A Degeneracy Criterion for $A_\infty$-Structures

In the paper conditions are found under which the $A_\infty$-structure on the homology of a differential algebra degenerates. Examples of such algebras are given. In particular, it is shown that the $A_\infty$-algebra structure on the Milnor coalgebra degenerates, and that if the cohomology of a topological space is a polynomial algebra, then the $A_\infty$-algebra structure on it degenerates.