The topology of asymptotically locally flat gravitational instantons

AbstractIn this Letter we demonstrate that the intersection form of the Hausel–Hunsicker–Mazzeo compactification of a four-dimensional ALF gravitational instanton is definite and diagonalizable over the integers if one of the Kähler forms of the hyper-Kähler gravitational instanton metric is exact. This leads to their topological classification.The proof exploits the relationship between L2 cohomology and U(1) anti-instantons over gravitational instantons recognized by Hitchin. We then interprete these as reducible points in a singular SU(2) anti-instanton moduli space over the compactification leading to the identification of its intersection form.This observation on the intersection form might be a useful tool in the full geometric classification of various asymptotically locally flat gravitational instantons