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In section 3 of the paper https://www.sciencedirect.com/science/article/pii/S0723086907000151

The author constructs a fiber bundle $(\rho_n)\zeta$ by taking the pullback of the diagram

$S^8\xrightarrow{\rho_n} S^8$ with $\zeta$ above the rightmost $S^8$ (sorry dont know who to draw it online). We're treating the sphere's as the one point compactifications of the octonions. Here the map $\rho_n$ is just the map sending an octonion to it's $n$th power.
In the next paragraph the author starts talking about a vector bundle $(\rho_n)^!\zeta$. It doesn't seem to be explained what this exclamation notation means, and I haven't seen it before. Any ideas?

EDIT/UPDATE: I'm tempted to believe that the exclamation is meant to indicate the isomorphism class of $(\rho_n)\zeta$ in $KO(S^8)$, however if that were the case, wouldn't the author also be using the notation $n^!\zeta$ to talk about the class of $n\zeta$ in $KO(S^8)$? Instead they just use $n\zeta$...

EDIT #2: I'm nearly convinced that there is a typo and the initial definition of $(\rho_n)\zeta$ should have included an exclamation. I have attempted to contact the author and will update when and if he responds.
The professor who passed this paper on to me is currently out of the country, but when he returns I will also ask him if I don't have answer yet, and share his take.

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    $\begingroup$ Since I have no evidence of the following, I will just comment. I believe that in the pullback diagram the author meant to use the ! notation, thus defining $(\rho_n)^! \zeta$ as the pullback of $\zeta$. As you observed, the notation refers to a specific bundle since he speaks of the clutching function, so it should not refer to an isomorphism class of bundles. The author also makes distinctions later if he means that two bundles are equivalent in some KO group or when they are stably equivalent. $\endgroup$ – Michael Harrison Dec 19 '17 at 16:38
  • $\begingroup$ I suppose I should have made an update sooner, but I've reached the same conclusion. I still can't be %100 sure, but it seems to be the most reasonable explanation. I emailed the author a few days ago, but have received no response, and will post when and if I do. $\endgroup$ – Kristaps John Balodis Dec 20 '17 at 3:03
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I think it's marking a "shriek map". I can't currently view your paper, but I would guess you're looking at an integration along fibers or a Gysin homomoprhism. (But you might not.)

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  • $\begingroup$ And they say mathematicians have no sense of humor. $\endgroup$ – Tiwa Aina Dec 13 '17 at 5:14
  • $\begingroup$ I glanced over your links, and I don't think that's it. They talk about $(\rho_n)^!\zeta$ having the clutching function $S^7\rightarrow SO(8)$ where $a$ maps to $a^nx$, so they're definitely talking a specific bundle. $\endgroup$ – Kristaps John Balodis Dec 13 '17 at 6:53

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