Additivity of Frøyshov’s q3 invariant

Determine whether the invariant q3 is additive under connected sum for arbitrary integer homology spheres.

Background

The paper proves an additivity formula for connected summing an integer homology sphere with the Poincaré homology sphere and notes a broader additivity statement when one summand is homology cobordant to a connected sum of Seifert spaces and manifolds with Dehn surgery number one. These results do not establish additivity for all integer homology spheres.

The unresolved issue is whether q3(Y#Y')=q3(Y)+q3(Y') holds for every pair of integer homology spheres Y and Y'. Such a formula would extend the partial additivity results discussed in the paper and clarify the behavior of q3 under connected sum.