An algebraic model for finite loop spaces
When Dwyer and Wilkerson introduced \(p\)-compact groups in [\textit{W. G. Dwyer} and \textit{C. W. Wilkerson}, Ann. Math. (2) 139, No. 2, 395--442 (1994; Zbl 0801.55007)], they managed to provide a \(p\)-local version of finite loop spaces. A compact Lie group defines a \(p\)-compact group if it is connected, and more generally if its group of connected components is a \(p\)-group. The technical issue here is that the classifying space of a \(p\)-compact group is required to be \(p\)-complete. The theory of \(p\)-local compact groups, [the authors, Geom. Topol. 11, 315--427 (2007; Zbl 1135.55008)], built in analogy with that of \(p\)-local finite groups, [\textit{C. Broto} et al., J. Am. Math. Soc. 16, No. 4, 779--856 (2003; Zbl 1033.55010)], solves this problem as the authors proved that one can associate to \textit{any} compact Lie group a \(p\)-local compact group (consisting of a discrete \(p\)-toral ``Sylow'' subgroup and fusion data).NEWLINENEWLINEHowever, if the original motivation behind this is the study of \(p\)-local analogues of finite loop spaces, there is still a natural question left to answer: Can one construct, at the prime \(p\), any finite loop space as a \(p\)-local compact group? This is precisely what is achieved in this article. If \(X\) is a space such that \(\Omega X\) is mod \(p\) finite, consider the universal cover fibration \(\tilde X \rightarrow X \rightarrow B \pi\), where \(\pi= \pi_1 X\) is not necessarily a \(p\)-group. What one knows is that \(\tilde X^\wedge_p\) is the classifying space of a \(p\)-compact group, hence that of a \(p\)-local compact group. A careful study of \textit{extensions} of finite groups by \(p\)-local compact groups allows the authors to conclude that so is \(X^\wedge_p\). This goes through a classification of principal bundles over \(B\pi\) with fiber a \(p\)-local compact group, generalizing the study in [\textit{C. Broto} et al., Trans. Am. Math. Soc. 359, No. 8, 3791--3858 (2007; Zbl 1145.55013)] for \(p\)-local finite groups. The link with the combinatorial nature of \(p\)-local compact groups is given by simplicial twisting functions.
scientific article; zbMATH DE number 6369101
scientific article