A Trace Map on Higher Scissors Congruence Groups

Author:

Bohmann Anna Marie1,Gerhardt Teena2,Malkiewich Cary3,Merling Mona4,Zakharevich Inna5

Affiliation:

1. Department of Mathematics, Vanderbilt University , Nashville, TN 37240 , USA

2. Department of Mathematics, Michigan State University , East Lansing, MI 48824 , USA

3. Department of Mathematics and Statistics, Binghamton University , Binghamton, NY 13905 , USA

4. Department of Mathematics, University of Pennsylvania , Philadelphia, PA 19104 , USA

5. Department of Mathematics, Cornell University , Ithaca, NY 14853 , USA

Abstract

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.

Funder

National Science Foundation

Publisher

Oxford University Press (OUP)

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