ANALYTIC SURGERY AND ANALYTIC TORSION Andrew Hassell B. Sc. (Hons), Australian National University, 1989 Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology May 1994 @1994 Massachusetts Institute of Technology All rights reserved Signatureo f Author .. - , Department of Mathematics April 20, 1994 Certified by . , -- Richard Melrose Professor of Mathematics Thesis Advisor Accepted by J>lC,rAavYui 1 LVv ocaI Professor of Mathematics Science Director of Graduate Studies .i . 1;4 ' AUG 1 1 1994 3 Analytic surgery and analytic torsion Andrew Hassell Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology Abstract Let (M, h) be a compact manifold in which H is an embedded hypersurface which separates M into two pieces M+ and M_. If h is a metric on M and x is a defining function for H consider the family of metrics dx2 +-h gE = X2 2+- 2 +h where e > 0 is a parameter. The limiting metric, go, is an exact b-metric on the disjoint union M = M+ U M_, i.e. it gives M+ asymptotically cylindrical ends with cross-section H. We investigate the behaviour of the analytic torsion of the Laplacian on forms with values in a flat bundle, with respect to the family of metrics g,. We find a surgery formula for the analytic torsion in terms of the 'b'-analytic torsion on M±. By comparing this to the surgery formula for Reidemeister torsion, we obtain a new proof of the Cheeger-Miiller theorem asserting the equality of analytic and Reidemeister torsion for closed manifolds, and compute the difference between b-analytic and Reidemeister torsion on manifolds with cylindrical ends. We also present a glueing formula for the eta invariant of the Dirac operator on an odd dimensional spin manifold M. This generalizes a result of Mazzeo and Melrose, who obtained a similar glueing formula under the assumption that the induced Dirac operator 3H on H is invertible. In both cases there is an 'extra' term in the glueing formula coming from the long time asymptotics of the heat kernel. The term can be expressed in terms of a one dimensional Laplacian associated to the null space of the Laplacian on M. This operator is determined by scattering data on M at zero energy, and controls the leading behaviour of small eigenvalues as e 0. Thesis Supervisor: Richard Melrose Title: Professor of Mathematics To my parents, Jenny and Cleve 5 C 7 ACKNOWLEDGEMENTS I would like to express my deep gratitude to my advisor Richard Melrose for suggesting a terrific thesis problem, for sharing with me his mathematical insight, and for his encouragement. His influence is evident on every page. I am also grateful to Rafe Mazzeo for sharing his ideas with me. As noted below, this thesis is in part joint work with both of them. I am indebted to many other graduate students at MIT, present and former, for friendship and mathematical discussions, particularly Tanya Christiansen, Alan Blair, Richard Stone and Mark Joshi. To all my other friends here, I wish to express my appreciation for their support. The financial support of the Australian-American Educational Foundation and the Alfred P. Sloan Foundation is gratefully acknowledged. This work is dedicated to my parents in appreciation for their constant love and encouragement. 8 DECLARATION Some of this thesis is joint work. The single and double logarithmic spaces were developed jointly with Richard Melrose and Rafe Mazzeo. Much of chapters 2, 3 and 4 is due in part to Richard Melrose, and several other ideas in the thesis were influenced or suggested by him. It is intended to publish chapters one through nine in a joint paper with Mazzeo and Melrose. Otherwise, except where noted in the text, the work described here is my own. 10