nLab On PL deRham theory and rational homotopy type

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• Aldridge Bousfield, Victor Gugenheim:

  On PL deRham theory and rational homotopy type,

  Memoirs of the AMS, vol. 179 (1976)

  ams:memo-8-179

on rational homotopy theory via a model structure on dg-algebras (hence on objects dual to the dg-Lie algebras originally considered in Quillen 69).

Related references:

• Sergei Gelfand, Yuri Manin, Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original, Springer 1996. xviii+372 pp. 2nd corrected ed. 2002 (doi:10.1007/978-3-662-12492-5)

Review:

• Kathryn Hess, Rational homotopy theory: a brief introduction, contribution to Summer School on Interactions between Homotopy Theory and Algebra, University of Chicago, July 26-August 6, 2004, Chicago (arXiv:math.AT/0604626), chapter in Luchezar Lavramov, Dan Christensen, William Dwyer, Michael Mandell, Brooke Shipley (eds.), Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436, AMS 2007 (doi:10.1090/conm/436)

• Joshua Moerman, Rational Homotopy Theory, 2015 (pdf, pdf)

Generalization to equivariant rational homotopy theory (fundamental theorem of equivariant rational homotopy theory?):

• Laura Scull, A model category structure for equivariant algebraic models, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 (doi:10.1090/S0002-9947-07-04421-2)

1. The simplicial Algebra $\nabla$

• polynomial differential forms on simplices

• PL de Rham complex

2. The polynomial de Rham Theory

• PL de Rham theorem

3. Multiplicative structure and the Eilenberg-Moore theorem

4. A Quillen Homotopy Theory for DG Algebras

• model structure on connective dgc-algebras

5. Function Spaces for DG Algebras

6. The Homotopy Relation for DG Algebras

7. Minimal DG Algebras

• minimal Sullivan model

8. The de Rham functors and their Adjoints

• Quillen adjunction between simplicial sets and connective dgc-algebras

9. The Sullivan-de Rham Equivalence Therem

• fundamental theorem of dg-algebraic rational homotopy theory

10. Proof of the Sullivan-de Rham Equivalence Therem

11. The Sullivan-de Rham Localization and Homotopy Theorems

• rationalization via the fundamental theorem of dg-algebraic rational homotopy theory

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category: reference