Minc's generating function and a Segal conjecture for Thom spectra. La fonction generatrice de Minc et une conjecture de Segal pour certains spectres de Thom

TL;DR

This paper constructs minimal injective resolutions for unstable modules related to Thom spectra's mod 2 cohomology, revealing combinatorial properties and homotopical applications involving Brown-Gitler and Steinberg modules.

Contribution

It introduces a new method for constructing injective resolutions using tensor products of Brown-Gitler and Steinberg modules, with applications to homotopy theory.

Findings

Alternating sum of Poincare series of modules is zero

Homotopical applications derived from combinatorial results

Resolution terms involve tensor products of specific modules

Abstract

One constructs minimal injective resolutions for certain unstable modules that appears to be the mod 2 cohomology of Thom spectra. The terms of the resolution are tensor products of Brown-Gitler modules and Steinberg modules introduced by S. Mitchell and S. Priddy. A combinatorial result of Andrews shows that the alternating sum of the Poincare series of the considered modules is zero. One gives homotopical applications of this result.