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A rational homotopy form of a differential step algebra (DGA) can be represented by a family of cohomological tensors that constitute the $A_\infty$ minimal model of this DGA. A DGA is called formal if only cohomology is required to determine the rational homotopy type. By Miller’s theorem, a compact $k$ connected manifold is formal if the dimension is less than or equal to $4k+2$. Expanding on this theorem and the result of Crowley-Nordstr \”{o}m, we can say that the compact $k$ connected manifold $N\leq (l+1)k+2$ has dimension complex where all $For p\geq l$ we have a $A_\infty$-minimum model with $m_p=0$. We prove that the field $has an A_\infty$-minimal model where only $m_2$ and $m_3$ are non-trivial. gives a formality requirement that is sufficient if the dimension of the elementary symplectic manifold is less than or equal to 6.

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