The Karpelevič Compactification

Abstract

This compactification was introduced by Karpelevic in [K3]. The original inductive definition of ${\overline X ^K}$ is recalled in § 5.3. By examining the closure of a flat A · o in ${\overline X ^K}$, a non-inductive characterization of the closure ${\overline {{A^ + } \cdot o} ^K}$ of ${\overline {A \cdot o} ^K}$ is obtained (see Theorem 5.6). The nature of the Karpelevič topology restricted to the flat is clarified by the introduction of the class of K- fundament al sequences. Using this concept, one shows that (mathtype) is isomorphic to a compactification of a determined by its polyhedral structure. This compactification of a, referred to as the Karpelevič compactification of a, is used to give a new proof that the Karpelevič topology is compact. Lemma 5.26, Proposition 5.27, and Corollary 5.28 explain the relations between the Karpelevič compactification and the conic and dual cell compactifications. Finally, in Remark 5.32 a new way to define the Karpelevič compactification is presented. It consists of fitting together the Karpelevič compactifications of the flats kA · o, k ⊀ K, in exactly the same way that the dual cell compactification is obtained from the polyhedral compactifications of the flats kA · o.