Closed-manifold 3D index via the Gang–Yonekura formula

Establish that, for any closed 3-manifold M obtained by Dehn filling on a singly cusped 3-manifold N admitting a 1‑efficient triangulation T, applying the Gang–Yonekura formula yields a well-defined 3D index I_M(q) independent of auxiliary choices, and prove the proposed value classification: I_M(q)=1+P(q) with P(q)∈q^{1/2} Z[[q^{1/2}]] if M is hyperbolic; I_M(q)=0 if M is a lens space; I_M(q)=1 if M is a Seifert fibered space with base orbifold S^2(2,3,5), S^2(2,3,4), or S^2(2,3,3), or a Seifert fibered space with hyperbolic base or a manifold with Sol geometry; and I_M(q)=2 if M is a Seifert fibered space with base orbifold S^2(2,2,n).

Background

The paper develops and proves a rigorous Gang–Yonekura formula for the transformation of the 3D index under Dehn filling in the cusped setting. Motivated by extensive computations, the authors conjecture a closed-manifold version: that the Gang–Yonekura prescription extends to define a coherent index for closed 3-manifolds produced by filling, and that this index assumes simple canonical values depending on the geometric type of the manifold.

A proof would furnish the first mathematically sound definition of the 3D index for closed 3-manifolds, establish its invariance under choices of triangulations and surgery descriptions, and link index values to Thurston–Perelman geometries and Seifert fibered classifications.