arXiv subjects
Publications and source records attributed to Victor M. Buchstaber.
We prove that the integral cohomology algebra of the moment-angle complex Z_K, or of the corresponding coordinate subspace arrangement complement U(K), is isomorphic to the Tor-algebra of the face ring Z[K] of simplicial complex K.
The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of simplicial and cubical subdivisions of manifolds and, especially, spheres. We describe important constructions which allow to study all these combinatorial objects by means of methods of commutative and homological algebra. The proposed approach to combinatorial problems relies on the theory of moment-angle complexes, currently being developed by the authors. The theory centres around the construction that assigns to each simplicial complex $K$ with $m$ vertices a $T^m$-space $\zk$ with a special bigraded cellular decomposition. In the framework of this theory, the well-known non-singular toric varieties arise as orbit spaces of maximally free actions of subtori on moment-angle complexes corresponding to simplicial spheres. We express different invariants of simplicial complexes and related combinatorial-geometrical objects in terms of the bigraded cohomology rings of the corresponding moment-angle complexes. Finally, we show that the new relationships between combinatorics, geometry and topology result in solutions to some well-known topological problems.
We extend work of Davis and Januszkiewicz by considering {\it omnioriented} toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples $B_{i,j}$, which are toric manifolds over products of simplices, and verify that their natural stably complex structure is induced by an omniorientation. Studying connected sums of products of the $B_{i,j}$ allows us to deduce that every complex cobordism class of dimension >2 contains a toric manifold, necessarily connected, and so provides a positive answer to the toric analogue of Hirzebruch's famous question for algebraic varieties. In previous work, we dealt only with disjoint unions, and ignored the relationship between the stably complex structure and the action of the torus. In passing, we introduce a notion of connected sum $#$ for simple $n$-dimensional polytopes; when $P^n$ is a product of simplices, we describe $P^n# Q^n$ by applying an appropriate sequence of {\it pruning operators}, or hyperplane cuts, to $Q^n$.
Let $ρ:(D^2)^m\to I^m$ 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 $T^m$-invariant bigraded cellular decomposition of the subset $ρ^{-1}(C)\subset(D^2)^m$ with cells corresponding to the faces of $C$. Different combinatorial problems concerning cubical complexes and related combinatorial objects can be treated by studying the equivariant topology of corresponding moment-angle complexes. Here we consider moment-angle complexes defined by canonical cubical subdivisions of simplicial complexes. We describe relations between the combinatorics of simplicial complexes and the bigraded cohomology of corresponding moment-angle complexes. In the case when the simplicial complex is a simplicial manifold the corresponding moment-angle complex has an orbit consisting of singular points. The complement of an invariant neighbourhood of this orbit is a manifold with boundary. The relative Poincare duality for this manifold implies the generalized Dehn-Sommerville equations for the number of faces of simplicial manifolds.
We show that the cohomology algebra of the complement of a coordinate subspace arrangement in m-dimensional complex space is isomorphic to the cohomology algebra of Stanley-Reisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) Then we calculate the latter cohomology algebra by means of the standard Koszul resolution of polynomial ring. To prove these facts we construct an equivariant with respect to the torus action homotopy equivalence between the complement of a coordinate subspace arrangement and the moment-angle complex defined by the simplicial complex. The moment-angle complex is a certain subset of a unit poly-disk in m-dimensional complex space invariant with respect to the action of an m-dimensional torus. This complex is a smooth manifold provided that the simplicial complex is a simplicial sphere, but otherwise has more complicated structure. Then we investigate the equivariant topology of the moment-angle complex and apply the Eilenberg-Moore spectral sequence. We also relate our results with well known facts in the theory of toric varieties and symplectic geometry.
An n-dimensional polytope P^n is called simple if exactly n codimension-one faces meet at each vertex. The lattice of faces of a simple polytope P^n with m codimension-one faces defines an arrangement of even-dimensional planes in R^{2m}. We construct a free action of the group R^{m-n} on the complement of this arrangement. The corresponding quotient is a smooth manifold Z_P invested with a canonical action of the compact torus T^m with the orbit space P^n. For each smooth projective toric variety M^{2n} defined by a simple polytope P^n with the given lattice of faces there exists a subgroup T^{m-n}\subset T^m acting freely on Z_P such that Z_P/T^{m-n}=M^{2n}. We calculate the cohomology ring of Z_P and show that it is isomorphic to the cohomology ring of the face ring of P^n regarded as a module over the polynomial ring. In this way the cohomology of Z_P acquires a bigraded algebra structure, and the additional grading allows to catch the combinatorial invariants of the polytope. At the same time this gives an example of explicit calculation of the cohomology of the complement of an arrangement of planes, which is of independent interest.
We investigate geometrical interpretations of various structure maps associated with the Landweber-Novikov algebra S^* and its integral dual S_*. In particular, we study the coproduct and antipode in S_*, together with the left and right actions of S^* on S_* which underly the construction of the quantum (or Drinfeld) double D(S^*). We set our realizations in the context of double complex cobordism, utilizing certain manifolds of bounded flags which generalize complex projective space and may be canonically expressed as toric varieties. We discuss their cell structure by analogy with the classical Schubert decomposition, and detail the implications for Poincare duality with respect to double cobordism theory; these lead directly to our main results for the Landweber-Novikov algebra.