On the Van Est homomorphism for Lie groupoids (Q5963021)

Let \(G \rightrightarrows M\) be a Lie groupoid for which \(G\) and \(M\) are smooth manifolds and its source and target maps \(\text{s, t} : G \rightrightarrows M\) are surjective submersions onto a submanifold \(M \subset G\) of units, Then the normal bundle \(A \to M\) of \(M\) in \(G\) forms a Lie algebroid. Consider the space \(B_pG\) of \(p\)-composable arrows, then putting \(B_0G=M\), we can define the simplicial manifold \(B_\bullet G\). We write \(C^\infty(B_\bullet G)\) for its de Rham complex. Let \({\mathsf C}^\bullet(A)\) denote the Chevalley-Eilenberg complex for \(A\). Then there is a morphism NEWLINE\[NEWLINE\text{VE} : C^\infty(B_\bullet G) \to {\mathsf C}^\bullet(A)NEWLINE\]NEWLINE of cochain complexes, called the Van Est map. This map induces an isomorphism in cohomology in all degrees if we suppose, for simplicity, the \(\text{t}\)-fibers are contractible. In this paper the authors propose an alternative approach for constructing this VE map. In order to do this the authors adopt a method to go through the the bidifferential algebra \({\mathsf C}^\bullet(T_{\mathcal F}E_\bullet G)\) based on the homological perturbation theory where \(T_{\mathcal F}E_p G\) is the tangent bundle to the fibers of the principal bundle \(\kappa_p : E_p \to B_p\). By considering an intertwining map of the total differential \(\text{d}+\delta\) with the Chevalley-Eilenberg differential the authors succeed in constructing a homotopy equivalence between \(\text{Tot}^\bullet{\mathsf C}(T_{\mathcal F}EG)\) and \({\mathsf C}^\bullet(A)\). Composing this equivalence with the map \(\kappa^\ast_\bullet : C^\infty(B_\bullet G) \to \text{Tot}^\bullet{\mathsf C}(T_{\mathcal F}EG)\) we get the desired map. Similarly to this we obtain its inverse cochain map. In addition, it is shown that the method described above can be generalized to that of the construction of the Van Est map NEWLINE\[NEWLINE\text{VE} : \Omega^\bullet(B_\bullet G) \to W^{\bullet, \bullet}(A)NEWLINE\]NEWLINE for the Bott-Shulman-Stasheff double complex.