H^{{{\rm d i m}\,}X + {{\bullet}}}(X)\,.$$ Similarly, for the homology spectral sequence associated to an increasing filtration $F^{p-1}X \subset F^p X \subset \dots $ on a space $X$ we have $$\label{E1}
E^1_{p,q}(X) = H_{p+q} (F^{-p} X , F^{-p-1} X)$$ and $$\label{d1}
d_1 :  E^1_{p,q}(X)  \to E^1_{p+1,q}(X)\,.$$

Hochschild cochain complex of an $A_{\infty}$-algebra
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It is known that for every graded vector space $A$ $$\label{Hom-A-A}
\bigoplus_{k \ge 0} {{\bf s}}^k\, {{\rm Hom}\,}(A^{\otimes \, k}, A)$$ is equipped with a ${{\Lambda}}{{\bf Lie}}$-algebra structure. The corresponding bracket is the Gerstenhaber bracket $[\,,\,]_G$. It is given by the formula $$[Q_1, Q_2]_{G} =$$ $$\label{Gerst}
\sum_{i=1}^{k_1}(-1)^{{{\varepsilon}}_{i, k_2}}
Q_1(a_1,\,\dots , Q_2 (a_i,\, \dots, a_{i+k_2-1}),\, \dots,
a_{k_1+k_2-1}) -
(-1)^{(k_1 + 1)(k_2 + 1)} (1 \leftrightarrow 2)\,,$$ where $k_1$ (resp. $k_2$) is the degree of $Q_1$ (resp. $Q_2$) in (\[Hom-A-A\]) and $${{\varepsilon}}_{i,k_2} = (k_2+1)(|a_1| + \dots + |a_{i-1}| + i -1 )\,.$$

Let us recall that an $A_{\infty}$-algebra structure on $A$ is an element $$\label{flat}
m \in \bigoplus_{k \ge 1} {{\bf s}}^k\, {{\rm Hom}\,}(A^{\otimes \, k}, A)$$ satisfying the Maurer-Cartan equation: $$\label{MC}
[m,m]_{G} = 0\,.$$

Equation (\[MC\]) implies that the formula $$\label{Hoch}
{{\partial}}^{Hoch} = [m, ~~ ]_{G}$$ defines a differential on the ${{\Lambda}}{{\bf Lie}}$-algebra (\[Hom-A-A\]). In particular, restricting $m$ 