Homotopy idempotents on manifolds and Bass' conjectures

TL;DR

This paper explores the Bass trace conjectures within the context of homotopy idempotent selfmaps on manifolds, providing new formulations and comparisons involving Lefschetz numbers to advance understanding of these conjectures.

Contribution

It introduces novel reformulations of the Bass trace conjectures using homotopy idempotents and Lefschetz numbers, connecting topological and analytical perspectives.

Findings

Formulated the strong Bass conjecture via Geoghegan's approach.

Reformulated the weaker conjecture as a comparison of Lefschetz numbers.

Provided new insights into the relationship between algebraic K-theory and topological fixed point theory.

Abstract

The Bass trace conjectures are placed in the setting of homotopy idempotent selfmaps of manifolds. For the strong conjecture, this is achieved via a formulation of Geoghegan. The weaker form of the conjecture is reformulated as a comparison of ordinary and L^2-Lefschetz numbers.