TITLE: Units of MO and MU
QUESTION [13 upvotes]: Real (or complex) cobordism is described by a symmetric ring spectrum MO (or MU respectively) as explained in examples 2.8 and 2.9 here. Associated to such a ring spectrum $R$, we have a unit spectrum $GL_1(R)$ (see for example chapter 22 in the book by May and Sigurdsson. 

Is there a nice description or interpretation of the units $GL_1(MO)$ or $GL_1(MU)$? 

Let me "define" what I mean by nice by giving an analogy. For the units of $K$-theory, we can identify $GL_1(KU) = BU_{\otimes} \times \mathbb{Z}/2\mathbb{Z}$ and we know that this is the classifying space for virtual vector bundles of virtual dimension $\pm 1$, similarly for $KO$.

REPLY [13 votes]: This is really just a comment, but a bit too long. Thanks Tyler.  Ulrich, that is a good question.  I once worked hard to understand the zeroth space of MU (and more simply BP), one prime at a time, calculating Dyer Lashof operations.  I threw away my notes because the answers I was getting seemed unhelpful.  I seem to recall a paper of Steve Wilson that considers the Hopf ring of BP_0, but my memory may be failing me.  The problem is that these spaces are very  large and seem to have nothing like the economical and intuitive description of their K-theory analogues.  It could be a good problem to try to understand what these infinite loop spaces really look like and what they mean geometrically.  A short modern summary of early work on GL_1(R), alias FR, in general is in [114] on my web page. There is work in progress by John Lind that makes honest the idea that BGL_1R is a classifying space for a reasonably concrete kind of ``principal GL_1R-bundle''.