Question

I read in a paper by Christopher Douglas that third cohomology twists of $K$-theory may be interpreted as TMF-classes via a map $K(\mathbb{Z},3) \to TMF$, which is related to String orientations. How exactly is this map constructed? Could it be that there is an extension to higher twists, i.e. is there an extension to $BBU_{\otimes} \to TMF$?

EDIT: I know that $BBU_{\otimes}$ splits off $K(\mathbb{Z},3)$ as a factor. Therefore there is of course a map $BBU_{\otimes} \to TMF$, which factors over $K(\mathbb{Z},3)$. The corresponding classes of TMF, however, only see the ordinary third-cohomology twists. So, I reshould restate the second question as something like: Is there an extension $BBU_{\otimes} \to TMF$, which "sees" higher twists?

Answer 1

Inspired by the title of your question, you should look at Twists of K-theory and TMF by Ando-Blumberg-Gepner. For twists of $TMF$, there is a map $K(\mathbb{Z},4) \to BGL_1(TMF)$, and the latter classifies the most general twists that can occur. The discussion of what you're interested in starts on page 21, and twists of K-theory are indeed related to $BBU_\otimes$.

It depends on the existence of an orientation constructed by Ando, Hopkins, and Rezk in "Multiplicative orientatons of KO-theory and the spectrum of topological modular forms".

Answer 2

You can construct the map $K(\mathbb{Z},3) \to tmf$ as follows: first there is the String orientation of tmf, which you already mention. This is a map $$ MString \to tmf$$ Then String is by definition a $K(\mathbb{Z},2)$-fibration over Spin. This yields in particular a map $$ K(\mathbb{Z},3) \to MString $$ Then you can construct the map $K(\mathbb{Z},3) \to tmf$ as the composition of the above two maps. In order to extend this constuction you had to find a map $BBU_\otimes \to MString$. I think such a map does not exist apart from the one you describe, but I am not entirely sure.