Intrinsic formality and certain types of algebras

Lupton, Gregory

doi:10.1090/S0002-9947-1990-1005081-0

Intrinsic formality and certain types of algebras
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Trans. Amer. Math. Soc. 319 (1990), 257-283 Request permission

Abstract:

In this paper, a type of algebra is introduced and studied from a rational homotopy point of view, using differential graded Lie algebras. The main aim of the paper is to establish whether or not such an algebra is the rational cohomology algebra of a unique rational homotopy type of spaces. That is, in the language of rational homotopy, whether or not such an algebra is intrinsically formal. Examples are given which show that, in general, this is not so—7.8 and 7.9. However, whilst it is true that not all such algebras are intrinsically formal, some of them are. The main results of this paper show a certain class of these algebras to be intrinsically formal—Theorem $2$ (6.1); and a second, different type of algebra also to be intrinsically formal—Theorem $1$ (5.2), which type of algebra overlaps with the first type in many examples of interest. Examples are given in $\S 7$.

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