Simplicial polytope complexes and deloopings of $K$-theory

This paper is a continuation of the author’s previous paper on scissors congruence, in which we defined the notion of a polytope complex and its $K$-theory. In this paper we produce formulas for the delooping of a simplicial polytope complex and the cofiber of a morphism of simplicial polytope complexes. Along the way we also prove that the (classical and higher) scissors congruence groups of polytopes in a homogeneous $n$-manifold (with sufficient geometric data) are determined by its local properties.

Keywords

Waldhausen $K$-theory, scissors congruence

2010 Mathematics Subject Classification

13D15, 18D05, 18F25, 19D99

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Published 4 December 2014