Whitehead torsion and the kernel of assembly

TL;DR

This paper establishes a new categorical framework linking Whitehead torsion to K-theory for finite simplicial complexes, providing a novel perspective on classical invariants.

Contribution

It introduces the Whitehead category and torsion cosheaf, connecting Whitehead torsion with K-theory and assembly functors in a new way.

Findings

The assembly functor has a fully faithful right adjoint for finite simplicial complexes.

A Whitehead category is defined with K-theory identified with the Whitehead spectrum.

Classical Whitehead torsion is recovered via K-theory classes of torsion cosheaves.

Abstract

For a topological space that is homeomorphic to a finite simplicial complex, we prove that the Bartels--Nikolaus assembly functor has a fully faithful right adjoint. Using this, we define for each such topological space $X$ a {\em Whitehead category}, whose K-theory is canonically identified with the Whitehead spectrum of $X$; and for a homotopy equivalence between two such spaces, we define an object in the Whitehead category of $X$ called the {\em torsion cosheaf} of the map, whose K-theory class recovers the classical Whitehead torsion.