Two-dimensional Euler insulators are novel kind of systems that host multi-gap topological phases, quantified by a quantised first Euler number in their bulk. This topological invariant is protected by the spacetime inversion symmetry.
Recently, these phases have been experimentally realised in suitable two-dimensional synthetic matter setups. In this talk, I introduce the second Euler invariant, a familiar invariant in both differential topology (Chern-Gauss-Bonnet theorem) and in four-dimensional Euclidean gravity, whose
existence has not been explored in condensed matter systems. Specifically, I firstly define two specific novel models in four dimensions that support a non-zero second Euler number in the bulk together with peculiar gapless boundary states. Secondly, I discuss its robustness in general spacetime-inversion invariant phases and its role in the classification of topological degenerate real bands through real Grassmannians. Finally, I show how to engineer these new topological phases in a fourdimensional ultracold atom setup.