From $(\mathbb{Z},X)$-modules to homotopy cosheaves

TL;DR

This paper establishes a functorial equivalence between $( Z,X)$-modules and homotopy cosheaves of chain complexes, linking algebraic $L$-theory with topological invariants and providing a new perspective on Ranicki's work.

Contribution

It constructs a functor from $( Z,X)$-modules to homotopy cosheaves, bridging algebraic and topological $L$-theories and translating Ranicki's ideas into homotopy chain complexes.

Findings

Established an equivalence on $L$-theory between the categories.

Reproved the topological invariance of rational Pontryagin classes.

Provided a translation of Ranicki's ideas into homotopy chain complexes.

Abstract

We construct a functor from the category of $(\mathbb{Z},X)$-modules of Ranicki (cf. \cite{Ra92}) to the category of homotopy cosheaves of chain complexes of Ranicki-Weiss (cf. \cite{RaWei10}) inducing an equivalence on $L$-theory. The $L$-theory of $(\mathbb{Z},X)$-modules is central in the algebraic formulation of the surgery exact sequence and in the construction of the total surgery obstruction by Ranicki, as described in \cite{Ra79}. The symmetric $L$-theory of homotopy cosheaf complexes is used by Ranicki-Weiss in \cite{RaWei10}, to reprove the topological invariance of rational Pontryagin classes. The work presented here may be considered as an addendum to the latter article and suggests some translation of ideas of Ranicki into the language of homotopy chain complexes of cosheaves.