# Units of MO and MU

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 $KO$.

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One comment is that if $S$ is the sphere spectrum there is a map $U\to GL_1(S)$ (resp. $O \t GL_1(S)$) and the description of MU explicitly makes the composite $U\to GL_1(S) \to GL_1(MU)$ nullhomotopic as a map of infinite loop spaces (similarly for $MO$). Another is that even though $MO$ is equivalent to a product of Eilenberg-Mac Lane objects, $GL_1(MO)$ is (as an infinite loop space) not. $GL_1$ is tough, and a good chunk of literature in orientation theory (much by Peter May) is devoted to determining information about it. – Tyler Lawson Mar 28 '13 at 12:41
Given the title, I thought you were gonna discuss "units of Mathoverflow and Mathunderflow (aka math.SE) ---- but this turned out to be bait-and-switch ;-) – Suvrit Mar 28 '13 at 18:41

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''.
Can you be more specific about what you mean when you say you were calculating Dyer-Lashof operations? Do you mean on $H_* MU$ etc.? – Sean Tilson Mar 28 '13 at 17:43