 
Summary: Annals of Global Analysis and Geometry 27: 333340 (2005)
C 2005 Springer.
333
Eta Invariant and Conformal Cobordism
XIANZHE DAI
Department of Mathematics, University of California, Santa Barbara, California 93106, U.S.A.
email: dai@math.ucsb.edu
(Received 20 January 2004; accepted 9 November 2004)
Abstract. In this note we study the problem of conformally flat structures bounding conformally
flat structures and show that the eta invariants give obstructions. These lead us to the definition of
an Abelian group, the conformal cobordism group, which classifies the conformally flat structures
according to whether they bound conformally flat structures in a conformally invariant way. The eta
invariant gives rise to a homomorphism from this group to the circle group, which can be highly
nontrivial. It remains an interesting question of how to compute this group.
Mathematics Subject Classification (2000): 55N22.
Key words: eta invariant, conformal cobordism, conformally flat structures.
1. Introduction
This note is inspired by a question from my colleague Darren Long (see the very
interesting paper [4]). Motivated in part by considerations in physics, Long and
Reid considered the question of whether a closed orientable hyperbolic 3manifold
