Coverings and automorphisms of chamber systems
In recent work \textit{J. Tits} [The geometric vein. The Coxeter Festschr. 519-547 (1982; Zbl 0496.51001)] has introduced a new concept of chamber system and its ramified covers which allows him to obtain a local characterization of buildings. The paper under review presents a systematic, elementary and most interesting theory of ramified covers of chamber systems. A chamber system over a set I is a set C whose elements are called chambers together with a system of partitions of C indexed by I which are viewed as maps \(p_ i:\) \(C\to C^ i\), \(i\in I\), where \(C^ i\) is some set. If \(J\subset I\), \(p_ j:\) \(C\to C^ J\) denotes the finest partition such that all \(p_ j\), \(j\in J\), are refinements of it. An element \(\sigma \in C^ j\) is called a cell of cotype J and St \(\sigma\) denotes the chamber system over J whose set of chambers is \(p_ J^{- 1}(\sigma)\) with partition maps \(p_ j| p_ J^{-1}(\sigma)\), \(j\in J\). Let \(A\subset P(I)\) where P(I) is the power set of I. If C is a chamber system over I then an A-cover is a chamber system \(\bar C\) together with a morphism p: \(\bar C\to C\) such that \(p| St \sigma\) is an isomorphism for all \(\sigma \in C^ J\), \(J\in A\). If \(\sigma\) is a cell for which \(p| St \sigma\) is not an injection then p is said to be ramified at p \(\sigma\). The author associates a cell complex \(\Delta\) C with every chamber system C and he shows among other things that if I is finite then p: \(\bar C\to C\) is unramified if and only if the induced map \(\Delta\) \(p: \Delta\) \(\bar C\to \Delta C\) is a topological cover. He also introduces a 2-dimensional complex \(\Gamma_ AC\) and shows that there is a bijection between topological covers of \(\Gamma_ AC\) and A- covers of C. A theory of A-homotopy and universal A-covers is then developed for chamber systems without any prior knowledge of topology. Finally, necessary and sufficient conditions for an automorphism of C to lift to an automorphism of some A-cover \(\bar C\) are obtained as well as conditions allowing the group Aut \(\bar C\) to lift to a section of Aut \(\bar C.\)
scientific article; zbMATH DE number 3881381