Yves Felix – författare

Rational homotopy is a very powerful tool for differential topology and geometry. This text aims to provide graduates and researchers with the tools necessary for the use of rational homotopy in geometry. Algebraic Models in Geometry has been written for topologists who are drawn to geometrical problems amenable to topological methods and also for geometers who are faced with problems requiring topological approaches and thus need a simple and concrete introductionto rational homotopy. This is essentially a book of applications. Geodesics, curvature, embeddings of manifolds, blow-ups, complex and Kähler manifolds, symplectic geometry, torus actions, configurations and arrangements are all covered. The chapters related to these subjects act as an introduction tothe topic, a survey, and a guide to the literature. But no matter what the particular subject is, the central theme of the book persists; namely, there is a beautiful connection between geometry and rational homotopy which both serves to solve geometric problems and spur the development of topological methods.

Rational homotopy is a very powerful tool for differential topology and geometry. This text aims to provide graduates and researchers with the tools necessary for the use of rational homotopy in geometry. Algebraic Models in Geometry has been written for topologists who are drawn to geometrical problems amenable to topological methods and also for geometers who are faced with problems requiring topological approaches and thus need a simple and concrete introduction to rational homotopy. This is essentially a book of applications. Geodesics, curvature, embeddings of manifolds, blow-ups, complex and Kähler manifolds, symplectic geometry, torus actions, configurations and arrangements are all covered. The chapters related to these subjects act as an introduction to the topic, a survey, and a guide to the literature. But no matter what the particular subject is, the central theme of the book persists; namely, there is a beautiful connection between geometry and rational homotopy which both serves to solve geometric problems and spur the development of topological methods.

Rational homotopy is a very powerful tool for differential topology and geometry. This text aims to provide graduates and researchers with the tools necessary for the use of rational homotopy in geometry. Algebraic Models in Geometry has been written for topologists who are drawn to geometrical problems amenable to topological methods and also for geometers who are faced with problems requiring topological approaches and thus need a simple and concrete introduction to rational homotopy. This is essentially a book of applications. Geodesics, curvature, embeddings of manifolds, blow-ups, complex and Kähler manifolds, symplectic geometry, torus actions, configurations and arrangements are all covered. The chapters related to these subjects act as an introduction to the topic, a survey, and a guide to the literature. But no matter what the particular subject is, the central theme of the book persists; namely, there is a beautiful connection between geometry and rational homotopy which both serves to solve geometric problems and spur the development of topological methods.

as well as by the list of open problems in the final section of this monograph. The computational power of rational homotopy theory is due to the discovery by Quillen [135] and by Sullivan [144] of an explicit algebraic formulation. In each case the rational homotopy type of a topological space is the same as the isomorphism class of its algebraic model and the rational homotopy type of a continuous map is the same as the algebraic homotopy class of the correspond­ ing morphism between models. These models make the rational homology and homotopy of a space transparent. They also (in principle, always, and in prac­ tice, sometimes) enable the calculation of other homotopy invariants such as the cup product in cohomology, the Whitehead product in homotopy and rational Lusternik-Schnirelmann category. In its initial phase research in rational homotopy theory focused on the identi­ of these models. These included fication of rational homotopy invariants in terms the homotopy Lie algebra (the translation of the Whitehead product to the homo­ topy groups of the loop space OX under the isomorphism 11'+1 (X) ~ 1I.(OX», LS category and cone length. Since then, however, work has concentrated on the properties of these in­ variants, and has uncovered some truly remarkable, and previously unsuspected phenomena. For example • If X is an n-dimensional simply connected finite CW complex, then either its rational homotopy groups vanish in degrees 2': 2n, or else they grow exponentially.

as well as by the list of open problems in the final section of this monograph. The computational power of rational homotopy theory is due to the discovery by Quillen [135] and by Sullivan [144] of an explicit algebraic formulation. In each case the rational homotopy type of a topological space is the same as the isomorphism class of its algebraic model and the rational homotopy type of a continuous map is the same as the algebraic homotopy class of the correspond­ ing morphism between models. These models make the rational homology and homotopy of a space transparent. They also (in principle, always, and in prac­ tice, sometimes) enable the calculation of other homotopy invariants such as the cup product in cohomology, the Whitehead product in homotopy and rational Lusternik-Schnirelmann category. In its initial phase research in rational homotopy theory focused on the identi­ of these models. These included fication of rational homotopy invariants in terms the homotopy Lie algebra (the translation of the Whitehead product to the homo­ topy groups of the loop space OX under the isomorphism 11'+1 (X) ~ 1I.(OX», LS category and cone length. Since then, however, work has concentrated on the properties of these in­ variants, and has uncovered some truly remarkable, and previously unsuspected phenomena. For example • If X is an n-dimensional simply connected finite CW complex, then either its rational homotopy groups vanish in degrees 2': 2n, or else they grow exponentially.

as well as by the list of open problems in the final section of this monograph. The computational power of rational homotopy theory is due to the discovery by Quillen [135] and by Sullivan [144] of an explicit algebraic formulation. In each case the rational homotopy type of a topological space is the same as the isomorphism class of its algebraic model and the rational homotopy type of a continuous map is the same as the algebraic homotopy class of the correspond­ ing morphism between models. These models make the rational homology and homotopy of a space transparent. They also (in principle, always, and in prac­ tice, sometimes) enable the calculation of other homotopy invariants such as the cup product in cohomology, the Whitehead product in homotopy and rational Lusternik-Schnirelmann category. In its initial phase research in rational homotopy theory focused on the identi­ of these models. These included fication of rational homotopy invariants in terms the homotopy Lie algebra (the translation of the Whitehead product to the homo­ topy groups of the loop space OX under the isomorphism 11''+1 (X) ~ 1I.(OX», LS category and cone length. Since then, however, work has concentrated on the properties of these in­ variants, and has uncovered some truly remarkable, and previously unsuspected phenomena. For example • If X is an n-dimensional simply connected finite CW complex, then either its rational homotopy groups vanish in degrees 2'': 2n, or else they grow exponentially.

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.

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.

This book introduces a novel framework to study the rational homotopy of a space through the construction of enriched differential graded Lie algebras (dgls), extending Quillen rational homotopy to non-simply connected spaces in a way that is compatible with the Sullivan minimal models approach. Part I contains the basic theory of enriched Lie algebras and associated quadratic Sullivan algebras. Minimal Sullivan algebras and Sullivan rationalizations are then described in Part II. Part III explores the relations between enriched dgl models, Sullivan models, and topological spaces. The connection between enriched dgls and commutative differential graded algebras (cdgas) is realized using a generalization of the cochain algebra functor. This part contains all the theory necessary for computation of explicit examples and for developing interesting applications. Finally, Part IV concerns inert cell attachments and their applications.

This book introduces a novel framework to study the rational homotopy of a space through the construction of enriched differential graded Lie algebras (dgls), extending Quillen rational homotopy to non-simply connected spaces in a way that is compatible with the Sullivan minimal models approach. Part I contains the basic theory of enriched Lie algebras and associated quadratic Sullivan algebras. Minimal Sullivan algebras and Sullivan rationalizations are then described in Part II. Part III explores the relations between enriched dgl models, Sullivan models, and topological spaces. The connection between enriched dgls and commutative differential graded algebras (cdgas) is realized using a generalization of the cochain algebra functor. This part contains all the theory necessary for computation of explicit examples and for developing interesting applications. Finally, Part IV concerns inert cell attachments and their applications.

This proceedings volume centers on new developments in rational homotopy and on their influence on algebra and algebraic topology. Most of the papers are original research papers dealing with rational homotopy and tame homotopy, cyclic homology, Moore conjectures on the exponents of the homotopy groups of a finite CW-c-complex and homology of loop spaces. Of particular interest for specialists are papers on construction of the minimal model in tame theory and computation of the Lusternik-Schnirelmann category by means articles on Moore conjectures, on tame homotopy and on the properties of Poincaré series of loop spaces.

This proceedings volume centers on new developments in rational homotopy and on their influence on algebra and algebraic topology. Most of the papers are original research papers dealing with rational homotopy and tame homotopy, cyclic homology, Moore conjectures on the exponents of the homotopy groups of a finite CW-c-complex and homology of loop spaces. Of particular interest for specialists are papers on construction of the minimal model in tame theory and computation of the Lusternik-Schnirelmann category by means articles on Moore conjectures, on tame homotopy and on the properties of Poincaré series of loop spaces.