Smooth Differential Forms

  • Yves Félix
  • Stephen Halperin
  • Jean-Claude Thomas
Part of the Graduate Texts in Mathematics book series (GTM, volume 205)


The construction \(A_{PL} (--;\Bbbk )\) of polynomial differential forms in §10 was suggested by the classical cochain algebra A DR (M) of smooth differential forms on a smooth manifold M. In this section we review the construction of A DR (M) and establish a chain of quasi-isomorphisms
$$ A_{PL} (M)\xrightarrow{ \simeq } \cdots \xleftarrow{ \simeq }A_{PL} (M;\mathbb{R}) $$
of commutative cochain algebras. This implies (§12) that A DR (M) and A PL (M; ℝ) have the same minimal Sullivan algebras and hence that many rational homotopy invariants (e.g. dim πk(M) ⊗ ℚ, \(\Bbbk \) ≥ 2 and the rational LS category of M) can be computed directly from A DR (M).




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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Yves Félix
    • 1
  • Stephen Halperin
    • 2
  • Jean-Claude Thomas
    • 3
  1. 1.Institut MathematiquesUniversite de Louvain La NeuveLouvain-la-NeuveBelgium
  2. 2.College of Computer, Mathematical, and Physical ScienceUniversity of MarylandCollege ParkUSA
  3. 3.Faculte des SciencesUniversite d’AngersAngersFrance

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