Almost formality of manifolds of low dimension

In this paper we introduce the notion of Poincar'e DGCAs of Hodge type, which is a subclass of Poincar'e 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 Poincar'e DGCA of Hodge type. Using these concepts, we investigate the equivalence class of $(r-1)$ connected $(r>1)$ Poincar'e DGCAs of Hodge type. In particular, we show that a $(r-1)$ connected Poincar'e DGCA of Hodge type ${\mathcal{A}}^{\ast }$ of dimension $n\le 5r-3$ is $A_{\mathrm{\infty }}$-quasi-isomorphic to an $A_3$-algebra and prove that the only obstruction to the formality of ${\mathcal{A}}^{\ast }$ is a distinguished Harrison cohomology class $[\mu_3] \in {$\mathsf{H}\mathsf{a}\mathsf{r}\mathsf{r}$}^{3,-1} (H^({\mathcal A}^\ast), H^({\mathcal A}^\ast))$\mathrm{.}Moreover,thecohomologyclass$[\mu_3]$ and the DGCA isomorphism class of $H^{\ast }({\mathcal{A}}^{\ast })$ determine the $A_{\mathrm{\infty }}$-quasi-isomorphism class of ${\mathcal{A}}^{\ast }$. This can be seen as a Harrison cohomology version of the Crowley-Nordstr"om results [D. Crowley, J. Nordstr"om, The rational homotopy type of $(n-1)$-connected manifolds of dimension up to $5n-3$, arXiv:1505.04184v2] 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 $G_2$-manifolds, which have been discovered recently by Chan-Karigiannis-Tsang in [K.F. Chan, S. Karigiannis and C.C. Tsang, The ${\mathcal{L}}_B$-cohomology on compact torsion-free ${\mathrm{G}}_2$ manifolds and an application to `almost’ formality, arXiv:1801.06410, to appear in Ann. Global Anal. Geom.], from our results and the Cheeger-Gromoll splitting theorem.