Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Poincaré duality spaces related to the Joker

Andrew Baker

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 Poincaré 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.

Poincaré duality spaces related to the Joker

Abstract

The well known Joker -module of Adams and Priddy is known to be realisable as the cohomology of a -connected space. By attaching an extra cell we obtain an -dimensional Poincaré duality space whose mod~ cohomology realising is an unstable -algebra. We use obstruction theory to show that this admits a -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.

Paper Structure

This paper contains 5 sections, 6 theorems, 37 equations.

Table of Contents

  1. Poincaré duality algebras over the Steenrod algebra
  2. Some Poincaré duality complexes
  3. Obstruction theory for $J^8$
  4. Some calculations
  5. A homogeneous space realisation

Key Result

Proposition 1.3

The characteristic classes of the SPDA $P^*$ have the following properties. (a) If $k>\lfloor d/2\rfloor$ then $v_k=0$. (b) If $d$ is even then $w_d=v_{d/2}^2$. (c) For $1\leqslant k\leqslant d$ the Stiefel-Whitney classes satisfy

Theorems & Definitions (14)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • proof
  • Remark 1.4
  • Proposition 1.5
  • proof
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • ...and 4 more