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This question came back to my mind while pondering this MO question. The classification of principal bundles is seriously difficult because of our lack of understanding of homotopy groups of compact Lie groups. Although I only understand very little of it, I have seen that the work of Bendersky, Davis, Mahowald, Mimura and others has resulted in the complete computation of $v_1$-periodic homotopy groups of compact Lie groups. Moreover, these seem to describe pretty well the torsion in the homotopy of compact Lie groups in suitably low dimensions. So here are some questions:

  1. is it possible to compute the homotopy set $[M,v_1BG]$ where $v_1BG$ should be the $v_1$-periodization of the classifying space $BG$ of a compact Lie group $G$ (i.e. the space whose homotopy are the $v_1$-periodic homotopy groups of $BG$)?

  2. what can be said about the relation between the above set and the homotopy set $[M,BG]$ where $M$ is a finite CW-complex? I would expect that this is possible for $M$ of suitably low dimension, but is it possible to quantify "suitably low-dimensional" here?

  3. would the above points 1 and 2 allow for improvements in our understanding of classification of principal bundles? For instance, it seems that complete classification of principal $SO(n)$-bundles is possible up to dimensions around 6 or 7. Do we get classification results from $v_1$-periodic homotopy groups on CW-complexes of larger dimension?

I would be grateful for any comments, and I apologize if what I have said above does not capture the meaning of $v_1$-periodic homotopy groups. As I said, I only know very little about these matters.

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