Related papers: Two transformations of simple polytopes preserving…

We describe here two simple polytopes that have different Betti numbers and whose moment-angle manifolds are diffeomorphic.

In this note we give the definition of the "doubling operation" for simple polytopes, find the formula for the h-polynomial of new polytope.As an application of this operation we establish the relationship between moment-angle manifolds and…

The paper is devoted to the well-known problem of smooth structures on moment-angle manifolds. Each real or complex moment-angle manifold has an equivariant smooth structure given by an intersection of quadrics corresponding to a geometric…

We construct a family of manifolds, one for each $n\geq 2$, having a nontrivial Massey $n$-product in their cohomology. These manifolds turn out to be smooth closed 2-connected manifolds with a compact torus action called moment-angle…

Let $P$ be a simple polytope of dimension $n$ with $m$ facets and $P_{v}$ be a polytope obtained from $P$ by cutting off one vertex $v$. Let $Z=Z(P)$ and $Z_{v}=Z(P_{v})$ be the corresponding moment-angle manifolds. In \cite{[GL]} S.Gitler…

We answer a problem posed by Panov, which is to describe the relationship between the wedge summands in a homotopy decomposition of the moment-angle complex corresponding to a disjoint union of k points and the connected sum factors in a…

Two simple polytopes of dimension 3 having the identical bigraded Betti numbers but non-isomorphic Tor-algebras are presented. These polytopes provide two homotopically different moment-angle manifolds having the same bigraded Betti…

We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly $(d+1)$-colored) triangulation of a combinatorial $d$-manifold into another balanced triangulation. These moves form a natural analog of bistellar…

We introduce a new operation, double point surgery, on immersed surfaces in a 4-manifold, and use it to construct knotted configurations of surfaces in many 4-manifolds. Taking branched covers, we produce smoothly exotic actions of Z/m x…

A simple polytope $P$ is called $B$-rigid if its combinatorial type is determined by the cohomology ring of the moment-angle manifold $\mathcal{Z}_P$ over $P$. We show that any tensor product decomposition of this cohomology ring is…

We construct simple geometric operations on faces of the Cayley sum of two polytopes. These operations can be thought of as convex geometric counterparts of divided difference operators in Schubert calculus. We show that these operations…

The mixing operation for abstract polytopes gives a natural way to construct the minimal common cover of two polytopes. In this paper, we apply this construction to the regular convex polytopes, determining when the mix is again a polytope,…

We generalize the notion of moment-angle manifold over a simple convex polytope to an arbitrary nice manifold with corners. For a nice manifold with corners Q, we first compute the stable decomposition of the moment-angle manifold Z_Q via a…

This paper investigates the moment-angle manifolds whose cohomology ring is isomorphic to that of a connected sum of sphere products. We first give a example of moment-angle manifolds corresponding to a 4 dimentional simplicial polytope. It…

Let $\rho:(D^2)^m\to I^m$ be the orbit map for the diagonal action of the torus $T^m$ on the unit poly-disk $(D^2)^m$, $I^m=[0,1]^m$ is the unit cube. Let $C$ be a cubical subcomplex in $I^m$. The moment-angle complex $\ma(C)$ is a…

For a directed polytope, we construct a colored operad whose Poincare-Hilbert series encodes certain operations on the cellular complex of the polytope. We conjecture that for a class of short polytopes the constructed operads are Koszul…

We show that a moment-angle manifold associated to a neighbourly triangulation of an odd dimensional sphere is homotopy equivalent to a connected sum of products of two spheres, resolving a problem of Buchstaber and Panov. The methods are…

We show that two orientable, four-dimensional folded symplectic toric manifolds are isomorphic provided that their orbit spaces have trivial degree-two integral cohomology and there exists a diffeomorphism of the orbit spaces (as manifolds…

We define a certain merging operation that given two $d$-polytopes $P$ and $Q$ such that $P$ has a simplex facet $F$ and $Q$ has a simple vertex $v$ produces a new $d$-polytope $P\hspace{0.1em}\triangleright Q$ with $f_0(P)+f_0(Q)-(d+1)$…

We introduce a notion of moment map adapted to actions of Lie groups that preserve a closed three-form. We show existence of our multi-moment maps in many circumstances, including mild topological assumptions on the underlying manifold.…