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

Using configuration space level Pontryagin–Thom constructions, we construct a simple Koszul self duality map for the little disks operad ${\mathrm{\Sigma }}_{+}^{\mathrm{\infty }}{E}_n$. For a framed $n$-manifold $M$, we show that a compatible self duality map exists for ${\mathrm{\Sigma }}_{+}^{\mathrm{\infty }}{E}_M$.