In 1999 Florian Deloup and I were attempting to find closed formulae for all abelian quantum invariants. These invariants can be expressed in terms of generalized Gauss sums, which depend on a quadratic form obtained from the linking form of the $3$-manifold. Toward this end  we formulated a conjecture that was intended to refine a theorem by Kawauchi and Kojma that demonstrated all linking pairings  on finite abelian groups (i.e. symmetric, non-degenerate bilinear forms into $\mathbb{Q}/\mathbb{Z}$)  arise as a "linking form" of some $3$-manifold. Their construction involves taking the connected sum of three different types of $3$-manifolds. The basis of this theorem was Wall's work and the subsequent work of Kawauchi and Kojma that classified all linking pairings on finite abelian groups.

Our  conjecture that was supposed to refine this theorem stated that any linking pairing on a finite abelian group arises from the linking form of a Seifert fibered rational homology sphere. We proved this result in the case when the abelian group has no $2$-torsion by 2004. In 2010 Jonathan Hillman gave counterexamples in the $2$-torsion case. The underlying reason for the failure of the linking form conjecture is that there are homology cobordism classes of $3$-manifolds that do not contain any Seifert manifolds.

It is possible to reformulate the linking form conjecture so it fulfils its original purpose. A corollary of this "new" linking form "theorem"  is that every homology cobordism class has a representative that arises from a "generalized Seifert presentation".

Furthermore, there are some interesting applications of these abelian quantum invariants to physics.