The affirmative answer to Singer's conjecture on the algebraic transfer of rank four

TL;DR

This paper proves Singer's conjecture that the algebraic transfer is injective for homological degree four, advancing understanding of the mod-2 cohomology of the Steenrod algebra in algebraic topology.

Contribution

It provides a proof of Singer's conjecture specifically for the case of homological degree four, a significant step forward in the field.

Findings

Confirmed injectivity of the algebraic transfer at degree four

Advances the understanding of the structure of the Steenrod algebra's cohomology

Supports the broader conjecture for all degrees

Abstract

In recent decades, the structure of the mod-2 cohomology of the Steenrod ring $\mathscr A$ has become a major subject of study in the field of Algebraic Topology. One of the earliest attempts to study this cohomology through the use of modular representations of the general linear groups was the groundbreaking work [Math. Z. 202 (1989), 493-523] by W.M. Singer. In that work, Singer introduced a homomorphism, commonly referred to as the "algebraic transfer," which maps from the coinvariants of a certain representation of the general linear group to the mod-2 cohomology group of the ring $\mathscr A.$ Singer's conjecture, in particular, which states that the algebraic transfer is a monomorphism for all homological degrees, remains a highly significant and unresolved problem in Algebraic Topology. In this research, we take a major stride toward resolving the Singer conjecture by establishing its truth for the homological degree four.