Enhanced $$A_{\infty }$$-obstruction theory

Abstract

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.