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Urs has taught us a lot about lifting up structures through the Whitehead tower of the orthogonal group. What happens if you begin with a different Lie group, say, the symplectic group or $G_2$?
What do you mean by ’what happens’? The theory is non specific in general and interprets according to the example …. or are you looking form much more than that vague handwave. :-)
So for instance in the applications that involve lifts trough $B String \to B SO$ there is in general also the question of lifting through $(B SU)\langle 4\rangle \to B SU$.
(In the application to the heterotic string, neigther lift needs to exist by itself, but the obstructions to each need to cancel each other.)
I was just wondering whether for some reason there’s an intrinsically greater interest in lifting from the orthogonal group rather than from elsewhere. In the former case there are special names for increasingly connected covers, spin(n) and string(n), and a successful search for smooth models. Should the 3-connected cover of $SU(n)$, say, be given this treatment - a new name and a smooth model?
Edit: Just seen we have a section on this.
I was just wondering whether for some reason there’s an intrinsically greater interest in lifting from the orthogonal group rather than from elsewhere.
There is maybe not a very general reason, but there is a specific reason in that this case just so happens to be interesting for a fairly large class of phenomena that are already known to appear elsewhere.
One is this: the first few lifts through the Whitehead tower of $O$ correspond to orientations in generalized cohomology for important generalized cohomology theories: $SO$-structure is ordinary orientation, $Spin$-structure is orientation in $K O$-theory and then $String$-structure is orientation in tmf. That’s where much of the pure math interest in String-structures comes from.
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