Enhanced \(A_{\infty }\)-obstruction theory

An \(A_n\)-algebra \(A= (A,m_1, m_2, \ldots , m_n)\) is a special kind of \(A_\infty \)-algebra satisfying the \(A_\infty \)-relations involving just the \(m_i\) listed. We consider obstructions to extending an \(A_{n-1}\) algebra to an \(A_n\)-algebra. We enhance the known techniques by extending the Bousfield–Kan spectral sequence to apply to the homotopy groups of the space of minimal (i.e.?\(m_1=0)\)\(A_\infty \)-algebra structures on a given graded projective module. We also consider the Bousfield–Kan spectral sequence for the moduli space of \(A_\infty \)-algebras. We compute up to the \(E_2\) terms and differentials \(d_2\) of these spectral sequences in terms of Hochschild cohomology.