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ANALYTIC SURGERY ANALYTIC TORSION Andrew Hassell 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 Signatureof Author .. - , Department of Mathematics April 20, 1994 Certified by . , -- Richard Melrose Professor of Mathematics Thesis Advisor Accepted by J>,rAY1LlCaviu Vv caIo Science Professor of Mathematics Director of Graduate Studies .i . 14'; 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 gE = X2 2-+ 2 +h+-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 CONTENTS Chapter 1. Introduction ................ 13 1.1. Analytic surgery .................... ..13 1.2. Eta invariant and analytic torsion . 14 1.3. Statement of Results ... 15 1.4. Outline of the proof . ... 17 Chapter 2. Manifolds with corners, blowups and b-fibrations 20 2.1. Manifolds with corners . 20 ... 2.2. Blowups . 21 ... 2.3. Operations on conormal functions . 22 2.4. Two Blowup Lemmas .. 24 2.5. Logarithmic blow up . 27 ... 2.6. Total boundary blow up . .28 Chapter 3. The Single Space . .. ... 31 3.1. Definition .... ... ... 31 3.2. Densities ........... ... 32 3.3. Lift of V,(X) . ... ... 33 3.4. Models ....... ... ... 37 Chapter 4. The double space and the pseudodifferential calculus . 41 4.1. Preliminary remarks . .. 41 4.2. Logarithmic Double space .. .. .. 41 4.3. Densities ..................... ... 43 4.4. Logarithmic Surgery Pseudodifferential Operators . 44 4.5. Action on Distributions .. .. 45 4.6. The Triple Space . .. 46 4.7. Compositon and the Residual Space .. .. 46 4.8. Symbol Map . .. 48 4.9. Model Operators. ................... 49 4.10. Neumann Series for Residual Operators ... ... .. 50 4.11. Composition of small calculus with residual calculus . .. .. 51 Chapter 5. One dimensional surgery resolvent . 53 .. 5.1. Scaling property. 53 .... 5.2. Scattering Matrix ................ 54 ......... 5.3. Properties at the boundary . 54 ......... 5.4. Eigenvalues ............. 55 .......... 5.5. Heat kernel and large Izi asymptotics of . 55 ......... 5.6. Determinant and eta invariant. 57 ......... 11 12 CONTENTS Chapter 6. Resolvent with scaled spectral parameter ........ .......60 6.1. Preliminaries ................. ............... 60 6.2. Terms of order (ias)-l ......... ........ ...63 6.3. Terms of order (ias E) . ........ ...64 6.4. Terms of order (ias E). ........ ...67 6.5. Compatibility with the symbol . .. ........ ...67 6.6. From parametrix to resolvent . ........ ...67 6.7. Near the discrete spectrum of RN(A). ........ ...68 6.8. In the presence of L2 null space . ........ .......70 6.9. Very small eigenvalues . ........ .......71 Chapter 7. Full Resolvent ....... 73 ...... ........ 7.1. Resolvent spaces ....... 73 ...... ........ 7.2. Operator Calculus ............ ........ .......75 7.3. Full Parametrix ............ 75 7.4. Full Parametrix to Full Resolvent . ........ .......78 Chapter8.Heat Kernel ...... ... ... .. .80 8.1. The Heat Space and the Heat-Resolvent Space .. ..... 80 8.2. Contour spaces. ....... .... 81 8.3. Behaviour as t - . ... .. 84 8.4. Very small eigenvalues .. .. .. .. 87 8.5. Full Heat Kernel. .. ....... 88 Chapter 9. Limit of Eta Invariant . 89 ... 9.1. Eta invariant. 89 ... 9.2. The diagonal of the Logarithmic heat space . 89 ... 9.3. Asymptotic expansion of iq as e - 0 . 90 ... Chapter 10. A Hodge Mayer-Vietoris cohomology sequence . 93 10.1. Mayer-Vietoris sequence . ... .. 93 10.2. b-Hodge theory ..... 93 10.3. Surgery Hodge theory ..... 94 Chapter 11. Analytic Torsion and Reidemeister torsion . .. 97 11.1. Analytic torsion. .. 97 11.2. Surgery formula for analytic torsion. 99 11.3. Reidemeister torsion .. .. .. 102 11.4. Cheeger-Miiller Theorem. 104 Chapter 1. Introduction 1.1. Analytic surgery. In this thesis, we continue the study of analytic surgery initiated in [15]. By "analytic surgery" we mean a singular deformation of a Riemannian metric on a closed manifold M that models the cutting of M along a hypersurface H (possibly disconnected), forming a manifold with boundary M ("surgery"). For simplicity we assume that H separates M; thus M is the disjoint union of two manifolds with boundary M±. We consider a specific deformation which degenerates to a complete metric on M of the form dx2 / x2 + h, where x is a boundary defining function for H (that is, > 0, H = {x = 0} and dxz 0 on H) and h is a smooth metric on M. This form of metric on a manifold with boundary, called an "exact b-metric" and studied in some detail in [17], gives M asymptotically cylindrical ends, with log x approximately arc length along the end. Specifically, we consider a family of the form dx2 ge - 2 e2 + h; this is a smooth metric on M for every e > 0, which develops a long neck of length 2 log 1/e + 0(1) -- oo as e -- 0 and whose singular limit is manifestly an exact b-metric on M. Similar deformations, usually phrased in terms of a family of manifolds which have a long cylindrical neck across H with length I - oo, have been studied by several authors. There are two main reasons for interest in this procedure. One is to understand the behaviour of geometric or topological invariants such as the index of a Dirac operator, eta invariant or analytic torsion under surgery, as in [4], [6], [10], and [8]. The other is to analyse the behaviour of the spectrum of operators (such as the Laplacian) under the transition from closed manifold to complete manifold. In Mazzeo and Melrose's paper [15] and the present thesis, both questions are investigated. Of course these two problems are closely related. In this thesis glueing formulae for the eta invariant and analytic torsion under surgery are presented, but these are obtained by studying the full resolvent family of generalized Laplacians under surgery, including the analysis of accumulation of eigenvalues at the bottom of the continuous spectrum of A0. Closely related are the papers of McDonald [16] and Seeley and Singer [28], who studied metric degeneration to incomplete conic metrics, and Ji [13], who studied degeneration of Riemann surfaces to surfaces with hyperbolic cusps. It should be remarked that the approach of McDonald inspired [15]and the present work. There are two motivations for the choice of a "cylindrical ends" metric for M. One is that Atiyah, Patodi and Singer obtained their well-knownglobal boundary condition for the Dirac operator on a manifold with boundary in [1] heuristically by considering a cylindrical end attached to the boundary. The other is that Richard Melrose has presented a detailed analysis of the Laplacian associated to an exact b-metric in [17], as well as a proof of the APS index theorem in the 'b' context.
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