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Author: Martin Majewski
Publisher: American Mathematical Soc.
ISBN: 0821819208
Category : Mathematics
Languages : en
Pages : 175
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Book Description
The main goal of this paper is to prove the following conjecture of Baues and Lemaire: the differential graded Lie Tlgebra associated with the Sullivan model of a space is homotopy equivalent to its Quillen model. In addition we show the same for the cellular Lie algebra model which we build from the simplicial analog of the classical Adams-Hilton model. It turns out that this cellular Lie algebra model is one link in a chain of models connecting the models of Quillen and Sullivan.The key result which makes all this possible is Anick's correspondence between differential graded Lie algebras and Hopf algebras up to homotopy. In addition we show that the Quillen model is a rational homotopical equivalence, and we conclude the same for the other models using our main result. Theconstruction of the three models is given in detail. The background from homotopy theory, differential algebra, and algebra is presented in great generality.
Author: Martin Majewski
Publisher: American Mathematical Soc.
ISBN: 0821819208
Category : Mathematics
Languages : en
Pages : 175
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Book Description
The main goal of this paper is to prove the following conjecture of Baues and Lemaire: the differential graded Lie Tlgebra associated with the Sullivan model of a space is homotopy equivalent to its Quillen model. In addition we show the same for the cellular Lie algebra model which we build from the simplicial analog of the classical Adams-Hilton model. It turns out that this cellular Lie algebra model is one link in a chain of models connecting the models of Quillen and Sullivan.The key result which makes all this possible is Anick's correspondence between differential graded Lie algebras and Hopf algebras up to homotopy. In addition we show that the Quillen model is a rational homotopical equivalence, and we conclude the same for the other models using our main result. Theconstruction of the three models is given in detail. The background from homotopy theory, differential algebra, and algebra is presented in great generality.
Author: Yves Felix
Publisher: Springer Science & Business Media
ISBN: 146130105X
Category : Mathematics
Languages : en
Pages : 574
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Book Description
Rational homotopy theory is a subfield of algebraic topology. Written by three authorities in the field, this book contains all the main theorems of the field with complete proofs. As both notation and techniques of rational homotopy theory have been considerably simplified, the book presents modern elementary proofs for many results that were proven ten or fifteen years ago.
Author: Wen-tsün Wu
Publisher: Springer
ISBN: 3540390251
Category : Mathematics
Languages : en
Pages : 228
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Book Description
This comprehensive monograph provides a self-contained treatment of the theory of I*-measure, or Sullivan's rational homotopy theory, from a constructive point of view. It centers on the notion of calculability which is due to the author himself, as are the measure-theoretical and constructive points of view in rational homotopy. The I*-measure is shown to differ from other homology and homotopy measures in that it is calculable with respect to most of the important geometric constructions encountered in algebraic topology. This approach provides a new method of treatment and leads to various new results. In particular, an axiomatic system of I*-measure is formulated, quite different in spirit from the usual Eilenberg-Steenrod axiomatic system for homology, and giving at the same time an algorithmic method of computation of the I*-measure in concrete cases. The book will be of interest to researchers in rational homotopy theory and will provide them with new ideas and lines of research to develop further.
Author: Phillip Griffiths
Publisher: Springer Science & Business Media
ISBN: 1461484685
Category : Mathematics
Languages : en
Pages : 228
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Book Description
This completely revised and corrected version of the well-known Florence notes circulated by the authors together with E. Friedlander examines basic topology, emphasizing homotopy theory. Included is a discussion of Postnikov towers and rational homotopy theory. This is then followed by an in-depth look at differential forms and de Tham’s theorem on simplicial complexes. In addition, Sullivan’s results on computing the rational homotopy type from forms is presented. New to the Second Edition: *Fully-revised appendices including an expanded discussion of the Hirsch lemma *Presentation of a natural proof of a Serre spectral sequence result *Updated content throughout the book, reflecting advances in the area of homotopy theory With its modern approach and timely revisions, this second edition of Rational Homotopy Theory and Differential Forms will be a valuable resource for graduate students and researchers in algebraic topology, differential forms, and homotopy theory.
Author: Steve Halperin
Publisher: World Scientific
ISBN: 9814651451
Category : Mathematics
Languages : en
Pages : 449
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Book Description
This research monograph is a detailed account with complete proofs of rational homotopy theory for general non-simply connected spaces, based on the minimal models introduced by Sullivan in his original seminal article. Much of the content consists of new results, including generalizations of known results in the simply connected case. The monograph also includes an expanded version of recently published results about the growth and structure of the rational homotopy groups of finite dimensional CW complexes, and concludes with a number of open questions.This monograph is a sequel to the book Rational Homotopy Theory [RHT], published by Springer in 2001, but is self-contained except only that some results from [RHT] are simply quoted without proof.
Author: Urtzi Buijs
Publisher: Springer Nature
ISBN: 3030544303
Category : Mathematics
Languages : en
Pages : 302
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Book Description
Since the birth of rational homotopy theory, the possibility of extending the Quillen approach – in terms of Lie algebras – to a more general category of spaces, including the non-simply connected case, has been a challenge for the algebraic topologist community. Despite the clear Eckmann-Hilton duality between Quillen and Sullivan treatments, the simplicity in the realization of algebraic structures in the latter contrasts with the complexity required by the Lie algebra version. In this book, the authors develop new tools to address these problems. Working with complete Lie algebras, they construct, in a combinatorial way, a cosimplicial Lie model for the standard simplices. This is a key object, which allows the definition of a new model and realization functors that turn out to be homotopically equivalent to the classical Quillen functors in the simply connected case. With this, the authors open new avenues for solving old problems and posing new questions. This monograph is the winner of the 2020 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.
Author: Henar Herrero
Publisher: Springer Nature
ISBN: 3031620259
Category :
Languages : en
Pages : 159
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Book Description
Author: David Chataur
Publisher: American Mathematical Soc.
ISBN: 1470428873
Category : Mathematics
Languages : en
Pages : 122
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Book Description
Let X be a pseudomanifold. In this text, the authors use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. The authors do it simplicially in the setting of a filtered version of face sets, also called simplicial sets without degeneracies, in the sense of C. P. Rourke and B. J. Sanderson. They define perverse local systems over filtered face sets and intersection cohomology with coefficients in a perverse local system. In particular, as announced above when X is a pseudomanifold, the authors get a perverse local system of cochains quasi-isomorphic to the intersection cochains of Goresky and MacPherson, over a field. We show also that these two complexes of cochains are quasi-isomorphic to a filtered version of Sullivan's differential forms over the field Q. In a second step, they use these forms to extend Sullivan's presentation of rational homotopy type to intersection cohomology.
Author: Mark Hovey
Publisher: American Mathematical Soc.
ISBN: 0821843613
Category : Mathematics
Languages : en
Pages : 229
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Book Description
Model categories are used as a tool for inverting certain maps in a category in a controllable manner. They are useful in diverse areas of mathematics. This book offers a comprehensive study of the relationship between a model category and its homotopy category. It develops the theory of model categories, giving a development of the main examples.
Author: Dorina Mitrea
Publisher: American Mathematical Soc.
ISBN: 082182659X
Category : Mathematics
Languages : en
Pages : 137
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Book Description
The general aim of the present monograph is to study boundary-value problems for second-order elliptic operators in Lipschitz sub domains of Riemannian manifolds. In the first part (ss1-4), we develop a theory for Cauchy type operators on Lipschitz submanifolds of co dimension one (focused on boundedness properties and jump relations) and solve the $Lp$-Dirichlet problem, with $p$ close to $2$, for general second-order strongly elliptic systems. The solution is represented in the form of layer potentials and optimal non tangential maximal function estimates are established.This analysis is carried out under smoothness assumptions (for the coefficients of the operator, metric tensor and the underlying domain) which are in the nature of best possible. In the second part of the monograph, ss5-13, we further specialize this discussion to the case of Hodge Laplacian $\Delta: =-d\delta-\delta d$. This time, the goal is to identify all (pairs of) natural boundary conditions of Neumann type. Owing to the structural richness of the higher degree case we are considering, the theory developed here encompasses in a unitary fashion many basic PDE's of mathematical physics. Its scope extends to also cover Maxwell's equations, dealt with separately in s14. The main tools are those of PDE's and harmonic analysis, occasionally supplemented with some basic facts from algebraic topology and differential geometry.
