Smooth Differential Forms

Abstract

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 π_k(M) ⊗ ℚ, $\Bbbk$ ≥ 2 and the rational LS category of M) can be computed directly from A _DR (M).