Induced cyclic cocycles and higher eta invariants
Let \(M\) be a closed, odd-dimensional Riemannian manifold with \(\text{spin}^c\) structure. Having a representation \(\alpha:\pi_1(M)\to G\) of the fundamental group of \(M\) into a compact connected Lie group \(G\), the product \(M\times G\) has two foliations, the obvious one \({\mathcal F}_G\) and, transversally, \({\mathcal F}_{\alpha}\) induced by the representation \(\alpha\). Furthermore, the \(\text{spin}^c\) Dirac operator \(\eth\) of \(M\) induces a leafwise elliptic Dirac operator \(\eth_{\alpha}\) over \(M\times G\). The paper gives a construction of a cyclic cocycle \(c_H\) on the foliation algebra \(C_c^{\infty}({\mathcal F}_{\alpha})\) for any closed normal subgroup \(H\) of \(G\), such that the pairing with the index of the induced Dirac operator \(\eth_{\alpha}\) gives a spectral invariant, and motivates the notion of these invariants as higher eta invariants.
leafwise elliptic Dirac operator on a foliated manifold
scientific article; zbMATH DE number 1046928
scientific article