A faithful unitary representation of the 2-compact group \(DI(4)\)
A \(p\)-compact group, as introduced by \textit{W. G. Dwyer} and \textit{C. W. Wilkerson} in [Ann. Math. (2) 139, No.~2, 395--442 (1994; Zbl 0801.55007)], is the homotopical analogue at a prime \(p\) of a compact Lie group. In fact, the \(p\)-completion of a compact Lie group with a \(p\)-group of components yields such an object and basically the only other, exotic, \(2\)-compact group is \(DI(4)\) (a complete classification has been obtained by \textit{K. K. S. Andersen} and \textit{J. Grodal} after this article has been written [``The classification of \(2\)-compact groups'', J. Am. Math. Soc. 22, No.~2, 387--436 (2009)]). The article under review is concerned with finding a faithful unitary homotopy representation of \(DI(4)\), i.e. a map \(BDI(4) \rightarrow B\text{U}(n)^{\wedge}_2\) with mod \(2\) finite homotopy fibre, which makes therefore \(H^*(BDI(4);\mathbb{F}_2)\) a finitely generated module over \(H^*(B\text{U}(n);\mathbb{F}_2)\). The main result is the construction of such a representation for \(n=2^{46}\). Together with previous work of \textit{N. Castellana} at odd primes [``Representacions homotòpiques de grups \(p\)-compactes'', Ph.D. Thesis, Univ. Autón. Barcelona, Bellatera (2000) and Trans. Am. Math. Soc. 358, No.~7, 2799--2819 (2006; Zbl 1145.55014)] this shows by a case-by-case investigation based on the classification that every \(p\)-compact group admits a faithful unitary representation. The strategy is to consider the centralizer decomposition diagram used by \textit{W. G. Dwyer} and \textit{C. W. Wilkerson} in [J. Am. Math. Soc. 6, No.~1, 37--64 (1993; Zbl 0769.55007)] to construct \(BDI(4)\). The largest centralizer is \(\text{Spin}(7)\) and the author carefully chooses a representation of the (discrete approximation of the) maximal torus normalizer which he shows extends to a homotopy compatible family of maps from the \(p\)-stubborn subgroup decomposition diagram of \(\text{Spin}(7)\). Such a map was proven by the author in [J. Pure Appl. Algebra 212, No.~6, 1525--1541 (2008; Zbl 1138.55009)] to factor through a map \(B\text{Spin}(7) \rightarrow B\text{U}(2^{46})^{\wedge}_2\) (it is unnecessary to refer to the Ph. D. Thesis of the author here). The hardest part is then to prove that this homotopy representation of \(\text{Spin}(7)\) extends to a homotopy compatible family of maps from the centralizer decomposition diagram and the last step is the computation of obstructions, the triviality of which yields the desired representation of \(DI(4)\).
scientific article; zbMATH DE number 5544378
scientific article