A decomposition theorem for the spectral sequence of Lie foliations

Abstract:

For a Lie $\mathfrak {g}$-foliation $\mathcal {F}$ on a closed manifold $M$, there is an "infinitesimal action of $\mathfrak {g}$ on $M$ up to homotopy along the leaves", in general it is not an action but defines an action of the corresponding connected simply connected Lie group $\mathfrak {S}$ on the term ${E_1}$ of the spectral sequence associated to $\mathcal {F}$. Even though ${E_1}$ in general is infinite-dimensional and non-Hausdorff (with the topology induced by the ${\mathcal {C}^\infty }$-topology), it is proved that this action can be averaged when $\mathfrak {S}$ is compact, obtaining a tensor decomposition theorem of ${E_2}$. It implies duality in the whole term ${E_2}$ for Riemannian foliations on closed oriented manifolds with compact semisimple structural Lie algebra.