Buchstaber invariants of skeleta of a simplex

A moment-angle complex $\mathcal{Z}_K$ is a compact topological space associated with a finite simplicial complex $K$. It is realized as a subspace of a polydisk $(D^2)^m$, where $m$ is the number of vertices in $K$ and $D^2$ is the unit disk of the complex numbers \C, and the natural action of a torus $(S^1)^m$ on $(D^2)^m$ leaves $\mathcal{Z}_K$ invariant. The Buchstaber invariant $s(K)$ of $K$ is the maximum integer for which there is a subtorus of rank $s(K)$ acting on $\mathcal{Z}_K$ freely. The story above goes over the real numbers $\R$ in place of \C and a real analogue of the Buchstaber invariant, denoted $s_\R(K)$, can be defined for $K$ and $s(K)\leqq s_\R(K)$. In this paper we will make some computations of $s_\R(K)$ when $K$ is a skeleton of a simplex. We take two approaches to find $s_\R(K)$ and the latter one turns out to be a problem of integer linear programming and of independent interest

The cohomology ring of the GKM graph of a flag manifold of classical type

If a closed smooth manifold $M$ with an action of a torus $T$ satisfies certain conditions, then a labeled graph \mG_M with labeling in $H^2(BT)$ is associated with $M$, which encodes a lot of geometrical information on $M$. For instance, the "graph cohomology" ring \mHT^*(\mG_M) of \mG_M is defined to be a subring of \bigoplus_{v\in V(\mG_M)}H^*(BT), where V(\mG_M) is the set of vertices of \mG_M, and is known to be often isomorphic to the equivariant cohomology $H^*_T(M)$ of $M$. In this paper, we determine the ring structure of \mHT^*(\mG_M) with $\Z$ (resp. $\Z[1/2]$) coefficients when $M$ is a flag manifold of type A, B or D (resp. C) in an elementary way.Comment: 22 page

Topological toric manifolds

We introduce the notion of a topological toric manifold and a topological fan and show that there is a bijection between omnioriented topological toric manifolds and complete non-singular topological fans. A topological toric manifold is a topological analogue of a toric manifold and the family of topological toric manifolds is much larger than that of toric manifolds. A topological fan is a combinatorial object generalizing the notion of a simplicial fan in toric geometry. Prior to this paper, two topological analogues of a toric manifold have been introduced. One is a quasitoric manifold and the other is a torus manifold. One major difference between the previous notions and topological toric manifolds is that the former support a smooth action of an $S^1$-torus while the latter support a smooth action of a \C^*-torus. We also discuss their relation in details.Comment: 42 pages, 4 figure