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Homology Theory Of Submersions
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Book Synopsis Homology Theory of Submersions by : Junpei Sekino
Download or read book Homology Theory of Submersions written by Junpei Sekino and published by . This book was released on 1974 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this dissertation we construct a homology theory on the category of submersions which generalizes the homology of the base space with coefficients in the homology of the fiber as given by the E2-terms of the Serre spectral sequence of a fiber bundle. The main motivation for this new homology theory is the fact that it permits a generalization of the Serre spectral sequence to arbitrary submersions. The homology theory in question is first defined on a category of combinatorial objects called simplicial bundles which at once generalize the notion of fiber bundles (over polyhedra) and simplicial complexes. We next enlarge the category of submersions to include all direct limits of simplicial bundles and extend the homology functor by a category-theoretic construction. The resultant theory is shown to satisfy axioms of Eilenberg-Steenrod type, and we prove a uniqueness theorem.
Book Synopsis Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture by : Peter B. Gilkey
Download or read book Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture written by Peter B. Gilkey and published by CRC Press. This book was released on 1999-07-27 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: This cutting-edge, standard-setting text explores the spectral geometry of Riemannian submersions. Working for the most part with the form valued Laplacian in the class of smooth compact manifolds without boundary, the authors study the relationship-if any-between the spectrum of Dp on Y and Dp on Z, given that Dp is the p form valued Laplacian and pi: Z ® Y is a Riemannian submersion. After providing the necessary background, including basic differential geometry and a discussion of Laplace type operators, the authors address rigidity theorems. They establish conditions that ensure that the pull back of every eigenform on Y is an eigenform on Z so the eigenvalues do not change, then show that if a single eigensection is preserved, the eigenvalues do not change for the scalar or Bochner Laplacians. For the form valued Laplacian, they show that if an eigenform is preserved, then the corresponding eigenvalue can only increase. They generalize these results to the complex setting as well. However, the spinor setting is quite different. For a manifold with non-trivial boundary and imposed Neumann boundary conditions, the result is surprising-the eigenvalues can change. Although this is a relatively rare phenomenon, the authors give examples-a circle bundle or, more generally, a principal bundle with structure group G where the first cohomology group H1(G;R) is non trivial. They show similar results in the complex setting, show that eigenvalues can decrease in the spinor setting, and offer a list of unsolved problems in this area. Moving to some related topics involving questions of positive curvature, for the first time in mathematical literature the authors establish a link between the spectral geometry of Riemannian submersions and the Gromov-Lawson conjecture. Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture addresses a hot research area and promises to set a standard for the field. Researchers and applied mathematicians interested in mathematical physics and relativity will find this work both fascinating and important.
Book Synopsis Simplicial Bundles and the Homology Structure of Submersions by : Patrick C. Endicott
Download or read book Simplicial Bundles and the Homology Structure of Submersions written by Patrick C. Endicott and published by . This book was released on 1977 with total page 126 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this dissertation we construct a homology spectral sequence attached to a submersion whose E2 term takes values in a certain homology with local coefficients. The motivation for this work is that the spectral sequence provides an effective tool for the conjecture and proof of theorems regarding the global structure of submersions. The spectral sequence is first derived for certain combinatorial objects known as simplicial bundles which at once generalize the notion of fiber bundles (over polyhedra) and simplicial complexes. The spectral sequence of a submersion is then obtained by taking the direct limit of the spectral sequences associated with an approximating system of simplicial bundles.
Book Synopsis Introduction to Differential Topology by : Theodor Bröcker
Download or read book Introduction to Differential Topology written by Theodor Bröcker and published by Cambridge University Press. This book was released on 1982-09-16 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended as an elementary introduction to differential manifolds. The authors concentrate on the intuitive geometric aspects and explain not only the basic properties but also teach how to do the basic geometrical constructions. An integral part of the work are the many diagrams which illustrate the proofs. The text is liberally supplied with exercises and will be welcomed by students with some basic knowledge of analysis and topology.
Book Synopsis Morse Theory and Floer Homology by : Michèle Audin
Download or read book Morse Theory and Floer Homology written by Michèle Audin and published by Springer Science & Business Media. This book was released on 2013-11-29 with total page 595 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an introduction to modern methods of symplectic topology. It is devoted to explaining the solution of an important problem originating from classical mechanics: the 'Arnold conjecture', which asserts that the number of 1-periodic trajectories of a non-degenerate Hamiltonian system is bounded below by the dimension of the homology of the underlying manifold. The first part is a thorough introduction to Morse theory, a fundamental tool of differential topology. It defines the Morse complex and the Morse homology, and develops some of their applications. Morse homology also serves a simple model for Floer homology, which is covered in the second part. Floer homology is an infinite-dimensional analogue of Morse homology. Its involvement has been crucial in the recent achievements in symplectic geometry and in particular in the proof of the Arnold conjecture. The building blocks of Floer homology are more intricate and imply the use of more sophisticated analytical methods, all of which are explained in this second part. The three appendices present a few prerequisites in differential geometry, algebraic topology and analysis. The book originated in a graduate course given at Strasbourg University, and contains a large range of figures and exercises. Morse Theory and Floer Homology will be particularly helpful for graduate and postgraduate students.
Book Synopsis Differential Topology by : Victor Guillemin
Download or read book Differential Topology written by Victor Guillemin and published by American Mathematical Soc.. This book was released on 2010 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. In the years since its first publication, Guillemin and Pollack's book has become a standard text on the subject. It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. The text is mostly self-contained, requiring only undergraduate analysis and linear algebra. By relying on a unifying idea--transversality--the authors are able to avoid the use of big machinery or ad hoc techniques to establish the main results. In this way, they present intelligent treatments of important theorems, such as the Lefschetz fixed-point theorem, the Poincaré-Hopf index theorem, and Stokes theorem. The book has a wealth of exercises of various types. Some are routine explorations of the main material. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem. An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. The book is suitable for either an introductory graduate course or an advanced undergraduate course.
Book Synopsis Singular Intersection Homology by : Greg Friedman
Download or read book Singular Intersection Homology written by Greg Friedman and published by Cambridge University Press. This book was released on 2020-09-24 with total page 823 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first expository book-length introduction to intersection homology from the viewpoint of singular and piecewise linear chains.
Book Synopsis Topics in the Homology Theory of Fibre Bundles by : Armand Borel
Download or read book Topics in the Homology Theory of Fibre Bundles written by Armand Borel and published by Springer. This book was released on 2014-03-12 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Foundations of Vietoris Homology Theory with Applications to Non-compact Spaces by : Robert E. Reed
Download or read book Foundations of Vietoris Homology Theory with Applications to Non-compact Spaces written by Robert E. Reed and published by . This book was released on 1980 with total page 54 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Introduction to Homology Theory by : William S. Massey
Download or read book Introduction to Homology Theory written by William S. Massey and published by . This book was released on 1977 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis An Introduction to Manifolds by : Loring W. Tu
Download or read book An Introduction to Manifolds written by Loring W. Tu and published by Springer Science & Business Media. This book was released on 2010-10-05 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt: Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.
Download or read book Mathematical Chronicle written by and published by . This book was released on 1982 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Lectures on Morse Homology by : Augustin Banyaga
Download or read book Lectures on Morse Homology written by Augustin Banyaga and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a detailed presentation of results needed to prove the Morse Homology Theorem using classical techniques from algebraic topology and homotopy theory. The text presents results that were formerly scattered in the mathematical literature, in a single reference with complete and detailed proofs. The core material includes CW-complexes, Morse theory, hyperbolic dynamical systems (the Lamba-Lemma, the Stable/Unstable Manifold Theorem), transversality theory, the Morse-Smale-Witten boundary operator, and Conley index theory.
Book Synopsis Homology of Analytic Sheaves and Duality Theorems by : V.D. Golovin
Download or read book Homology of Analytic Sheaves and Duality Theorems written by V.D. Golovin and published by Springer. This book was released on 2012-03-14 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: The homology of analytic sheaves is a natural apparatus in the theory of duality on complex spaces. The corresponding apparatus in algebraic geometry was developed by Grothendieck in the fifties. In complex ana lytic geometry the apparatus of homology was missing until recently, and in its stead the hypercohomology of complex sheaves (the hyper-Ext func tors) and the Aleksandrov-Cech homology with coefficients in co presheaves were used. The homology of analytic sheaves, sheaves of germs of homology and homology groups of analytic sheaves, were intro duced and studied in the mid-seventies in a number of papers by the author. The main goal of this book is to give a systematic and detailed account of the homology theory of analytic sheaves and some of its applications to duality theory on complex spaces and to the theory of hyperfunctions. In order to read this book one must be acquainted with the foundations of ho mological algebra and the theory of topological vector spaces. Only the most elementary concepts and results from the theory of functions of sev eral complex variables are assumed to be known. The information needed about sheaves and complex spaces is recounted briefly at the beginning of the fIrst chapter. v. D. Golovin v CONTENTS Chapter 1. ANALYTIC SHEA YES .................................... 1 1. Prelirriinary Information .................................... 1 2. Injectivity Test................................................ 16 3. Local Duality . ....... ... ........ ....... ........... ... ... ..... 24 4. Injective and Global Dimension ........................... 36 5. Properties of Fine Sheaves ................................. 46 Chapter 2. HOMOLOGY THEORY ................................ " .. 63 1. Sheaves of Germs of Homology. . . . . . . . . . . . . . .. . . . . . . . . 63 . . .
Book Synopsis Differential Topology by : Morris W. Hirsch
Download or read book Differential Topology written by Morris W. Hirsch and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 230 pages. Available in PDF, EPUB and Kindle. Book excerpt: "A very valuable book. In little over 200 pages, it presents a well-organized and surprisingly comprehensive treatment of most of the basic material in differential topology, as far as is accessible without the methods of algebraic topology....There is an abundance of exercises, which supply many beautiful examples and much interesting additional information, and help the reader to become thoroughly familiar with the material of the main text." —MATHEMATICAL REVIEWS
Book Synopsis Advanced Classical Field Theory by : G. Giachetta
Download or read book Advanced Classical Field Theory written by G. Giachetta and published by World Scientific. This book was released on 2009 with total page 393 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contemporary quantum field theory is mainly developed as quantization of classical fields. Therefore, classical field theory and its BRST extension is the necessary step towards quantum field theory. This book aims to provide a complete mathematical foundation of Lagrangian classical field theory and its BRST extension for the purpose of quantization. Based on the standard geometric formulation of theory of nonlinear differential operators, Lagrangian field theory is treated in a very general setting. Reducible degenerate Lagrangian theories of even and odd fields on an arbitrary smooth manifold are considered. The second Noether theorems generalized to these theories and formulated in the homology terms provide the strict mathematical formulation of BRST extended classical field theory. The most physically relevant field theories ? gauge theory on principal bundles, gravitation theory on natural bundles, theory of spinor fields and topological field theory ? are presented in a complete way.This book is designed for theoreticians and mathematical physicists specializing in field theory. The authors have tried throughout to provide the necessary mathematical background, thus making the exposition self-contained.
Book Synopsis Noether's Theorems by : Gennadi Sardanashvily
Download or read book Noether's Theorems written by Gennadi Sardanashvily and published by Springer. This book was released on 2016-03-18 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book provides a detailed exposition of the calculus of variations on fibre bundles and graded manifolds. It presents applications in such area's as non-relativistic mechanics, gauge theory, gravitation theory and topological field theory with emphasis on energy and energy-momentum conservation laws. Within this general context the first and second Noether theorems are treated in the very general setting of reducible degenerate graded Lagrangian theory.
