One point compactifications of configuration spaces and the self duality of the little disks operad

TL;DR

This paper constructs a Koszul self-duality map for the little disks operad using Pontryagin--Thom constructions, and extends this to framed manifolds, revealing a new duality structure in operad theory.

Contribution

It introduces a simple, explicit self-duality map for the little disks operad and generalizes it to framed manifolds, advancing understanding of operad dualities.

Findings

Constructed a Koszul self duality map for the little disks operad.

Extended the self duality to framed manifolds.

Provided a new perspective on operad duality structures.

Abstract

Using configuration space level Pontryagin--Thom constructions, we construct a simple Koszul self duality map for the little disks operad $\Sigma^\infty_+ E_n$. For a framed $n$-manifold $M$, we show that a compatible self duality map exists for $\Sigma^\infty_+ E_M$.