Split Injectivity ofA-Theoretic Assembly Maps

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.

On the Farrell-Jones conjecture for localising invariants

“…The construction of transfer classes requires adequate point-set models for objects in V c,perf Spc op * . We will be able to obtain such models using the notion of controlled CW-complexes that we previously considered in [BKW19] and that goes back to work of Weiss [Wei02]. Section 4.1 recalls the definition of the category CW(X) of controlled CW-complexes over a coarse space X.…”

“…There is a canonical notion of weak equivalence between objects in CW(X), namely that of a controlled homotopy equivalence; these are the weak equivalences of the Waldhausen structure on CW(X) considered in [BKW19]. We close this subsection by recalling the definition and showing that r inverts controlled homotopy equivalences.…”

“…The construction of transfer classes relies on point-set input data which need to be imported into our setting. Section 4.1 recalls the notion of controlled CW-complexes over a bornological coarse space from [BKW19]. Sections 4.2 and 4.3 then construct a natural transformation from these categories of controlled CW-complexes to the categories of controlled objects introduced in Section 3.…”

Coarse assembly maps

“…But most of them can be extended to all of GBornCoarse and admit a model in the sense defined above. We refer to [BEKW20a] for coarse equivariant ordinary homology (see also Section 6), to [BEKW20a], [BC20] for coarse algebraic K with coefficients in an additive category, to [BKW21] and [BCKW19] for coarse algebraic K-theory of spaces and coarse algebraic K with coefficients in a left-exact ∞-category, to [BEc] for equivariant coarse topological K-theory (see also Section 7), and to [Cap20] for equivariant coarse cyclic and Hochschild homology. In the non-equivariant case every locally finite homology theory admits a coarsification [Roe03, Sec.…”

Injectivity results for coarse homology theories

“…Then P gives rise to a GOrb-spectrum A P sending a transitive G-set S to the spectrum A(P × G S). By [14,Theorem 5.17], A P is a CP-functor.…”