Homotopy equivalences of \(p\)-completed classifying spaces of finite groups
The paper deals with the homotopy properties of the \(p\)-completed classifying space \(BG^\wedge_p\) of a finite group \(G\) for a given prime \(p\). More precisely the following two questions are studied: when are two \(p\)-completed classifying spaces \(BG^\wedge_p\) and \(BG^{\prime\wedge}_p\) for finite groups \(G\) and \(G'\) homotopy equivalent, and what is the homotopy type of the topological monoid \(\Aut(BG^\wedge_p)\) of homotopy self equivalences of \(BG^\wedge_p\). Concerning the first question, \textit{J. Martino} and \textit{S. Priddy} stated in [Math. Proc. Camb. Philos. Soc. 119, 119--137 (1996; Zbl 0855.55008)] that \(BG^\wedge_p\) and \(BG^{\prime\wedge}_p\) are homotopy equivalent if and only if there is a fusion preserving isomorphism \(S \to S'\) between the \(p\)-Sylow subgroups of \(G\) and \(G'\), where an isomorphism \(\varphi:S \to S'\) is called fusion preserving if for any two subgroups \(P\) and \(Q\) of \(S\) a group isomorphism \(\alpha:P \to Q\) is induced by conjugation in \(G\) if and only if the corresponding isomorphism \(\varphi\alpha\varphi^{-1}:\varphi(P) \to \varphi(Q)\) is induced by a conjugation in \(G'\). The proof Martino and Priddy gave in [loc. cit.] however turned out to be incorrect, as found out by the authors of the paper under consideration. Instead of fusion preserving isomorphisms the authors investigate linking categories. The linking category \({\mathcal L}_p(G)\) of a group \(G\) is a special case of a ``categorie associée à une \(p\)-localité'' in the sense of \textit{L. Puig} [Structure locale dans les groupes finis, Bull. Soc. Math. Fr., Suppl., Mém. 47 (1976; Zbl 0355.20024)]. The objects of \({\mathcal L}_p(G)\) are the \(p\)-subgroups of \(G\), and for two \(p\)-subgroups \(P\) and \(Q\) the set of morphisms \(\text{Mor}_{{\mathcal L}_p(G)}(P,Q)\) is given by the quotient of \(N_G(P,Q)=\{x\in G\mid xPx^{-1} \leq Q\}\) by the action of \(O^p(C_G(P))\), where \(O^p(C_G(P))\) denotes the smallest normal subgroup of \(p\)-power index contained in the centralizer \(C_G(P)\) of \(P\) in \(G\). Relevant for the present context is the full subcategory \({\mathcal L}^c_p(G)\) of \({\mathcal L}_p(G)\) whose objects are the \(p\)-centric subgroups of \(G\), i.e., those \(p\)-subgroups \(P\) of \(G\) for which the center \(Z(P)\) is contained in the \(p\)-Sylow subgroup of the centralizer \(C_G(P)\). The first main result of the paper says that \(BG^\wedge_p\) and \(BG^{\prime\wedge}_p\) are homotopy equivalent if and only if the two centric linking categories \({\mathcal L}^c_p(G)\) and \({\mathcal L}^c_p(G')\) are equivalent. In fact the authors also show that \(BG^\wedge_p\) is homotopy equivalent to the \(p\)-completed classifying space \(B{\mathcal L}^c_p(G)^\wedge_p\). The answer to the second question given in the paper says that the topological monoid \(\Aut(BG^\wedge_p)\) is homotopy equivalent to the classifying space of the category of isotypical self equivalences of \({\mathcal L}^c_p(G)\), where a self equivalence \(\psi\) of \({\mathcal L}^c_p(G)\) is called isotypical if the forgetful functor \(\lambda\) from \({\mathcal L}^c_p(G)\) to the category of finite groups is naturally isomorphic to the composite \(\lambda\circ\psi\). The homotopy type of the identity component of \(\Aut(BG^\wedge_p)\) was already determined in [\textit{C. Broto} and \textit{R. Levi}, Topology 41, 229--255 (2002; Zbl 1059.55007)], so calculating the group of components of \(\Aut(BG^\wedge_p)\) was of primary interest here. The authors also investigate a natural homomorphism from the group of components of \(\Aut(BG^\wedge_p)\) to the group of fusion preserving automorphisms of a \(p\)-Sylow subgroup of \(G\) and identify kernel and cokernel of this homomorphism with certain higher limits terms. While the kernel in general does not vanish, recent work of the third author shows that the cokernel is always trivial, giving a new proof of the statement of Martino and Priddy. Along the way the authors introduce the new concept of linking category of a space which also motivates considering linking categories from an abstract point of view. In fact, the classifying space of an abstract linking category can be regarded as a homotopy theoretic version of a finite group, similarly as a connected \(p\)-compact group can be regarded as a homotopy theoretic version of a connected compact Lie group. Recently much activity arose in studying abstract linking categories which finally led to the concept of \(p\)-local finite groups [cf. \textit{C. Broto, R. Levi} and \textit{B. Oliver} [J. Am. Math. Soc. 16, No. 4, 779--856 (2003; Zbl 1033.55010)] and further to the concept of \(p\)-local compact groups, which unifies the concept of \(p\)-local finite groups on the one hand and the concept of \(p\)-compact groups on the other.
scientific article; zbMATH DE number 1965474