  We construct a graded Lie algebra in which a solution to the vacuum Einstein
equations is any element of degree 1 whose bracket with itself is zero. Each
solution generates a cochain complex, whose first cohomology is linearized
gravity about that solution. We gauge-fix to get a smaller cochain complex with
the same cohomologies (deformation retraction). The new complex is much
smaller, it consists of the solution spaces of linear homogeneous wave
equations (symmetric hyperbolic equations). The algorithm that produces these
gauges and wave equations is both for linearized gravity and the full Einstein
equations. The gauge groupoid is the groupoid of rank 2 complex vector bundles.
