Rational isomorphisms of \(p\)-compact groups
The notion of a \(p\)-compact group was introduced by \textit{W. G. Dwyer} and \textit{C. W. Wilkerson} as a homotopy theoretic candidate for a replacement of compact Lie groups. They defined in [Ann. Math., II. Ser. 139, 395-442 (1994; Zbl 0801.55007)] a \(p\)-compact group to be an \(\mathbb{F}_p\)-complete loop space with finite \(\mathbb{F}_p\)-cohomology and with a finite \(p\)-group as component group. Using some ideas and results of \textit{P. F. Baum} [Pac. J. Math. 22, 197-204 (1967; Zbl 0178.02802)] about the local isomorphism of compact connected Lie groups, the author proves in this paper the following main results: Theorem 1.1. Suppose that \(X_1\) and \(X_2\) are two locally isomorphic connected \(p\)-compact groups and that \(X_1 @<q_1<< Y \times S @>q_2>> X_2\) are special isogenies, an isogeny between two \(p\)-compact groups is an epimorphism whose kernel is a finite \(p\)-group. For any isogeny \(f : X_1 \to X_2\) there exist an automorphism \(g\) of \(Y\) and a self-isogeny \(h\) of \(S\) such that the diagram of \(p\)-compact group homomorphisms \[ \begin{tikzcd} Y \times S \ar[r,"g \times h"]\ar[d,"q_1"'] & Y \times S \ar[d,"q_2"]\\ X_1 \ar[r,"f"'] & X_2 \end{tikzcd} \] commutes up to conjugacy; \(g\) and \(h\) are uniquely determined up to conjugacy. Furthermore, \(f\) is an isomorphism if and only if both \(g\) and \(h\) are automorphism. Theorem 1.2. Let \(X_1\) and \(X_2\) be two connected \(p\)-compact groups with maximal tori \(i_1 : T_1 \to X_1\) and \(i_2 : T_2 \to X_2\). For any homomorphism \(f : X_1 \to X_2\) there exists a homomorphism \(\varphi : T_1 \to T_2\) such that the diagram of \(p\)-compact group homomorphisms \[ \begin{tikzcd} T_1 \ar[r,"\varphi"]\ar[d,"i_1"'] & T_2\ar[d,"i_2"]\\ X_1 \ar[r,"f"'] & X_2\end{tikzcd} \] commutes up to conjugacy; the conjugacy class of the homomorphism \(\varphi\) is unique up to the action of the Weyl group \(X_2\). Assume additionally that \(X_1\) and \(X_2\) are locally isomorphic so that we may take \(T_1 = T_2\). Then \(\varphi\) is an isogeny which is an automorphism if and only if \(f\) is an isomorphism. Theorem 1.3. Any nontrivial endomorphism \(f : X \to X\) of a connected simple \(p\)-compact group \(X\) is a rational isomorphism, and if \(p\) divides the order of the Weyl group, even an automorphism. The paper contains any more results and the author and D. Notbohm have in press [J. Reine Angew. Math.] a related paper used also in the present paper.
scientific article; zbMATH DE number 863700