Disk Embedding Conjecture

The unrestricted four-dimensional disk embedding conjecture asks for disjoint proper locally flat embedded disks with prescribed boundary maps from immersed disk data and suitable algebraic dual spheres, without a restriction on the fundamental group. For an oriented four-manifold and based surfaces, the equivariant intersection pairing takes values in the group ring . The input disks have disjoint embedded boundary circles and dual sphere maps with trivialized normal bundles satisfying

Here is the reduced self-intersection invariant, with values in the additive quotient

The conjectured conclusion preserves the induced trivializations of the boundary normal bundles and supplies geometric duals. The quotient defining is additive, rather than by a group-ring ideal.