Rational Homotopy Theory II 1st Edition by Steve Halperin, Yves Felix, Jean claude Thomas 9814651427 9789814651424
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ISBN 10: 9814651427
ISBN 13: 9789814651424
Author: Steve Halperin; Yves Felix; Jean-claude Thomas
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.
Table of contents:
1. Basic definitions and constructions
1.1 Graded algebra
1.2 Differential graded algebra
1.3 Simplicial sets
1.4 Polynomial differential forms
1.5 Sullivan algebras
1.6 The simplicial and spatial realizations of a Λ-algebra
1.7 Homotopy and based homotopy
1.8 The homotopy groups of a minimal Sullivan algebra
2. Homotopy Lie algebras and Sullivan Lie algebras
2.1 The homotopy Lie algebra of a minimal Sullivan algebra
2.2 The fundamental Lie algebra of a Sullivan 1-algebra
2.3 Sullivan Lie algebras
2.4 Primitive Lie algebras and exponential groups
2.5 The lower central series of a group
2.6 The linear isomorphism (˄sV)# = UL
2.7 The fundamental group of a 1-finite minimal Sullivan algebra
2.8 The homology Hopf algebra of a 1-finite minimal Sullivan algebra
2.9 The action of GL on πn(| ˄ V, d|, *)
2.10 Formal Sullivan 1-algebras
3. Fibrations and Λ-extensions
3.1 Fibrations, Serre fibrations and homotopy fibrations
3.2 The classifying space fibration and Postnikov decompositions of a connected CW complex
3.3 Λ-extensions
3.4 Existence of minimal Sullivan models
3.5 Uniqueness of minimal Sullivan models
3.6 The acyclic closure of a minimal Sullivan algebra
3.7 Sullivan extensions and fibrations
4. Holonomy
4.1 Holonomy of a fibration
4.2 Holonomy of a Λ-extension
4.3 Holonomy representations for a Λ-extension
4.4 Nilpotent and locally nilpotent representations
4.5 Connecting topological and Sullivan holonomy
4.6 The holonomy action on the homotopy groups of a fibre .
5. The model of the fibre is the fibre of the model
5.1 The main theorem
5.2 The holonomy action of π1(Y, *) on π*(F)
5.3 The Sullivan model of a universal covering space
5.4 The Sullivan model of a spatial realization
6. Loop spaces and loop space actions
6.1 The loop cohomology coalgebra of (˄V, d)
6.2 The transformation map ηL
6.3 The graded Hopf algebra, H*(|˄ U|; Q)
6.4 Connecting Sullivan algebras with topological spaces
7. Sullivan spaces
7.1 Sullivan spaces
7.2 The classifying space BG
7.3 The Sullivan 1-model of BG
7.4 Malcev completions
7.5 The morphism m|˄V,d| : (˄V, d) → APL(|˄ V, d|)
7.6 When BG is a Sullivan space
8. Examples
8.1 Nilpotent and rationally nilpotent groups
8.2 Nilpotent and rationally nilpotent spaces
8.3 The groups Z# … #Z
8.4 Semidirect products
8.5 Orientable Riemann surfaces
8.6 The classifying space of the pure braid group Pn is a Sullivan space
8.7 The Heisenberg group
8.8 Seifert manifolds
8.9 Arrangement of hyperplanes
8.10 Connected sum of real projective spaces
8.11 A final example
9. Lusternik-Schnirelmann category
9.1 The LS category of topological spaces and commutative cochain algebras
9.2 The mapping theorem
9.3 Module category and the Toomer invariant
9.4 cat = mcat
9.5 cat = e(–)#
9.6 Jessup’s Theorem
9.7 Example
10. Depth of a Sullivan algebra and of a Sullivan Lie algebra
10.1 Ext, Tor and the Hochschild-Serre spectral sequence
10.2 The depth of a minimal Sullivan algebra
10.3 The depth of a Sullivan Lie algebra
10.4 Sub Lie algebras and ideals of a Sullivan Lie algebra
10.5 Depth and relative depth
10.6 The radical of a Sullivan Lie algebra
10.7 Sullivan Lie algebras of finite type
11. Depth of a connected graded Lie algebra of finite type
11.1 Summary of previous results
11.2 Modules over an abelian Lie algebra
11.3 Weak depth
12. Trichotomy
12.1 Overview of results
12.2 The rationally elliptic case
12.3 The rationally hyperbolic case
12.4 The gap theorem
12.5 Rationally infinite spaces of finite category
12.6 Rationally infinite CW complexes of finite dimension
13. Exponential growth
13.1 The invariant log index
13.2 Growth of graded Lie algebras
13.3 Weak exponential growth and critical degree
13.4 Approximation of log index L
13.5 Moderate exponential growth
13.6 Exponential growth
14. Structure of a graded Lie algebra of finite depth
14.1 Introduction
14.2 The spectrum
14.3 Minimal sub Lie algebras
14.4 The weak complements of an ideal
14.5 L-equivalence
14.6 The odd part of a graded Lie algebra
15. Weight decompositions of a Sullivan Lie algebra
15.1 Weight decompositions
15.2 Exponential growth of L
15.3 The fundamental Lie algebra of 1-formal Sullivan algebra
16. Problems
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Tags: Steve Halperin, Yves Felix, Jean claude Thomas, Rational