External Spanier-Whitehead duality and homology representation theorems for diagram spaces

TL;DR

This paper develops a duality framework for diagram spectra, establishing a homology representation theorem and constructing Chern characters for rational homology theories, advancing the understanding of structured homotopy theories.

Contribution

It introduces a Spanier-Whitehead type duality functor for diagram spectra and proves a homology representation theorem, enabling new constructions of Chern characters.

Findings

Established a duality functor relating finite -spectra and their opposites

Proved that -homology theories are represented by homotopy groups of a balanced smash product

Constructed Chern characters for rational -homology theories

Abstract

We construct a Spanier-Whitehead type duality functor relating finite $\mathcal{C}$-spectra to finite $\mathcal{C}^{\mathrm{op}}$-spectra and prove that every $\mathcal{C}$-homology theory is given by taking the homotopy groups of a balanced smash product with a fixed $\mathcal{C}^{\mathrm{op}}$-spectrum. We use this to construct Chern characters for certain rational $\mathcal{C}$-homology theories.