Rational homotopy via Sullivan models and enriched Lie algebras

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Rational homotopy theory originated in the late 1960s and the early 1970s with the simultaneous but distinct approaches of Quillen (1969), Sullivan (1977) and Bousfield–Kan (1972). Each approach associated to a path connected space X an “algebraic object” A which is then used to construct a rational completion of X , X \to X_{\mathbb Q} . These constructions are homotopy equivalent for simply connected CW complexes of finite type, in which case H_*(X_{\mathbb Q})\cong H_*(X)\otimes \mathbb Q and \pi_*(X_{\mathbb Q}) \cong \pi_*(X)\otimes \mathbb Q . Otherwise, they may be different; in fact, Quillen’s construction is only available for simply connected spaces. In this review, discussion is limited to Sullivan’s completions, and the notation X\to X_{\mathbb Q} is reserved for these. We briefly review the construction, and follow that with a review of developments and examples over the subsequent decades, but often without the proofs. Since the explicit form of Sullivan’s completion has lent itself to a wide variety of applications in a range of fields, this survey will necessarily be modest in scope.