UC Irvine

For a formal differential graded algebra, if extended by an odd degree element, we prove that the extended algebra has an $A_\infty$-minimal model with only $m_2$ and $m_3$ non-trivial. As an application, the $A_\infty$-algebras constructed by Tsai, Tseng and Yau on formal symplectic manifolds satisfy this property. Separately, we expand the result of Miller and Crowley-Nordstr\"{o}m for $k$-connected manifold. In particular, we prove that if the dimension of the manifold $n\leq (l+1)k+2$, then its de Rham complex has an $A_\infty$-minimal model with $m_p=0$ for all $p\geq l$.