Properadic coformality of spheres
Coline Emprin, Alex Takeda
TL;DR
The paper develops a properadic framework for coformality questions of loop-space algebras by introducing the dioperad $\mathcal{Y}^{(n)}$ encoding $n$-pre-Calabi--Yau structures with vanishing copairing and showing how these lift to properadic algebras via $\mathcal{Y}^{(n)}_\infty$. It proves that based loop spaces of spheres are intrinsically coformal over any characteristic zero ring, using refined obstruction sequences and Hochschild theory to show all higher obstructions vanish, while establishing that even-dimensional spheres fail intrinsic coformality in characteristic two due to cyclic-symmetry obstructions. The work connects noncommutative Calabi–Yau-type deformation theory with classical string-topology structures, clarifying how Koszul duality and filtration-enhanced obstructions govern formality phenomena. This yields a robust criterion for intrinsic coformality in a broad noncommutative setting and explains characteristic-dependent rigidity phenomena observed in loop-space homology and associated BV-type structures.
Abstract
We define a properad that encodes $n$-pre-Calabi-Yau algebras with vanishing copairing. These algebras include chains on the based loop space of any space $X$ endowed with a fundamental class $[X]$ such that $(X,[X])$ satisfies Poincaré duality with local system coefficients, such as oriented manifolds. We say that such a pair $(X,[X])$ is coformal when $C_*(ΩX)$ is formal as an $n$-pre-Calabi-Yau algebra with vanishing copairing. Using a refined version of properadic Kaledin classes, we establish the intrinsic coformality of all spheres in characteristic zero. Furthermore, we prove that intrinsic formality fails for even-dimensional spheres in characteristic two.