Poincar\'e duality spaces related to the Joker

TL;DR

This paper constructs an 8-dimensional Poincaré duality space with a specific cohomology algebra related to the Joker module, and proves it admits a PL-structure using obstruction theory.

Contribution

It introduces a new Poincaré duality space with a cohomology algebra linked to the Joker module and establishes its PL-structure.

Findings

The space's cohomology is realizable as an unstable algebra.

The space admits a PL-structure.

Cohomology can be realized as that of a homogeneous space.

Abstract

The well known Joker $\mathcal{A}(1)$-module of Adams and Priddy is known to be realisable as the cohomology of a $1$-connected space. By attaching an extra cell we obtain an $8$-dimensional Poincar\'e duality space whose mod~$2$ cohomology realising is an unstable $\mathcal{A}$-algebra. We use obstruction theory to show that this admits a $PL$-structure. Although we are unable to show it is smoothable, it turns out that the cohomology can be realised as that of a homogeneous space.