On the Differentials of the Spectral Sequence of a Fibre Bundle

Abstract

Let $\xi=(X,p,B,G)$ be a principal $G$-bundle, $F$ be a $G$ space and $\eta=(E,p,B,F)$ be the associated bundle with the fiber $F$. Generally $\xi$ and the action $H_*(G)\otimes H_*(F)\to H_*(F)$ of the Pontriagin ring $H_*(G)$ on $H_*(F)$ do not define homologies of $E$. In this paper we define a two sequences of operations $\{f^i:H_*(G)^{\otimes i}\to H_*(G), i=3,4,...\}$, which we call Hochschild twisting cochain (with respect to Gerstenhaber product), and which in fact form on $H_*(G)$ an $A(\infty$-algebra structure), and $\{\bar{f}^i:H_*(G)^{\otimes (i-1)}\otimes H*(F)\to H_*(F), i=3,4,...\}$ (which in fact form on $H_*(F)$ an $A(\infty)$-module structure over the $A(\infty)$-algebra $(H_*(G),\{f^i\})$) and show that $\xi$ and these higher structures define $H_*(E)$.