Toward an exotic S²×S² and an exotic ℂP²#ℂP̄²

Abstract

We study a specified Luttinger filling in the construction of Lidman and Piccirillo whose simple connectivity would yield exotic $S^2\times S^2$ and $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$. An earlier version of this manuscript claimed simple connectivity from incorrectly derived group relations; that claim is withdrawn. We derive the complete fundamental-group presentation and express the group as a one-relator quotient of an iterated HNN extension containing the genus-two surface group. The filled group is trivial exactly when the final relation kills a specified fiber commutator. We also determine its universal central extension with cyclic kernel, construct geometric realizations of it, and extend the boundary involution over the filling. Proposed symplectic extensions construct a torus with a negative area–Maslov relation and exact tori detecting a nonzero transfer kernel for a choice of auxiliary handles. These constructions do not establish survival of the central element. The final filling problem remains unresolved.