Chain duality for categories over complexes

TL;DR

This paper proves that the category of chain complexes over a finite simplicial complex has a chain duality, which is crucial for algebraic surgery theory and offers a new geometric perspective.

Contribution

It establishes the chain duality for categories of chain complexes over complexes, filling a gap in foundational theory and providing a new geometric approach.

Findings

Confirmed the existence of chain duality in this setting.

Connected chain duality to algebraic surgery and Poincaré duality.

Provided a new geometric interpretation of chain duality.

Abstract

We show that the additive category of chain complexes parametrized by a finite simplicial complex $K$ forms a category with chain duality. This fact, never fully proven in the original reference, is fundamental for Ranicki's algebraic formulation of the surgery exact sequence of Sullivan and Wall, and his interpretation of the surgery obstruction map as the passage from local Poincar\'e duality to global Poincar\'e duality. Our paper also gives a new, conceptual, and geometric treatment of chain duality on $K$-based chain complexes.