Homology manifolds via six functor formalisms

We study homology manifolds through the eyes of the six functor formalism of spectral sheaves on locally compact Hausdorff spaces. As main results, we characterize cohomologically smooth objects by adapting an argument of Scholze, deduce that any hypercomplete locally compact ANR homology manifold is cohomologically smooth, show that compact ANR homology manifolds $X$ are Poincar\'e duality complexes whose Spivak tangent fibration identifies with the dualizing sheaf of $X$, and prove a generalization of Wilder's monotone mapping theorem about cell-like maps. Moreover, we introduce the notion of homotopy manifolds for which we prove an unstable analog of Wilder's orientability conjecture and show that hypercomplete ANR homology manifolds are homotopy manifolds. As a consequence, we show that for a compact $d$-dimensional ANR homology manifold, the Spivak tangent fibration of its associated Poincar\'e duality complex canonically destabilizes to a pointed $S^d$-fibration. Finally, we introduce homotopy manifolds with conical singularities, a generalization of Cohen's triangulated homotopy manifolds, and show that such objects are in fact topological manifolds, generalizing a result of Siebenmann. Along the way, we obtain comparisons between sheaf and singular cohomology and between the shape and the weak homotopy type of a topological space, explore the relation between various notions of cohomological dimension and hypercompleteness, and study six functor formalisms satisfying the K\"unneth formula.