Vol. 74, No. 2, 1978

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Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces

Joseph Alvin Neisendorfer

Vol. 74 (1978), No. 2, 429–460
Abstract

This paper establishes that the homotopy category of rational differential graded commutative coalgebras is equivalent to the homotopy category of rational differential graded Lie algebras which have a nilpotent completion as homology. This generalizes a result which Quillen proved in the simply connected case. When combined with Sullivan’s work on rational homotopy theory, our result shows that the homotopy category of rational differential graded Lie algebras with nilpotent finite type homology is equivalent to the rational homotopy category of nilpotent topological spaces with finite type rational homology.

Our results include the construction of minimal Lie algebra models for simply connected spaces, and we show that the rational homotopy groups of a simply connected CW complex may be calculated from a free Lie algebra generated by the cells with a differential given on generators by the attaching maps.

Mathematical Subject Classification 2000
Primary: 55P62
Secondary: 55P65, 55U35
Milestones
Received: 4 October 1976
Published: 1 February 1978
Authors
Joseph Alvin Neisendorfer