Reconstruction of graphs and surfaces from derived categories

Abstract: A pair of topological spaces $X$ and $Y$ is a \emph{Fourier--Mukai (FM) pair} if there is an equivalence of $\infty$-categories of derived sheaves of abelian groups [\operatorname{Shv}(X;\mathrm D(\mathbb Z))\simeq \operatorname{Shv}(Y;\mathrm D(\mathbb Z)).] An FM pair $(X,Y)$ is \emph{non-trivial} if $X$ and $Y$ are not homeomorphic. We consider the question of whether non-trivial FM pairs $(X,Y)$ exist for which $X$ and $Y$ are both "nice" spaces, which in this article means admitting a triangulation. We show that the answer is no if $X$ is second-countable and either (i) one-dimensional, (ii) a surface, or (iii) the 3-sphere. In order to prove this, we define two invariants of dualizable categories. The first invariant is an extension of \emph{Tate--Hochschild cohomology} (aka singular Hochschild cohomology) to a class of dualizable categories satisfying a countability assumption. For a homology manifold $X$, we relate the Tate--Hochschild cohomology of $\operatorname{Shv}(X;\mathrm D(\mathbb Z))$ to the homology at infinity of $X$ (in the sense of proper homotopy theory). The second invariant takes the form of a subcategory $\mathcal D_{\mathrm{Gor}}\subseteq\mathcal D$ called the \emph{Gorenstein locus}, defined for every dualizable stable category $\mathcal D$. For a triangularizable space $X$ and a field $k$, we show that the Gorenstein locus of $\operatorname{Shv}(X;\mathrm D(k))$ is equal to the full subcategory of sheaves supported on the $k$-homology manifold locus of $X$.