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We completely determine the mod $2$ Seiberg-Witten invariants for any spinstructure on any closed, oriented, smooth $4$-manifold $X$. Our computationconfirms the validity of the simple type conjecture mod $2$ for spinstructures. Our proof also works for families of spin $4$-manifolds and thuscomputes the mod $2$ Seiberg-Witten invariants for spin families. The proof ofour main result uses $Pin(2)$-symmetry to define an enhancement of the mod $2$Seiberg-Witten invariants. We prove a connected sum formula for the enhancedinvariant using localisation in equivariant cohomology. Unlike the usualSeiberg-Witten invariant, the enhanced invariant does not vanish on takingconnected sums and by exploiting this property, we are able to compute theenhanced invariant.

The mod 2 Seiberg-Witten invariants of spin structures and spin families