Abstract: Consider an effective smooth action of a compact torus on a connected closed smooth manifold X having isolated fixed points. We introduce the finite graded poset S(X) called the face poset of the action. If X is a toric variety or a quasitoric manifold, then S(X) is the face poset of the moment polytope X/T. However, S(X) is defined for actions of any complexity, in which case the local structure of S(X) is determined by the linear matroids of tangent weights.

If an action on X is equivariantly formal, we prove that the geometrical realization |S(X)| has some degree of acyclicity, depending on tangent weights. This statement gives a homological obstruction for particular actions to be equivariantly formal. As a motivating example, we study canonical conjugation actions on the manifolds of isospectral Hermitian matrices, having zeroes at prescribed positions. We prove a complete classification, which of these manifolds are equivariantly formal.

This talk is based on several works written jointly with V.Buchstaber, V.Cherepanov, M.Masuda, G.Solomadin, and K.Sorokin.