For an equivariantly formal action of T k on a smooth manifold X 2n with isolated fixed points we investigate the global homological properties of the graded poset S X of face submanifolds. We prove that the condition of j-independency of tangent weights at each fixed point implies pj `1q-acyclicity of the skeleta pS X q r for r ą j `1. This result provides a necessary topological condition for a GKM-graph to be a GKM-graph of some GKM-manifold. We use particular acyclicity arguments to describe the equivariant cohomology algebra of an equivariantly formal manifold X 2n with an pn ´1q-independent action of T n´1 , for n ě 5. This description relates the equivariant cohomology algebra to the face algebra of a certain simplicial poset. This observation relates actions of complexity one to the theory of torus manifolds.

No abstract

By the classical result of Milnor and Novikov, the unitary cobordism ring is isomorphic to a graded polynomial ring with countably many generators:. In this paper we solve the well-known problem of constructing geometric representatives for a i among smooth projective toric varieties, a n = [X n ], dim C X n = n. Our proof uses a family of equivariant modifications (birational isomorphisms) B k (X) → X of an arbitrary complex manifold X of (complex) dimension n (n 2, k = 0, . . . , n − 2). The key fact is that the change of the Milnor number under these modifications depends only on the dimension n and the number k and does not depend on the manifold X itself.

The present paper generalises the results of Ray [15] and Buchstaber-Ray [1], Buchstaber-Panov-Ray [3] in unitary cobordism theory. I prove that any class x ∈ Ω * U of the unitary cobordism ring contains a quasitoric totally normally and tangentially split manifold.Second, a method of producing new totally normally split toric varieties is given in Section 3. Namely, this is blow-up of a totally normally split toric variety at an invariant complex codimension 2 subvariety.These are used to construct some totally normally split toric varieties which are then shown to be multiplicative generators of Ω * U . Finally, a possible adaptation of N. Ray's construction to Theorem 1.3 is discussed in Section 7.The author is grateful to V.M. Buchstaber and T.E. Panov for numerous fruitful discussions.2 Bounded flag fibre bundlesThe idea of the bounded flag manifold ([2], [4, §7.7]) can be globalised in terms of fiber bundles. In this Section, the corresponding construction is given. For the rest of this Section, X stands for a compact stably complex smooth manifold of real dimension 2n and ξ i → X, rk ξ i = 1, i = 1, . . . , k + 1, are complex linear vector bundles over X. Also let ξ := k+1 i=1 ξ i . Everywhere below pull-backs and tensor products of vector bundles are omitted, unless otherwise specified.