Minimal models in homotopy theory

Abstract

There are three basic constructions in literature which relate a 1-connected topological space X to a differential graded algebra. Adams and Hilton [2] constructed a chain algebra (with integer coefficients) for the loop space ~?X, a special version of which is Adams' cobar construction [1]. Later Quillen [13] associated a differential graded rational Lie algebra 2(X) to the space X, and Sullivan [14,15], using simplicial differential forms with rational coefficients, obtained a DG commutative cochain algebra for X. For these cochain algebras, Sullivan introduces the notion of minimal model, which corresponds to the Postnikov decomposition of a space.In this paper, we construct minimal models for chain algebras (over any field) and for rational DG Lie algebras. These minimal models correspond to the Eckmann-Hilton homology decomposition of a space. The algebraic construction of the minimal model uses algebraic versions of the Hurewicz and BlakersMassey theorems (2.6), (2.9). The corresponding minimal models for topological spaces are studied in § 3. § 0. Recollection of Notations and ResultsOur references for notations are [10] and [13]. For the reader's convenience, we here collect basic definitions and facts we shall use in this paper.

Twisting Cochains and Duality Between Minimal Algebras and Minimal Lie Algebras

“…Thus Chen's Lie algebra model of K determines a rational homology decomposition of K. Since Sullivan's model of K determines a Postnikov tower of K, the duality theory gives a method of computing the rational cell structure of K from a Postnikov tower of K and vice versa. This result is related to the Baues-Lemaire conjecture [2] which asserts that the minimal model of Quillen's Lie algebra model of K (see [1,11]) is isomorphic to the minimal model of the £-construction £(91t£) on the dual of Sullivan's model GJ\LK of K.…”

“…(Similar Lie algebra models have been studied by Quillen [14], Allday [1], Baues and Lamaire [2] and Neisendorfer [13].) Under the duality theory, 911^ and LK correspond, and the pairings (1.1) and (1.…”

“…The topological implication of Theorems (3.7) and (4.1) is that we can inductively compute the ^-invariants of K from the characteristic maps of a homology decomposition of K and vice versa (see [8 or 10] for example). [9] and Baues and Lemaire [2]. We begin by reviewing the basic properties of Quillen's £ and 6-constructions.…”

“…Introduction. Baues and Lemaire [2] have developed an algebraic duality theory between 1-connected minimal cochain algebras of finite type and connected minimal chain Lie algebras of finite type. The duality theory gives a one-to-one correspondence between isomorphism classes of 1-connected minimal cochain algebras of finite type and isomorphism classes of connected minimal chain Lie algebras of finite type.…”

The fibre of a cell attachment

“…Thus over the rationals the attachment is inert iff 5 is null homotopic. We extend this observation to a field of any characteristic in terms of Adams-Hilton models: recall [1,3] that an Adams-Hilton model for a simply-connected space X is a quasi-isomorphism from a differential graded tensor algebra…”