Applications contain a concrete exposition of these methods together with their main applications in mathematics and physics. This third volume, presented in highly accessible languages, concentrates in homology theory. It contains introductions to the contemporary methods for the calculation of homology groups and the classification of manifesto. Both scientists and students of mathematics as well as theoretical physics will find this book to be a valuable reference and text.

Seminar on Triples and Categorical Homology Theory

ETH 1966/67

Author: H. Appelgate,M. Barr,J. Beck,F. W. Lawvere,F. E. Linton,E. Manes,M. Tierney,F. Ulmer

Publisher: Springer

ISBN: 3540360913

Category: Mathematics

Page: 406

View: 6916