Locally finite approximation of Lie groups. I (Q1059304)

The authors prove for any compact Lie group G and almost every prime p that there exists a ''locally finite approximation away from p.'' Such a locally finite approximation is a map \(\Phi\) : \(B\Gamma\) \(\to BG\) from the classifying space \(B\Gamma\) of a countable union \(\Gamma\) of finite groups to the classifying space BG of G which satisfies various properties, including the property that \(\Phi\) induces an isomorphism in cohomology with coefficients any finite \(\pi_ 0(G)-module\) whose order is prime to p. This property had been earlier proved for compact connected Lie groups by the authors [Comment. Math. Helv. 59, 347-361 (1984; Zbl 0548.55016)]. The authors propose that such an approximation of the classifying space of a compact Lie group BG gives new homotopy theoretic information about BG, reversing the technique of D. Quillen and others who used homotopy theoretic information concerning BG to obtain similar information concerning the classifying spaces for finite groups. As an example of this point of view, the authors apply their approximation theorem in conjunction with \textit{H. Miller'}s recent proof of the Sullivan conjecture [Ann. Math., II. Ser. 120, 39-87 (1984; Zbl 0552.55014)] to provide a reasonably explicit description of the homotopy groups of mapping complexes with source BG and target a finite dimensional, connected complex.