The \(\eta\)-invariant for Lipschitz manifolds
The \(\eta\)-invariant has been defined for \(C^\infty\)-manifolds by \textit{M. F. Atiyah, V. K. Patodi} and \textit{I. M. Singer} in [Math. Proc. Camb. Philos. Soc. 77, 43--69 (1975; Zbl 0297.58008)] as follows: let \(W\) be an oriented and odd-dimensional \(C^\infty\)-manifold, \(g\) a Riemannian structure on \(W\), \((E,l,\nabla)\) a Hermitian bundle with a unitary connection and \(A\) the associated signature operator. Then, the function \(\eta(s,A)= \text{Tr} (A| A|^{-s-1})\) is analytic for \(\mathbb{R}(s)> \dim W\) and admits a meromorphic extension to \(\mathbb{C}\). By definition \(\eta(W,E)= \frac12 \eta(0,A)\). In the quoted paper one establishes the formula \[ \operatorname {Ind}D= \int_V \text{ch} (\widetilde{\nabla}) L(V,r)- \eta(W,E)- \frac12 \dim \ker A, \] where \(V\) is an oriented manifold with \(\partial V=W\), \(r\) is a Riemannian structure such that \(r= dt^2\oplus g\) in a tubular neighbourhood of \(W\), \((\widetilde{E}, \widetilde{l}, \widetilde{\nabla})\) is an extension of \((E,l,\nabla)\), \(L(V,r)\) is the Hirzebruch polynomial of Pontryagin forms of the Riemannian connection \(r\) and \(D\) is a selfadjoint extension of the closing \(D_0\) of \(A\) in \(\Omega(V)\), defined by the spectral projector of the positive part of the spectrum of \(A\) (\(\text{ch} (\widetilde{\nabla})\) is the Chern character). In the present paper the author defines the \(\eta\)-invariant for Lipschitz Riemannian manifolds by the above formula given sense to \(\operatorname {Ind}D\) and \(L(V,r)\). Moreover he extends from smooth to topological manifolds the Atiyah-Patodi-Singer index theorem for flat bundles.
scientific article; zbMATH DE number 1782648