The L 2 inner product can be viewed as a Riemannian metric on the innitedimensional manifold A (or A s ). Identifying vectors and covectors via this metric

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We identify the leading order term of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants for finite order mapping tori with classical invariants for all simple and simplyconnected compact Lie groups. The square root of the Reidemeister torsion is used as a density on the moduli space of flat connections and the leading order term is identified with the integral over this moduli space of this density weighted by a certain phase for each component of the moduli space. We also identify this phase in terms of classical invariants such as Chern-Simons invariants, eta invariants, spectral flow and the rho invariant. As a result, we show agreement with the semiclassical approximation as predicted by the method of stationary phase. * Supported in part by the center of excellence grant "Center for quantum geometry of Moduli Spaces" from the Danish National Research Foundation.

We state Asymptotic Expansion and Growth Rate conjectures for the Witten-ReshetikhinTuraev invariants of arbitrary framed links in 3-manifolds, and we prove these conjectures for the natural links in mapping tori of finite-order automorphisms of marked surfaces. Our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface, which we show coincides with the construction of the Witten-Reshetikhin-Turaev invariants using conformal field theory, as was recently completed by Andersen and Ueno.

We establish a splitting formula for the spectral flow of the odd signature operator on a closed 3-manifold M coupled to a path of SU (2) connections, provided M = S ∪ X , where S is the solid torus. It describes the spectral flow on M in terms of the spectral flow on S , the spectral flow on X (with certain Atiyah-Patodi-Singer boundary conditions), and two correction terms which depend only on the endpoints.Our result improves on other splitting theorems by removing assumptions on the non-resonance level of the odd signature operator or the dimension of the kernel of the tangential operator, and allows progress towards a conjecture by Lisa Jeffrey in her work on Witten's 3-manifold invariants in the context of the asymptotic expansion conjecture [17].