Polytopal Linear Groups and Balanced 3-Polytopes

An example of Bruns-Gubeladze K-theory defined by three dimensional polytope

“…This note appeared as a result of an attempt to find a counterexample to the Bruns-Gubeladze conjecture. In fact, we did not succeed, but instead we found a series of balanced Col -divisible polytopes which had not been known before and was omitted in the Faramarzi's classification theorem [6,Theorem 3.2]. We calculate the corresponding elementary groups and as a corollary obtain an expression of the polytopal K-groups in terms of the Quillen K-groups.…”

“…(2) for t > 1 the polytope F (t, 1) is E-equivalent to the polytope P e (t, 1) (for definition of P e4 (t) and P e (t, 1) see [6]), (3) the polytope F (1, 1) (though it is three dimensional) is E-equivalent to the polygon of type (c) (for definition see [4]). It appears that the condition t ≥ s is essential.…”

“…They proved that there are 6 classes of E-equivalence of the polygones and also they calculated the corresponding K-groups. The case of 3dimensional polytopes was investigated in [6]. In that work a classification of the balanced 3-dimensional polytopes up to E-equivalence was proposed and the stable elementary groups of Col -divisible 3-dimensional polytopes were identified.…”

“…We calculate the corresponding elementary groups and as a corollary obtain an expression of the polytopal K-groups in terms of the Quillen K-groups. Despite [6], a classification of balanced (and balanced Col -divisible) 3-dimensional polytopes remains open.…”