Hopf cyclic cohomology and Hodge theory for proper actions (Q380282)

This paper studies cyclic cohomology of a Hopf algebroid arising from a proper action of Lie group \(G\) on a manifold \(M\), and derives some geometric results on \(G\)-invariant differential forms and vector fields of \(M\).NEWLINENEWLINEAfter recalling the concept of Hopf algebroid initiated by Lu, the authors define a Hopf algebroid \(\mathcal{H}\left( G,M\right) \equiv \left( A,B,\alpha ,\beta ,\Delta ,\epsilon ,S\right) \), closely related to the transformation group groupoid \(G\ltimes M\), consisting of the algebra \(B\) of differential forms on \(M\), the algebra \(A\) of \(B\)-valued functions on \(G\), source and target maps \(\alpha ,\beta :B\rightarrow A\) defined by \(\alpha \left( b\right) \left( g\right) :=b\) and \(\beta \left( b\right) \left( g\right) :=g^{\ast }\left( b\right) \) the pullback of \(b\in B\) under the action by \(g\in G\), the bimodule map \(\Delta :A\rightarrow A\otimes _{B}A\) defined by \(\Delta \left( \phi \right) \left( g_{1},g_{2}\right) :=\phi \left( g_{1}g_{2}\right) \), the counit \(\epsilon :A\rightarrow B\) defined by \(\epsilon \left( \phi \right) :=\phi \left( 1\right) \), and the antipode \( S:A\rightarrow A\) defined by \(S\left( \phi \right) \left( g\right) :=g^{\ast }\left( \phi \left( g^{-1}\right) \right) \). Briefly recalling the definition of cyclic cohomology of Hopf algebroid introduced by Connes and Moscovici, the authors show that the cyclic cohomology \(HC^{\cdot }\left( \mathcal{H}\left( G,M\right) \right) \) is isomorphic to the differentiable cohomology \(\bigoplus _{k\geq 0}HC^{\cdot -2k}\left( G,\left( \Omega ^{\ast }M,d\right) \right) \) of \(G\) with coefficients in \(\Omega ^{\ast }M\), and hence by Crainic's result, is isomorphic to the de Rham cohomology \(\bigoplus _{k\geq 0}HC^{\cdot -2k}\left( \left( \Omega ^{\ast }M\right) ^{G},d\right) \) on the space \(\left( \Omega ^{\ast }M\right) ^{G}\) of \(G\)-invariant differential forms on \(M\).NEWLINENEWLINEIf the proper \(G\)-action is also cocompact, then a Hodge theorem for \(G\) -invariant differential forms on \(M\) is derived, giving that every cyclic cohomology class of \(\mathcal{H}\left( G,M\right) \) is represented by a generalized harmonic form. In particular, the cyclic cohomology of \(\mathcal{ H}\left( G,M\right) \) is finite-dimensional and hence the Euler characteristic can be well defined. The authors then prove the Poincaré duality for twisted de Rham cohomology of \(G\)-invariant differential forms and derive that the Euler characteristic of a proper cocompact action on \(M\) is \(0\) if \(\dim \left( M\right) \) is odd or if there exists a nowhere vanishing \(G\)-invariant vector field on \(M\).