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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.
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 Kahler 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 Kahler 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.
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