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.
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.)