The higher fixed point theorem for foliations: applications to rigidity and integrality (Q6628884)

The article under review is the sequel to the authors' previous work [J. Funct. Anal. 259, No. 1, 131--173 (2010; Zbl 1207.46063)], which gives several applications of the main results in op. cit. We highlight two of them.\N\NWe first recall the higher Lefschetz theorem proven in [loc. cit.]. Let \((V, g)\) be a closed Riemannian manifold and \(F\) an oriented foliation of \((V, g)\). Let \(h:(V, g)\to(V, g)\) be a holonomy isometry (so \(h\) preserves the leaves of \(F\)) which generates a compact Lie group \(H\) of isometries of \((V, g)\). Assume the fixed point submanifold \(V^h:=V^H\) of \(h\) is transverse to \(F\). Let \((E, d)\to(V, F)\) be an \(H\)-equivariant leafwise elliptic pseudodifferential complex. Denote by \(\textrm{Ind}^H(E, d)\) the \(H\)-equivariant analytic index of \((E, d)\), which is a class in the \(H\)-equivariant \(C^*\)-algebra \(K\)-theory group \(K^H(C^*(V, F))\), where \(C^*(V, F)\) is Connes' \(C^*\)-algebra of \((V, F)\). Let \(C\) be an even dimensional closed Haefliger current on \((V, F)\) and \(\tau_C\) the associated cyclic cocycle. The \(C\)-higher Lefschetz number \(L_C(h; E, d)\) is defined to be \N\[\NL_C(h; E, d):=\langle\tau_C, \textrm{Ind}^H(E, d)\rangle(h)\in\mathbb{C}.\N\]\NDenote by \(F^h\) the induced foliation, \(N^h\) the normal bundle to \(V^h\) in \(V\), \(TF^h=T(F\cap V^h)\) the induced integrable subbundle of \(TV^h\) and \(i:TF^h\to TF\) the \(H\)-inclusion map. The higher Lefschetz theorem states that \N\[\NL_C(h; E, d)=\textrm{Ind}_{C|_{V^h}}\bigg(\frac{i^*[\sigma(E, d)](h)}{\lambda_{-1}(N^h\otimes\mathbb{C})(h)}\bigg),\N\]\Nwhere \(C|_{V^h}\) is the closed Haefliger current on the fixed point foliation \((V^h, F^h)\) obtained by restricting \(C\) to \((V^h, F^h)\) and \(\textrm{Ind}_{C|_{V^h}}:K(TF^h)\otimes\mathbb{C}\to\mathbb{C}\) is the complexified higher \(C|_{V^h}\)-index map on \((V^h, F^h)\).\N\NUnder the assumption that the induced foliation \(F^h\) is oriented, the cohomological Lefschetz formula is given by \N\[\NL_C(h; E, d)=\bigg\langle[C|_{V^h}], \int_{F^h}\frac{\textrm{ch}_{\mathbb{C}}(i^*[\sigma(E, d)](h))}{\textrm{ch}_{\mathbb{C}}(\lambda_{-1}(N^h\otimes\mathbb{C})(h))}\textrm{Todd}(TF^h\otimes\mathbb{C})\bigg\rangle,\N\]\Nwhere the map \(\textrm{ch}_{\mathbb{C}}:K(TF^h)\otimes R(H)_h\to H^*(TF^h; \mathbb{C})\) is obtained by trivially extending the topological Chern character \(\textrm{ch}:K(TF^h)\to H^*_c(TF^h; \mathbb{R})\) for compactly supported \(K\)-theory.\N\NIn the article under review, the authors prove the following rigidity theorem, which is an application of the cohomological Lefschetz formula. Under the above setup, suppose furthermore that \(TF\to F\) is an even dimensional spin bundle with associated leafwise Dirac operator \(\mathsf{D}\). If there exists a closed holonomy invariant current \(C\) such that \N\[\N\bigg\langle[C], \int_F\widehat{A}(TF)\bigg\rangle\neq 0,\N\]\Nthen no compact connected Lie group \(H\) can act non-trivially as a group of isometries of \((V, g)\) preserving the leafs of \(F\) and their spin structure.\N\NThe authors also prove the following integrality theorem. Under the above setup, suppose furthermore that \(F\) is a Riemannian foliation and transversely spin. Denote by \(\phi^{CM}_{(V^h, F^h)}\) the even Connes-Moscovici residue cocycle in the \((b, B)\)-bicomplex associated with \((V^h, F^h)\), and by \(\textrm{Ind}^{CS}_{(V^h, F^h)}:K(TF^h)\to K(C^*(V^h, F^h))\) the Connes-Skandalis topological longitudinal index morphism for \((V^h, F^h)\). Then for any leafwise elliptic \(H\)-invariant pseudodifferential complex \((E, d)\to(V, F)\), \N\[\N\bigg\langle(\textrm{Ind}^{CS}_{(V^h, F^h)}\otimes\mathbb{C})\bigg(\frac{i^*[\sigma(E, d)](h)}{\lambda_{-1}(N^h\otimes\mathbb{C})(h)}\bigg), [\phi^{CM}_{(V^h, F^h)}]\otimes\textrm{id}_{\mathbb{C}}\bigg\rangle\in R(H)(h).\N\]