On the Non-Existence of J-Homomorphisms of Higher Height

TL;DR

This paper investigates the properties of certain spectra in algebraic topology, demonstrating that for many cases, specific mapping spaces remain unaffected by particular localizations and connective covers, revealing new structural insights.

Contribution

It establishes the non-existence of J-homomorphisms of higher height for a broad class of connective spectra, advancing understanding of their mapping space behavior.

Findings

Mapping spaces into units of the p-complete sphere spectrum are unaffected by L_1-localization and connective cover.

The non-existence of higher height J-homomorphisms is proven for a large class of spectra.

Provides new structural insights into the behavior of connective spectra and their mapping spaces.

Abstract

We show that for a large class of connective spectra, the mapping space into the units of the $p$-complete sphere spectrum is insensitive to $L_1$-localization followed by taking connective cover.