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Rational homotopy theory II

Author: Y Félix; Stephen Halperin; J -C Thomas
Publisher: New Jersey : World Scientific, [2015] ©2015
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Félix, Y. (Yves).
Rational homotopy theory II.
New Jersey : World Scientific, 2015
(DLC) 2014046652
(OCoLC)897510744
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Y Félix; Stephen Halperin; J -C Thomas
ISBN: 9789814651448 9814651443
OCLC Number: 904437886
Notes: Sequel to Rational homotopy theory (2001), but self contained--Introduction, following table of contents.
Description: 1 online resource (xxxvi, 412 pages)
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 [symbol]-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 [symbols]. 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 G[symbol] on [symbols]. 2.10. Formal Sullivan 1-algebras. 3. Fibrations and [symbol]-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. [symbol]-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 [symbol]-extension. 4.3. Holonomy representations for a [symbol]-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 [symbols] on [symbols]. 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 ([symbols]). 6.2. The transformation map [symbols]. 6.3. The graded Hopf algebra, [symbols]. 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 [symbols]. 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. 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 indexL. 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.
Other Titles: Rational homotopy theory two
Responsibility: Yves Félix (Catholic University of Louvain, Belgium), Steve Halperin (University of Maryland, USA), Jean-Claude Thomas (Universite d'Angers, France).

Abstract:

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  Read more...

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