On the Differentials of the Spectral Sequence of a Fibre Bundle
Abstract
Description
Let $ξ=(X,p,B,G)$ be a principal $G$-bundle, $F$ be a $G$ space and $η=(E,p,B,F)$ be the associated bundle with the fiber $F$. Generally $ξ$ 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 $ξ$ and these higher structures define $H_*(E)$.
This is the English version of the paper published originally in Russian where an A(infty) algebra structure in homology first show up in terms of Hochshild twisting cochains with respect to Gerstenhaber product
This is the English version of the paper published originally in Russian where an A(infty) algebra structure in homology first show up in terms of Hochshild twisting cochains with respect to Gerstenhaber product