The exact Calabi–Yau structure conjecture for acyclic manifold group algebras

Let $M$ be a $d$-dimensional compact orientable acyclic manifold, and let $k[\pi_1(M)]$ be its fundamental group algebra. Write $\operatorname{HH}_d(k[\pi_1(M)])$ for its degree-$d$ Hochschild homology, and call a unit central if it lies in the centre of $k[\pi_1(M)]$. Exact Calabi–Yau structure conjecture. The classes in $\operatorname{HH}_d(k[\pi_1(M)])$ corresponding to exact Calabi–Yau structures are given precisely by central units with zero constant coefficient. This proposed description is presented in the context of expected Calabi–Yau structures on loop-space algebras; the supplied text does not establish its status or provide a resolution.