Cobordism genera can often be refined to $E\_\infty$-orientations in the sense of Ando-Blumberg-Gepner-Hopkins-Rezk:


1) the mod 2 Euler characteristic $MO\to H\mathbb{F}\_2$;


2) the $\widehat A$-genus $MSpin\to KO$ (Ando-Hopkins-Rezk, Joachim);


3) the Todd genus $MSpin^c\to K$ (Joachim);


4) the Witten genus $MString\to tmf$ (Ando-Hopkins-Rezk).


Now, the signature (and Arf invariant, etc.) lifts to an $\mathbb{L}$-theory orientation for $PL$-bundles (e.g. Ranicki), and $MPL$ and $\mathbb{L}$ are both $E\_\infty$-ring spectra.


My question is the following: is it known whether this orientation has an $E\_\infty$ refinement? If the answer is yes, I would appreciate a reference.



Does the signature admit a homotopy coherent refinement?
[Since my comment above appears to have been helpful, I am repeating it here.]


I must admit I am unfamiliar with L-theory. Nevertheless, I came across a recent article on the arXiv which is related: [*Commutativity properties of Quinn spectra*](http://arxiv.org/abs/1304.4759) by Gerd Laures and James McClure. It states in remark 1.4 that the Sullivan–Ranicki orientation from $MSTop$ to L-theory is a ring map of symmetric ring spectra, which I assume to mean a map of associative or $A\_\infty$ monoids. Immediately before that remark, it is also stated that the authors are unaware of any previous result on the multiplicativity of the symmetric signature.


