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There is much activity around the study of highly connected covers of Lie groups (well, of their "infinite rank" versions like $\displaystyle{\lim_{N\to\infty}} \ O(N)$, say).

Looking at the resulting homotopy groups, one starts with a sequence of period 8 (Bott periodicity), replaces some starting terms with zeroes, and leaves the remaining tail intact. Current names for the results seem to be string, fivebrane, etc. groups.

On the other hand, in view of things like tmf and higher chromatic phenomena it is natural to replace with zeroes an infinite subsequence of some higher period (24 or 576 or...). This would correspond to taking homotopy fibers of maps from the group not to an Eilenberg-MacLane space but rather to some space which itself has periodic homotopy groups - either to an infinite product of Eilenberg-MacLane spaces (killing a periodic sequence of characteristic classes), or to something more subtle having nontrivial k-invariants, I have no idea what.

Have such constructions been carried out? What are the resulting groups/H-spaces and their classifying spaces?

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  • $\begingroup$ Googling "highly connected cover" points to a very small number of results. What do you refer to in your 1st sentence? $\endgroup$
    – YCor
    Commented Oct 29, 2016 at 20:11
  • $\begingroup$ @YCor I meant the stuff from subsequent links (they go to nLab). In short, these are Whitehead covers - having first few homotopy groups trivial and inducing isomorphism on all the higher ones. For example, a careful construction of a 3-connected cover for a simply connected topological group has been carried out by Stolz in the 5th section of "A conjecture concerning positive Ricci curvature and the Witten genus" (Math. Ann. 304 (1996) 785–800). Do you believe I should add some references at that place? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 29, 2016 at 20:22
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    $\begingroup$ Thanks, at least it gives me an idea of the topic (such as a correct interpretation of the word cover of a topological group $G$, namely realizing $G$ as quotient of a "larger" topological group). $\endgroup$
    – YCor
    Commented Oct 29, 2016 at 20:42

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