We consider certain natural (ℤ₂)ⁿ actions on real Grassmann and flag manifolds andS¹ actions on complex Grassmann manifolds with finite stationary point sets and determine completely which of them bound equivariantly.

We define equivariant cyclic and Hochschild cohomology modules of a cyclic objectX in the category ofG-sets and relate them with the Bredon cohomologies of the cyclic geometric realization ¦X¦^cy.

We investigate higher-order generalizations of well known results for Liealgebroids and bialgebroids. It is proved that $n$-Lie algebroid structures correspond to $n$-ary generalization of Gerstenhaber algebras and are implied by $n$-ary generalization of linear Poisson structures on the dual bundle. A Nambu–Poisson manifold (of order $n$ > 2) gives rise to a special bialgebroid structure which is referred to as a weak Lie–Filippov bialgebroid (of order $n$). It is further demonstrated that such bialgebroids canonically induce a Nambu–Poisson structure on the base manifold. Finally, the tangent space of a Nambu Lie group gives an example of a weak Lie–Filippov bialgebroid over a point.

We introduce the notion of the modular class of a Lie algebroid equipped with a Nambu structure. In particular, we recover the modular class of a Nambu-Poisson manifold $M$ with its Nambu tensor $\Lambda$ as the modular class of the tangent Lie algebroid $TM$ with Nambu structure $\Lambda$. We show that many known properties of the modular class of a Nambu-Poisson manifold and that of a Lie algebroid extend to the setting of a Lie algebroid with Nambu structure. Finally, we prove that for a large class of Nambu-Poisson manifolds considered as tangent Lie algebroids with Nambu structures, the associated modular classes are closely related to Evens-Lu-Weinstein modular classes of Lie algebroids.