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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...

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3 Answers

up vote 2 down vote accepted

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 $\Lambda_b B$.

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I would say such a map exists, but the target is the homology of the H-space of homotopy autoequivalences of the fiber. In the smooth case (i.e. when the fiber, the base and the total space are smooth manifolds) such a map exists on the level of H-spaces: one can equip the bundle with a connection and the resulting holonomy gives a map of H-spaces from $\Omega B$ to the autoequivalences of $F$. Then one can take the resulting homology map. I'm pretty sure there is a simplicial analog of this.

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I don't remember the details, but this is pretty much what K. Igusa does as the first step in his construction of higher Franz-Reidemeister torsion invariants. Have a look at his book and some of the papers on the arXiv (arXiv:math/0212383 and arXiv:math/0303047).

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This is one of the questions I wrote in my notes from reading that book last year. But I think the higher torsion construction is too fancy for this question and doesn't exactly address the above (at least I didn't think it did last year....). I'll look in there again, though. –  Romeo Sep 15 '10 at 21:11
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