# Does the signature admit a homotopy coherent refinement?

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.

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Welcome to Mathoverflow! I have no idea what the answer to this is, but it seems hard. We'd have to know something about, like... the units of L-theory...For KO and tmf we knew stuff about the K(n)−localizations and used Rezk′s magic juice; do we know anything about the K(n)−localizations of L-theory? This sounds hard. But also like buckets of fun. –  Dylan Wilson May 17 '13 at 19:17
I am unfamiliar with L-theory. Nevertheless, I came across a recent article on the arxiv which seems related: "Commutativity properties of Quinn spectra" (arxiv.org/abs/1304.4759). 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/$A_\infty$ 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. Perhaps this article and the references therein will be helpful. –  Ricardo Andrade May 18 '13 at 2:52
@Dylan: if I remember rightly, $L[1/2]$ is equivalent to an Eilenberg-MacLane spectrum, whereas $L_{(2)}$ is equivalent to $kO_{(2)}$, so the $K(n)$-localizations are not hard. However, these are not $E_\infty$ equivalences (or at least not obviously so). –  Neil Strickland May 19 '13 at 17:11
Neil: yes, $L$ is well-understood, but you have it backwards: actually, $L[1/2]=KO[1/2]$; $L_{(2)}$ is EM. A simple check: $L$ is 4-periodic, while $KO_{(2)}$ is only 8-periodic. –  Ben Wieland May 20 '13 at 20:08
@Ricardo: I think your comment answers the question. Thanks! –  Ben Knudsen Nov 9 '13 at 17:21

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 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.