of all complete Euclidean instantons of a requisite set of symmetries and set
of charges is of direct importance in the path integral approach to
semiclassical quantum gravity. In this talk, a proof of Birkhoff's theorem for
Euclidean Einstein-‐Maxwell gravity with positive, negative or zero
cosmological constant is presented, with particular attention payed to the fact
that the local spherical symmetry implies the existence of another local
Killing vector field orthogonal to the closed two dimensional orbits of SO(3).
Next, properties of the self-‐dual and anti-‐self-‐dual Weyl curvatures are
used to show that spherically symmetric spaces must have zero signature and
positive Euler characteristic. Finally, various theorems in topology are
applied to further restrict the possible covering spaces to a finite set,
providing a classification of all complete spherically symmetric Euclidean
instantons of the theory.