The topology and total curvature of a Riemann surface is determined by a single integer, the Euler characteristic (Leonhard Euler, 1707-1783). While this is not the case in dimension four, the Euler characteristic still gives an interesting invariant for finitely presented groups. For example, what is the minimum possible value for the Euler characteristic of a closed 4-manifold with a given fundamental group ? The talk will survey some recent joint work with Alejandro Adem on this theme.