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

## TL;DR

This paper studies $A_
abla$-minimal models of differential graded algebras, expanding formality criteria for manifolds and analyzing the structure of minimal models for sphere and circle bundles over formal manifolds.

## Contribution

It extends Miller's and Crowley-Nordstr"{o}m's theorems on formality and minimal models, providing new bounds and structural results for $A_
abla$-minimal models of manifolds.

## Key findings

- Manifolds with dimension ≤ (l+1)k+2 have $A_
abla$-minimal models with vanishing $m_p$ for p ≥ l.
- Sphere bundles over formal manifolds have $A_
abla$-minimal models with only $m_2$ and $m_3$ non-trivial.
- Necessary and sufficient conditions for the formality of circle bundles over formal symplectic manifolds in low dimensions.

## Abstract

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.