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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A decomposition theorem for the spectral sequence of Lie foliations

Author: Jesús A. Alvarez López
Journal: Trans. Amer. Math. Soc. 329 (1992), 173-184
MSC: Primary 57R30
MathSciNet review: 1041050
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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.

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Article copyright: © Copyright 1992 American Mathematical Society