Split injectivity of A-theoretic assembly maps

TL;DR

This paper develops a new equivariant coarse homology theory based on algebraic K-theory of spherical group rings, proving split injectivity of assembly maps and extending fundamental theorems to nonconnective K-theory.

Contribution

It introduces a novel equivariant coarse homology theory and demonstrates split injectivity of assembly maps, extending Waldhausen's K-theory results to nonconnective settings.

Findings

Established split injectivity for assembly maps in algebraic K-theory

Extended fundamental theorems of Waldhausen's K-theory to nonconnective K-theory

Constructed a new equivariant coarse homology theory from spherical group rings

Abstract

We construct an equivariant coarse homology theory arising from the algebraic $K$-theory of spherical group rings and use this theory to derive split injectivity results for associated assembly maps. On the way, we prove that the fundamental structural theorems for Waldhausen's algebraic $K$-theory functor carry over to its nonconnective counterpart defined by Blumberg--Gepner--Tabuada.