Realizing a complex of unstable modules

TL;DR

This paper demonstrates the topological realization of injective resolutions of mod 2 cohomology for a Thom spectrum, enabling geometric construction of dual Brown-Gitler spectra and advancing understanding of unstable modules.

Contribution

It provides a topological realization of injective resolutions previously constructed algebraically, linking homological algebra with geometric spectra.

Findings

Existence of cofibrations inducing exact sequences in mod 2 cohomology

Resolutions obtained by splicing short exact sequences

Potential for geometric construction of dual Brown-Gitler spectra

Abstract

In a preceding article the authors and Tran Ngoc Nam constructed a minimal injective resolution of the mod 2 cohomology of a Thom spectrum. A Segal conjecture type theorem for this spectrum was proved. In this paper one shows that the above mentioned resolutions can be realized topologically. In fact there exists a family of cofibrations inducing short exact sequences in mod 2 cohomology. The resolutions above are obtained by splicing together these short exact sequences. Thus the injective resolutions are realizable in the best possible sense. In fact our construction appears to be in some sense an injective closure of one of Takayasu. It strongly suggests that one can construct geometrically (not only homotopically) certain dual Brown-Gitler spectra. Contents