A note on Duality and the Atiyah-Hirzebruch spectral sequence

TL;DR

This paper demonstrates that Spanier-Whitehead duality creates an isomorphism between cohomological and homological Atiyah-Hirzebruch spectral sequences for finite spectra, with implications for Poincaré duality complexes.

Contribution

It establishes a duality-induced isomorphism between spectral sequences, linking homological and cohomological perspectives in stable homotopy theory.

Findings

Spanier-Whitehead duality induces an isomorphism between spectral sequences.

Poincaré duality complexes over a ring spectrum have isomorphic spectral sequences.

The result applies specifically to finite spectra and Poincaré duality complexes.

Abstract

We show that, for a finite spectrum $X$, Spanier-Whitehead duality induces an isomorphism between the cohomological and homological Atiyah-Hirzebruch spectral sequences. As an application, it follows that Poincar\'e duality for a Poincar\'e duality complex that is oriented over a ring spectrum $R$ induces an isomorphism between the two spectral sequences.