Cogroupoid structures on the circle and the Hodge degeneration

TL;DR

This paper reveals a new $E_2$-cogroupoid structure on the topological circle and connects it to the Hodge degeneration, Todd class, and Hochschild cohomology, offering insights into nonabelian Hodge theory and formal moduli problems.

Contribution

It introduces an $E_2$-cogroupoid structure on the circle and links it to Hodge degeneration, Todd classes, and rational homotopy theory, providing novel geometric and algebraic insights.

Findings

Identifies an $E_2$-cogroupoid structure on the circle.

Relates the Todd class to the formality of the pinch map.

Connects the structure to Hochschild cohomology consequences.

Abstract

We exhibit the Hodge degeneration from nonabelian Hodge theory as a $2$-fold delooping of the filtered loop space $E_2$-groupoid in formal moduli problems. This is an iterated groupoid object which in degree $1$ recovers the filtered circle $S^1_{fil}$ of [MRT19]. This exploits a hitherto unstudied additional piece of structure on the topological circle, that of an $E_2$-cogroupoid object in the $\infty$-category of spaces. We relate this cogroupoid structure with the more commonly studied "pinch map" on $S^1$, as well as the Todd class of the Lie algebroid $\mathbb{T}_{X}$; this is an invariant of a smooth and proper scheme $X$ that arises, for example, in the Grothendieck-Riemann Roch theorem. In particular we relate the existence of non-trivial Todd classes for schemes to the failure of the pinch map to be formal in the sense of rational homotopy theory. Finally we record some consequences of this bit of structure at the level of Hochschild cohomology.