Celebratio Mathematica — Neumann — Complete Bibliography

Complete Bibliography

Works connected to G. Robert Meyerhoff

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R. Meyerhoff and W. D. Neumann: “An asymptotic formula for the eta invariants of hyperbolic 3-manifolds,” Comment. Math. Helv. 67 : 1 (1992), pp. 28–46. MR 1144612 Zbl 0791.57009 article

Let \( M \) be an oriented complete finite-volume hyperbolic 3-manifold with one cusp. Suppose that an oriented basis \( \mathbf{l} \), \( \mathbf{m} \), for the first homology at the cusp has been chosen, so that we can speak of \( M(p,q) \), the result of \( (p,q) \)-Dehn filling the cusp — that is, we replace the cusp by a solid torus in which the class \( p\mathbf{m} + q\mathbf{l} \) is null-homologous. Thurston’s hyperbolic Dehn surgery theorem [Thurston 1978; Neumann and Zagier 1985] tells us that \( M(p,q) \) has a hyperbolic structure for \( p^2 + q^2 \) sufficiently large. In [1985] T. Yoshida proves a formula for the eta invariant \( \eta(M(p,q)) \) in terms of Thurston’s analytic Dehn surgery parameter \( u(p,q) \) and additional structure on \( M \) (various frame fields). For \( M \) equal to the figure-eight knot complement he gives a simpler and more explicit version of the formula which does not invoke the extra structure. The purpose of this note is to show that a formula of this simpler type can be derived in general from Yoshida’s result.