Vol. 113, No. 1, 1984

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ISSN: 0030-8730
Rational homotopy theory of fibrations

Flavio E. A. da Silveira

Vol. 113 (1984), No. 1, 1–34
Abstract

Let Y be a space and A a differential graded algebra over the field Q of rationals corresponding in Sullivan’s theory to the rational homotopy type of Y . Then to the rational homotopy type of a fibration over Y equipped with a given cross-section corresponds a differential graded Lie algebra L over A, free as an A-module. The differential graded Lie algebra Q AL corresponds in Quillen’s theory to the rational homotopy type of the fibre of the fibration. Furthermore, by restriction of scalars, L can be considered as a differential graded Lie algebra over Q. Then it contains a differential graded Lie sub-algebra over Q which corresponds to the rational homotopy type of the space of cross-sections which are homotopic to the given cross-section. Some examples illustrate this result.

Mathematical Subject Classification 2000
Primary: 55P62
Secondary: 55R05
Milestones
Received: 23 June 1980
Published: 1 July 1984
Authors
Flavio E. A. da Silveira