Almost formality of manifolds of low dimension

In this paper we introduce the notion of Poincaré DGCAs of Hodge type, which is a subclass of Poincaré DGCAs encompassing the de Rham algebras of closed orientable manifolds. Then we introduce the notion of the small algebra and the small quotient algebra of a Poincaré DGCA of Hodge type. Using these concepts, we investigate the equivalence class of (r-1)-connected (r > 1) Poincaré DGCAs of Hodge type. In particular, we show that an (r-1)-connected Poincar ́e DGCA of Hodge type A* of dimension n <= 5r - 3 is A-infinity-quasi-isomorphic to an A_3-algebra and prove that the only obstruction to the formality of A* is a distinguished Harrison cohomology class [μ3] in Harr^3-1(H*(A*), H*(A*)). Moreover, the cohomology class [μ3] and the DGCA isomorphism class of H*(A*) determine the A-infinity-quasi-isomorphism class of A*. This can be seen as a Harrison cohomology version of the Crowley- Nordstrom results on rational homotopy type of (r - 1)-connected (r > 1) closed manifolds of dimension up to 5r -3. We also derive the almost formality of closed G2 -manifolds, which have been discovered recently by Chan-Karigiannis- Tsang, from our results and the Cheeger-Gromoll splitting theorem.