= Turaev-torsion Dehn-filling formula
{c}
Let $Y$ be obtained by <Dehn filling> a torus boundary of $X$, with core class $h\in H_1(Y)$. Choose a one-dimensional coefficient <representation> $\rho:H_1(Y)\to F^\times$ over a <field>, with $\rho(h)\ne1$, and assume the associated twisted <chain complexes> are <acyclic>. Then, up to the usual torsion unit,
$$
\tau_\rho(Y)\doteq\frac{\tau_{\rho|X}(X)}{\rho(h)-1}.
$$
With compatible <Euler structures> and <homology orientations>, the refined formula fixes that ambiguity. The denominator comes from the attached <solid torus>'s circle complex. When $\rho(h)=1$, this division is unavailable; additional homology data are needed.<Surgery formula for torsions and Seiberg-Witten invariants of 3-manifolds by Vladimir Turaev>[Lemma 5.1 and Section 5.2, page 18]{ref}
Back to article page
