# Homology of bundles over a triangulated base and A∞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 the category of simplices in $B$ with inclusions.

Then $\sigma \hookrightarrow \tau$ in $\mathcal{S}$ gives us a map $\mathcal{S} \to H_{*}(p^{-1}(\sigma)) \to H_{*}(p^{-1}(\tau))$. Ie, a morphism in $\mathcal{S}$ gives us an element of $End(H_{*}(F))$

What I’d like to do in this set-up is now construct a map $H_{*}(\Omega B) \to End(H_{*}(F))$ using something like the monodromy representation.

(1) Does this map exist? I’d really love to see a construction.

(2) If the answer to (1) is “yes”, is this then a map of $A_{\infty}$-algebras?

Details would be most welcome – this kind of thing is hard to track down in the literature…

I think one can do something like the following. Let $M = map([0,1], B)$ and $e:M \to B$ be evaluation at 0: this is a Hurewicz fibration and a homotopy equivalence. Now form the pullback fibration $e^*E \to M$, and consider the composite $\bar{E} := e^*E \to M \to B$. This is a fibration fibrewise homotopy equivalent to your original one, and a point in the fibre $\bar{F}_b$ over $b \in B$ is a path $\gamma$ from $b$ to a $b_1$ and point in $p^{-1}(b_1)$. There is an evident $A_\infty$ action of the $A_\infty$ space $\Omega_b B$ on this fibre by composing $\gamma$ with loops at $b$.
Thus there is an $A_\infty$ map $\Omega_b B \to End(\bar{F}_b)$, which should give you what you want.
Doing the usual Moore loop tricks, one can find an equivalent fibration with an actual action of the grouplike monoid of Moore loops $\Lambda_b B$.