This is based on work with Domenico Fiorenza, Kotaro Kawai and Hong Van Le.

A classical result by Miller states that any $(k-1)$-connected closed oriented manifold of dimension $\leq 4k-2$ is formal. More recently, Crowley and Nordstr\”om defined the

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A classical result by Miller states that any $(k-1)$-connected closed oriented manifold of dimension $\leq 4k-2$ is formal. More recently, Crowley and Nordstr\”om defined the

*Bianchi-Massey tensor*and showed that its vanishing is the only obstruction to the formality of a $(k-1)$-connected closed oriented manifold of dimension $\leq 5k-3$. Moreover, recently Chan-Karigiannis-Tsang showed that closed $G_2$-manifolds are*almost formal*, meaning that their deRham algebra is equivalent to a DGA whose differential vanishes in all but one dimension.... more