On the homotopy groups of \(p\)-completed classifying spaces
Working one prime at a time has been a successful route into the structure of the homotopy groups of a space. As a consequence of Haynes Miller's solution to the Sullivan conjecture, McGibbon and Neisendorfer proved that a simply-connected CW-complex \(X\) with finite mod 2 cohomology has infinitely many homotopy groups \(\pi_i(X)\) with nontrivial 2-torsion. Generalizations of the algebraic condition, \(H^*(X; {\mathbb F}_2)\) finite, followed with the development of ideas like Lannes's \(T\) functor, \(p\)-local groups, topped off by a theorem of Dwyer and Wilkerson that a 2-connected CW-complex of finite type with nontrivial mod \(p\) cohomology and for which the indecomposable module of \(H^*(X;{\mathbb F}_p)\) is locally finite as a module over the Steenrod algebra has nontrivial \(p\)-torsion in infinitely many homotopy groups. The present paper concerns the gap between connected and 2-connected. The authors prove that a \(p\)-complete space \(X\) for which \(H^*(X;{\mathbb F}_p)\) is of finite type and for which the indecomposable module of \(H^*(X;{\mathbb F}_p)\) is locally finite as a module over the Steenrod algebra must satisfy either \(X\) is aspherical, \(X\) is a finite product of Eilenberg-Mac Lane spaces of the type \(K({\mathbb Z}_p^{^2})\), or \(X\) has infinitely many homotopy groups with nontrivial \(p\)-torsion. The authors further prove the same results for the particular case of \(X = BG\) for \(G\) a finite group, a \(p\)-local finite group, a finitely generated virtually nilpotent group, or a quasi-\(p\)-perfect group of finite virtual mod \(p\) cohomology.
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