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A note on the Segal conjecture for large objects

Robert Burklund, Vignesh Subramanian

Abstract

The Segal conjecture for $C_p$ (as proved by Lin and Gunawardena) asserts that the canonical map from the $p$-complete sphere spectrum to the Tate construction for the trivial action of $C_p$ on the $p$-complete sphere spectrum is an isomorphism. In this article we extend the collection of spectra for which the canonical map $X \to X^{tC_p}$ is known to be an isomorphism to include any $p$-complete, bounded below spectrum whose mod $p$ homology, viewed a module over the Steenrod algebra, is complete with respect to the maximal ideal $I \subseteq \mathcal{A}$.

A note on the Segal conjecture for large objects

Abstract

The Segal conjecture for (as proved by Lin and Gunawardena) asserts that the canonical map from the -complete sphere spectrum to the Tate construction for the trivial action of on the -complete sphere spectrum is an isomorphism. In this article we extend the collection of spectra for which the canonical map is known to be an isomorphism to include any -complete, bounded below spectrum whose mod homology, viewed a module over the Steenrod algebra, is complete with respect to the maximal ideal .
Paper Structure (4 sections, 7 theorems, 17 equations)

This paper contains 4 sections, 7 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Synthetic Spectra and the Adams spectral sequence
  3. The Segal Conjecture for $I$-complete spectra
  4. Examples of $I$-complete spectra

Key Result

Theorem 1.1

The canonical map $\mathbb{S}_{p} \to \mathbb{S}_p^{tC_p}$ is an isomorphism.

Theorems & Definitions (27)

  • Theorem 1.1: lin1980conjecturesgunawardena1980segal
  • Theorem 1.2: Theorem \ref{['segalconjIcmpl']}
  • Definition 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.5
  • Remark 2.6
  • ...and 17 more