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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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    $\begingroup$ 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. $\endgroup$
    – Tyler Lawson
    Commented Mar 28, 2013 at 12:41
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    $\begingroup$ 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 ;-) $\endgroup$
    – Suvrit
    Commented Mar 28, 2013 at 18:41
  • $\begingroup$ Dear Ulrich, did you ever find any useful information about these unit groups? $\endgroup$
    – Tyler Lawson
    Commented Dec 15, 2016 at 5:26

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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''.

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  • $\begingroup$ 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.? $\endgroup$
    – Sean Tilson
    Commented Mar 28, 2013 at 17:43
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    $\begingroup$ There are DL operations on H_*(BP) and they are easy to compute by comparison with the DL operations on Eilenberg MacLane spectra. That is in Mark Steinberger's contribution to "H_{\infty} ring spectra and their applications", [21] on my web site. So, no, I was not computing in H_*(MU) but rather in the homology of the zeroth space of BP. The point is that the unit components of that space (or of the zeroth space of MU) are what is relevant to the question. $\endgroup$
    – Peter May
    Commented Mar 29, 2013 at 0:56
  • $\begingroup$ Thanks for the clarification, I am a big fan of this corollary of Steinberger's. $\endgroup$
    – Sean Tilson
    Commented Mar 29, 2013 at 4:49

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