Bimonoidal operad-actions and the product in negative Tate-cohomology

TL;DR

This paper explores the interplay between Dyer--Lashof-operations in negative Tate-cohomology and the description of negative Tate-cohomology via joins, extending applicability beyond compact loop-spaces.

Contribution

It introduces a construction linking Dyer--Lashof-operations with join descriptions of negative Tate-cohomology, applicable to non-compact loop-spaces.

Findings

Provides a generalized framework for negative Tate-cohomology

Connects operad-actions with cohomological operations

Extends previous work to broader topological contexts

Abstract

We study a certain construction designed to bring together the following two topics: $i$) Dyer--Lashof-operations in negative Tate-cohomology, $ii$) the description of negative Tate-cohomology in terms of joins. It has the merit of making (some) sense in a more general context: where the loop-space of the space under study no longer has to be of compact homotopy-type. The exposition given here is a streamlined version of a previous version of mine, available here at the Arxiv.