Classifying 3-dimensional lens spaces by eta-invariants (Q753206)

Let p be a positive integer and q an integer relatively prime to p. There exist examples which show that the eta invariants \(\eta(p,q)\) of the three dimensional lens spaces \(L(p,q)\) are not complete isometric invariants. In the present paper the author studies the eta invariants \(\eta_{\alpha}(p,q)\) of \(L(p,q)\) which correspond to an irreducible representation \(\alpha\) of the fundamental group \(\pi_ 1(L(p,q))\cong {\mathbb{Z}}/p{\mathbb{Z}}\). He proves that two 3-dimensional lens spaces \(L(p,q)\) and \(L(p,\bar q)\) are isometric to each other if and only if some of their eta invariants corresponding to a representation of the fundamental group satisfy a certain set of equations. The proof uses the representation of \(\eta_{\alpha}(p,q)\) by a generalized Dedekind sum.