The topology of asymptotically locally flat gravitational instantons

TL;DR

This paper establishes that the intersection form of certain four-dimensional ALF gravitational instantons is definite and diagonalizable, aiding their topological classification, by linking cohomology, anti-instantons, and intersection forms.

Contribution

It provides a topological classification of ALF gravitational instantons using intersection forms and cohomology, revealing new structural insights.

Findings

Intersection form is definite and diagonalizable over integers.

Topological classification of ALF gravitational instantons achieved.

Relationship between L^2 cohomology and anti-instantons utilized.

Abstract

In 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 Kahler forms of the hyper-Kahler gravitational instanton metric is exact. This leads to the topological classification of these spaces. The proof exploits the relationship between L^2 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.