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](http://www.math.uni-bonn.de/~schwede/SymSpec.pdf). Associated to such a ring spectrum $R$, we have a unit spectrum $GL\_1(R)$ (see for example chapter 22 in the [book](http://books.google.de/books?id=9x7bUzDiojEC&printsec=frontcover&dq=Sigurdsson+May&hl=de&sa=X&ei=zAdUUfG5K8bBswa54YGgDw&ved=0CDAQ6AEwAA#v=onepage&q=Sigurdsson%20May&f=false) by May and Sigurdsson. 



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> Is there a nice description or interpretation of the units $GL\_1(MO)$ or $GL\_1(MU)$? 
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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 $KO$. 



Units of MO and MU
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''. 


