Steenrod Pseudocycles, Lifted Cobordisms, and Solomon's Relations for Welschinger's Invariants

Abstract

We establish two WDVV-style relations for the disk invariants of real symplectic fourfolds by implementing Georgieva's suggestion to lift homology relations from the Deligne-Mumford moduli spaces of stable real curves. This is accomplished by lifting judiciously chosen cobordisms realizing these relations. The resulting lifted relations lead to the recursions for Welschinger's invariants announced by Solomon in~2007 and have the same structure as his WDVV-style relations, but differ by signs from the latter. Our topological approach provides a general framework for lifting relations via morphisms between not necessarily orientable spaces.