Equivariant Floer groups for binary polyhedral spaces (Q1894814)

We compute the equivariant Floer groups for 3-manifolds \(Y_ \Gamma = S^ 3/\Gamma\) where \(\Gamma\) is a finite subgroup of \(\text{SU}(2)\). With the standard orientation induced from \(S^ 3\), we see that the equivariant Floer (co)homology generated by the irreducibles is killed off by that generated by the reducibles. This allows us to extend Donaldson's vanishing theorem for connected sums to a larger class of 4- manifolds which are split by some \(Y_ \Gamma\) with positive \(b^ +\) on both sides. A more general notion of indecomposability for algebraic surfaces results.