Lie Models for Spaces
Description
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.
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Content
Enriched and Pre-Enriched Lie Algebras.- Enriched Vector Spaces.- Lower Central Series.- The Quadratic Sullivan Model of an Enriched Lie Algebra.- Representations of Enriched Lie Algebras.- Profree Lie Algebras.- Sullivan Rational Homotopy Theory.- The Homotopy Lie Algebra L_V of a Minimal Sullivan Algebra.- The Sullivan Rationalization of X_Q of a Space X.- Sullivan Rational Spaces.- Enriched dgl's and Semi-quadratic Sullivan Algebras.- Profree dgl's and Profree dgl Models.- The Model Category of Enriched dgls.- The Profree dgl Model of a cdga and of a Topological Space.- Cylinder Objects and dgl Homotopy.- Topological Cell Attachments.- Inert Attachments.- Applications in Topology.
