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Homology of bundles over a triangulated base and $A_\infty$-algebras

Let $p:E \to B$ be a fiber bundle over a triangulated base $B$ with fiber $F$, $\sigma$ simplex in $B$, $\sigma \mapsto H_{*}(p^{-1}(\sigma)) \simeq H_{*}(F)$ the obvious map and let $\mathcal{S}$ be ...