Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Atiyah-Sutcliffe conjecture and $E_n$-algebras

Lorenzo Guerra, Paolo Salvatore

Abstract

We show that a certain conjecture by Atiyah and Sutcliffe implies the existence of an $ E_3 $-algebra (respectively $ E_2 $-algebra) structure on the disjoint union of all complex (respectively real) full flag manifolds modulo symmetric groups. Moreover, we show that these structures are liftings of exotic $ E_3 $ (respectively $ E_2 $) structures on the free $ E_\infty $-algebras on $ BU(1)+ $ (respectively $ BO(1)+ $), that do not extend to $ E_4 $ (respectively $ E_3 $) structures. We also provide some (co)homological calculations supporting the conjecture.

The Atiyah-Sutcliffe conjecture and $E_n$-algebras

Abstract

We show that a certain conjecture by Atiyah and Sutcliffe implies the existence of an -algebra (respectively -algebra) structure on the disjoint union of all complex (respectively real) full flag manifolds modulo symmetric groups. Moreover, we show that these structures are liftings of exotic (respectively ) structures on the free -algebras on (respectively ), that do not extend to (respectively ) structures. We also provide some (co)homological calculations supporting the conjecture.

Paper Structure

This paper contains 6 sections, 13 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The $E_3$- and $E_2$-actions
  3. The underlying $E_2$-structure on $D(BU(1))$ and its lifts to unordered flag manifolds
  4. Calculations for the Atiyah $E_3$-structure on $D(BU(1)$
  5. The $E_3$-structure on unordered flag manifolds and its non-extendibility.
  6. Stable cohomology of unordered flag manifolds and stable extendibility

Key Result

Proposition 2.2

Let $w \in \mathbb{N}^n$. The weighted Atiyah map extends uniquely to a continuous function $FM^3_n \to BU(1)^n$

Figures (1)

  • Figure 1: The category $\mathcal{D}(4)$

Theorems & Definitions (24)

  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • ...and 14 more