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In this paper we reduce the generalized Hilbert's third problem about Dehn invariants and scissors congruence classes to the injectivity of certain Cheeger–Chern–Simons invariants. We also establish a version of a conjecture of Goncharov relating scissors congruence groups of polytopes and the algebraic K-theory of C.
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A Trace Map on Higher Scissors Congruence Groups
Abstract Cut-and-paste $$K$$-theory has recently emerged as an important variant of higher algebraic $$K$$-theory. However, many of the powerful tools used to study classical higher algebraic $$K$$-theory do not yet have analogues in the cut-and-paste setting. In particular, there does not yet exist a sensible notion of the Dennis trace for cut-and-paste $$K$$-theory. In this paper we address the particular case of the $$K$$-theory of polyhedra, also called scissors congruence $$K$$-theory. We introduce an explicit, computable trace map from the higher scissors congruence groups to group homology, and use this trace to prove the existence of some nonzero classes in the higher scissors congruence groups. We also show that the $$K$$-theory of polyhedra is a homotopy orbit spectrum. This fits into Thomason’s general framework of $$K$$-theory commuting with homotopy colimits, but we give a self-contained proof. We then use this result to re-interpret the trace map as a partial inverse to the map that commutes homotopy orbits with algebraic $$K$$-theory.
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- PAR ID:
- 10537126
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2024
- Issue:
- 18
- ISSN:
- 1073-7928
- Format(s):
- Medium: X Size: p. 12683-12710
- Size(s):
- p. 12683-12710
- Sponsoring Org:
- National Science Foundation
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