BOUSFIELD AND GUGENHEIM Introduction In this paper we will develop and extend the rational PL de Rham theory of Sullivan. Several different approaches to rational homotopy theory have been devised over the years. In the early 1950*s, Serre introduced the idea of doing homotopy theory "modulo a class of groups." In practice his homo- topy theory "modulo torsion groups" amounted to investigating properties that showed up when working "over the rationals," and it was found to be enormously simpler than ordinary homotopy theory. A more recent approach involves "rational spaces," i.e., spaces whose homotopy groups are rational vector spaces. Each simply connected space can be approximated up to "rational equivalence" by a "rational space," and "rational homotopy theory" is essentially the ordinary homotopy theory of "rational spaces." Quillen made these ideas precise in his fundamental paper [Quillen (RHT)], and he proved that certain relatively simple algebraic categories are equivalent to the rational homotopy category in the simply connected case. The trouble with Quillen1s proof is its complexity: The desired equivalence is the composite of a long chain of intermediate equivalences. At the end of his program, Quillen solved a problem of Thorn—that of con- structing commutative cochains over the rationals. More recently Sullivan introduced his rational PL de Rham theory for simplicial complexes and applied it to show that the ordinary de Rham complex of a simply connected smooth manifold Mn determines the "real homotopy type" of Mn, e.g., it determines R ® TT^M11 as well as H*(Mn R). An exposition of this work is given in [Friedlander-Griffiths-Morgan]. Sullivan's theory starts with a solution to Thorn*s problem: He gives a de Rham-like construction of commutative cochains which works for any coefficient field of characteristic zero and any simplicial complex. We have learned from [Swan] that Thorn actually gave the same construction (over the reals) in an unpublished lecture in 1959. For X any simply iv
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