Fibrations And Their Classification
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Fibrations And Their Classification
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Author : Petar Pavešić
language : en
Publisher:
Release Date : 2013
Fibrations And Their Classification written by Petar Pavešić and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with Fiber bundles (Mathematics) categories.
The concept of fibration is one of the great unifying mathematical ideas. It was initially introduced around 1930 in geometry and topology, and gradually expanded into many other parts of mathematics. Together with fibre bundles (which precedeed fibrations), they give formal expression to the idea of a continuous family of spaces, and of operations on such families. This monograph contains an exposition of the fundamental ideas of the theory of fibrations with particular emphasis on their classification. It deals at length with various types of fibrations as defined by Hurewicz, Dold and Serre, as well as the quasifibrations of Dold and Thom. The relationship between these concepts is analyzed in depth, with examples and counter-examples given. One of the salient properties of fibre bundles is that they are classified by homotopy classes of maps into some special spaces called classifying spaces. The classifying theory for fibrations is presented both abstractly, through the theory of representable functors, and constructively, by describing various models, like those introduced by Dold and Lashof, and by Milgram and Steenrod. In the couple of decades following their intoduction, the growth of the theory of fibrations resulted in a plethora of similar and interrelated theories and classification results for vector bundles, general fibre bundles, and other types of fibre spaces. As a new organizational principle, Peter May invented the concept of F-fibrations that generalizes all of the above, and is at the same time sufficiently structured to admit workable classification objects. The second part of the book is dedicated to an in-depth discussion of the theory of F-fibrations. The book is reasonably self-contained and the reader is assumed to have only some knowledge of general topology and basic homotopy theory, including elementary properties of homotopy groups. However, one must be aware that the level of exposition is at some places more advanced, and for these a prior course in algebraic topology or in the theory of fibre bundles would be very helpful, both as a motivation for the problems that are studied, as well as a measure of the required mathematical sophistication. The book can be used both as a text-book or as a reference. Most chapters are concluded with historical notes, tracing the origins of the concepts and the developments related to the classification of fibre bundles and fibrations.
Fiber Bundles And Homotopy
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Author : Dai Tamaki
language : en
Publisher: World Scientific
Release Date : 2021-05-17
Fiber Bundles And Homotopy written by Dai Tamaki and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-05-17 with Mathematics categories.
This book is an introduction to fiber bundles and fibrations. But the ultimate goal is to make the reader feel comfortable with basic ideas in homotopy theory. The author found that the classification of principal fiber bundles is an ideal motivation for this purpose. The notion of homotopy appears naturally in the classification. Basic tools in homotopy theory such as homotopy groups and their long exact sequence need to be introduced. Furthermore, the notion of fibrations, which is one of three important classes of maps in homotopy theory, can be obtained by extracting the most essential properties of fiber bundles. The book begins with elementary examples and then gradually introduces abstract definitions when necessary. The reader is assumed to be familiar with point-set topology, but it is the only requirement for this book.
Classifying Spaces And Fibrations
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Author : J. Peter May
language : en
Publisher: American Mathematical Soc.
Release Date : 1975
Classifying Spaces And Fibrations written by J. Peter May and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1975 with Classifying spaces categories.
The basic theory of fibrations is generalized to a context in which fibres, and maps on fibres, are constrained to lie in any preassigned category of spaces [script capital] F. Then axioms are placed on [script capital] F to allow the development of a theory of associated principal fibrations and, under several choices of additional hypotheses on [script capital] F, a classification theorem is proven for such fibrations.
Classification Of Lagrangian Fibrations
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Author : Ricardo Castano Bernard
language : en
Publisher:
Release Date : 2002
Classification Of Lagrangian Fibrations written by Ricardo Castano Bernard and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with categories.
Classification Of Lagrangian Fibrations
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Author : Ricardo Castano Bernard
language : en
Publisher:
Release Date : 2002
Classification Of Lagrangian Fibrations written by Ricardo Castano Bernard and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with categories.
Classification Of Totally Real Elliptic Lefschetz Fibrations Via Necklace Diagrams
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Author : Nermin Salepci
language : de
Publisher:
Release Date : 2011
Classification Of Totally Real Elliptic Lefschetz Fibrations Via Necklace Diagrams written by Nermin Salepci and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with categories.
The Topology Of Classical Groups And Related Topics
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Author : S. Y. Husseini
language : en
Publisher: CRC Press
Release Date : 1969
The Topology Of Classical Groups And Related Topics written by S. Y. Husseini and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1969 with Mathematics categories.
Principal Fibrations
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Author : JEAN-PIERRE. MAYER
language : en
Publisher:
Release Date : 1960
Principal Fibrations written by JEAN-PIERRE. MAYER and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1960 with categories.
AD-2 0 5679 AD-260 568Div. 15U (3 Aug 61) OTS price $1.60 Johns Hopkins U., Baltimore, Md. P INCIPAL FIBRATIONS, b Jean-Pierre Mayer. 1960, 12p. (Contract DA 36-034-ORD-3299) (AROD-2799:1)Unclassified report DESCRIPTORS: *Algebraic topology, Groups (Mathematics). A fibration ith an Eilenberg-MacLane space of type (pi, n), n greater than 1, as fiber and simply-connected base is equivalent to one induced from a path-space fibration by a map of the base into an Eilenberg-MacLane space of type (pi, n + 1). For the purposes of this paper, and by analogy with the classification theory of fiber-bundles, this is called a fabrication principal. T e problem considered the conditions under which a fibration with a loop-space as fiber is equivalent to a principal one. (Author).
Skew Flat Fibrations And Totally Convex Immersions
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Author : Michael Harrison
language : en
Publisher:
Release Date : 2017
Skew Flat Fibrations And Totally Convex Immersions written by Michael Harrison and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with categories.
A fibration of $\R^n$ by oriented copies of $\R^p$ is called skew if no two fibers intersect nor contain parallel directions. Conditions on $p$ and $n$ for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of $\R^3$ by skew oriented lines was given by Salvai, in analogue with the classification of oriented great circle fibrations of $S^3$ by Gluck and Warner. We show that Salvai's classification has a topological variation which generalizes to characterize all continuous fibrations of $\R^n$ by skew oriented copies of $\R^p$. We show that the space of fibrations of $\R^3$ by skew oriented lines deformation retracts to the subspace of Hopf fibrations, and therefore has the homotopy type of a pair of disjoint copies of $S^2$. We discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of $\C^n$ (resp. $\Ham^n$) by skew oriented copies of $\C^p$ (resp. $\Ham^p$). \\An immersion $f: M \to \R^n$ is called totally convex if for each pair of distinct points $x$, $y$ of $M$, the tangent spaces at $f(x)$ and $f(y)$ do not contain parallel directions. We ask: given a smooth manifold $M$, what is the minimum dimension $n=n(M)$ such that there exists a totally convex immersion $M \to \R^n$? The corresponding question has been classically studied for immersions, embeddings, and many other maps satisfying distinguished differential conditions. The problem for totally convex immersions is especially difficult; for example, we do not know $n(S^2)$. We find that totally convex immersions are related to the generalized vector field problem, immersions and embeddings of real projective spaces, and the existence of nonsingular symmetric bilinear maps. We provide both upper and lower bounds on the number $n$. \\The first chapter, dedicated to skew fibrations, is almost entirely the original work of the author and appeared in similar form in Feb.\ 2016 in Math.\ Z. The second chapter, dedicated to totally convex immersions, is similar in spirit to the treatment of totally skew embeddings by Ghomi and Tabachnikov, but the statements are technically new. The results of Chapter 2, along with the future work outlined in the introduction of Chapter 2, will be the subject of a forthcoming article.
Obstructions For Seifert Fibrations And An Extension Of The Bolsinov Fomenko Theorem On Classification Of Hamiltonian Systems
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Author : Dušan Repovš
language : en
Publisher:
Release Date : 1998
Obstructions For Seifert Fibrations And An Extension Of The Bolsinov Fomenko Theorem On Classification Of Hamiltonian Systems written by Dušan Repovš and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998 with categories.