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The logarithm of the Dedekind \(\eta\)-function. (English) Zbl 0648.58035
The appearance of Dedekind sums in topology was first noted by F. Hirzebruch [Prospects Math., Ann. Math. Stud. 70, 3-31 (1971; Zbl 0252.58009)]. These sums were introduced by Dedekind to describe the transformation of \(\log\eta(\tau)\) under elements of \(SL(2,{\mathbb{Z}})\), where \(\eta(\tau)\) is the Dedekind eta-function. In the present paper the appearance of \(\log\eta(\tau)\) in topology is described in the general context of index theory and related to several other invariants. The paper deals with several generalizations of the signature theorem involving the case of manifolds with boundary, the equivariant case and families of elliptic operators. The results are applied to a fibration \(Z\to^{M}X\), where the fibre M is a torus, and X is a compact surface with boundary. The local coefficient system given by \(H^ 1(M)\) then arises from a representation \(\pi_ 1(X)\to SL(2,{\mathbb{Z}})\). The signature of Z is equal to the signature of the local coefficient system over X given by \(H^ 1(M)\), and there is a function \(\Phi\) : SL(2,\({\mathbb{Z}})\to {\mathbb{Q}}\), defined by W. Meyer [Math. Ann. 201, 239-264 (1973; Zbl 0241.55019)] such that \[ sign Z=sign (X,H^ 1(M))=- \sum \Phi (A), \] where the sum is over the monodromy matrices A around the boundary circles S of X by the action on \(H^ 1(M)\). There are several other invariants \(SL(2,{\mathbb{Z}})\to {\mathbb{Q}}\) defined in this context and described in the paper: The invariant \(\eta(A)\) which is the Atiyah-Patodi-Singer spectral invariant of the component W(A) of \(\partial Z\) corresponding to the circle with monodromy matrix A. The invariant \(\eta^ 0(A)\) which arises as an adiabatic limit from a family of \(\eta\)-invariants for W(A) as studied by J.-M. Bismut and D. S. Freed [Commun. Math. Phys. 107, 103-163 (1986)]. The invariant \(\chi(A)\) which describes essentially the transformation properties of \(\log\eta(\tau)\) under A. The signature defect \(\delta(A)\) defined by F. Hirzebruch. The invariant \(\mu(A)\) describing the logarithmic monodromy (divided by \(\pi^ i\)) of Quillen’s determinant bundle, which is a complex line bundle over X associated to the signature operator on Z. The value \(L_ A(0)\) of the Shimiuzu L-function. Some identities between these invariants were proved by different authors. The main result of the present paper is that they all coincide if \(A\in SL(2,{\mathbb{Z}})\) is hyperbolic.
Reviewer: K.H.Mayer

MSC:
58J20 Index theory and related fixed-point theorems on manifolds
11F11 Holomorphic modular forms of integral weight
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