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A note on Duality and the Atiyah-Hirzebruch spectral sequence

Maximilian David Hans

TL;DR

The paper shows that Spanier--Whitehead duality identifies the cohomological and homological Atiyah--Hirzebruch spectral sequences for finite spectra, with a pagewise isomorphism $(E^{p,q}_r,d_r) o (E_{-p,-q}^r,d^r)$. It then factorizes Poincaré duality for $d$-dimensional PD complexes oriented over a ring spectrum $\\mathscr{E}$ into a Thom--Dold orientation followed by SW duality, and proves that this yields an isomorphism of AHSS as well. The work unifies duality principles across cohomology and homology via filtrations and Thom constructions, enabling transfer of differential information between the two spectral sequences. It also clarifies how orientations on spherical fibrations and the Spivak normal fibration underpin the PD–AHSS correspondence, with broader implications for orientable PD complexes and related bordism theories.

Abstract

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

A note on Duality and the Atiyah-Hirzebruch spectral sequence

TL;DR

The paper shows that Spanier--Whitehead duality identifies the cohomological and homological Atiyah--Hirzebruch spectral sequences for finite spectra, with a pagewise isomorphism . It then factorizes Poincaré duality for -dimensional PD complexes oriented over a ring spectrum into a Thom--Dold orientation followed by SW duality, and proves that this yields an isomorphism of AHSS as well. The work unifies duality principles across cohomology and homology via filtrations and Thom constructions, enabling transfer of differential information between the two spectral sequences. It also clarifies how orientations on spherical fibrations and the Spivak normal fibration underpin the PD–AHSS correspondence, with broader implications for orientable PD complexes and related bordism theories.

Abstract

We show that, for a finite spectrum , the Spanier-Whitehead duality isomorphism induces an isomorphism between the cohomological and homological Atiyah-Hirzebruch spectral sequences. As an application, it follows that Poincaré duality for a Poincaré duality complex, which is oriented over a ring spectrum , induces an isomorphism between the two spectral sequences.

Paper Structure

This paper contains 5 sections, 6 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and conventions
  3. Spanier--Whitehead duality for spectra
  4. Spanier--Whitehead duality and the Atiyah--Hirzebruch spectral sequence
  5. Orientations on Poincaré duality complexes

Key Result

Theorem 1

Let $\mathscr{E}$ be a spectrum, and $X$ a finite spectrum. The Spanier--Whitehead duality isomorphism $\mathscr{S}\mathscr{W}\colon\mathscr{E}^*(X)\rightarrow \mathscr{E}_{-*}(\mathbb{D}(X))$ induces an isomorphism between the cohomological and homological Atiyah--Hirzebruch spectral sequences comp

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • proof : Proof of \ref{['thm:A']}
  • Remark 4.1
  • ...and 9 more