Absolute torsion and eta-invariant (Q1574459)

The paper gives a relation between the sign of absolute torsion \(\tau\) (an extension of Reidemeister torsion with a well defined sign) and the eta-invariant. The situation is the following: \(X\) is a closed oriented PL-manifold of odd dimension \(m\), \(F\) is a flat orientable Hermitian vector bundle over \(X\), the Stiefel-Whitney class \(w_{m-1}(X)\) vanishes and \(H_1(X,{\mathbb Z})\) has no \(2\)-torsion. Let now \(\nabla_t\), \(t\in (a,b)\), be a real analytic family of flat Hermitian connections on \(F\), and set \(F_t:=(F,\nabla_t)\). Assume \(H_*(X,F_t)=0\) for \(t\notin S\) with finite subset \(S\) of \((a,b)\). In this case, \(\tau(X,F_t)\in {\mathbb R}\) is defined for \(t\notin S\). Then the author proves: \[ \text{sign}(\tau(F_t))\exp(i\pi \eta(F_t)/2)V(t)\in{\mathbb C} \] is independent of \(t\in (a,b)-S\). Here, \(V(t)\) is an expression involving the Pontryagin classes of \(X\) and the monodromy of \(\nabla_t\). Moreover, if \(m\) is congruent to \(3\) modulo \(4\), or if \(\nabla_t\) is an analytic family of SU-connections, then the formula is valid also if \(H_1(X,{\mathbb Z})\) is arbitrary, and \(V(t)\) vanishes. The eta-invariant is the eta-invariant of the operator \(i^{(m+1)/2} (-1)^{N+1}(*\nabla-\nabla*)\) on forms of even degree (where \(N(\phi)\) is the degree of \(\phi\)).