$A_\infty$-Minimal Model on Differential Graded Algebras

The rational homotopy type of a differential graded algebra (DGA) can be represented by a family of tensors on its cohomology, which constitute an $A_\infty$-minimal model of this DGA. When only the cohomology is needed to determine the rational homotopy type, then the DGA is called formal. By a theorem of Miller, a compact $k$-connected manifold is formal if its dimension is not greater than $4k+2$. We expand this theorem and a result of Crowley-Nordstr\"{o}m to prove that if the dimension of a compact $k$-connected manifold $N\leq (l+1)k+2$, then its de Rham complex has an $A_\infty$-minimal model with $m_p=0$ for all $p\geq l$. Separately, for an odd-dimensional sphere bundle over a formal manifold, we prove that its de Rham complex has an $A_\infty$-minimal model with only $m_2$ and $m_3$ non-trivial. In the special case of a circle bundle over a formal symplectic manifold satisfying the hard Lefschetz property, we give a necessary condition for formality which becomes sufficient when the base symplectic manifold is of dimension six or less.