ANALYTIC SURGERY ANALYTIC TORSION Andrew Hassell

106 Pages·2012·6.11 MB·English

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

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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
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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.
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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.
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