241 research outputs found
Toric cohomological rigidity of simple convex polytopes
A simple convex polytope is \emph{cohomologically rigid} if its
combinatorial structure is determined by the cohomology ring of a quasitoric
manifold over . Not every has this property, but some important
polytopes such as simplices or cubes are known to be cohomologically rigid. In
this article we investigate the cohomological rigidity of polytopes and
establish it for several new classes of polytopes including products of
simplices. Cohomological rigidity of is related to the \emph{bigraded Betti
numbers} of its \emph{Stanley--Reisner ring}, another important invariants
coming from combinatorial commutative algebra.Comment: 18 pages, 1 figure, 2 tables; revised versio
Buchstaber numbers and classical invariants of simplicial complexes
Buchstaber invariant is a numerical characteristic of a simplicial complex,
arising from torus actions on moment-angle complexes. In the paper we study the
relation between Buchstaber invariants and classical invariants of simplicial
complexes such as bigraded Betti numbers and chromatic invariants. The
following two statements are proved. (1) There exists a simplicial complex U
with different real and ordinary Buchstaber invariants. (2) There exist two
simplicial complexes with equal bigraded Betti numbers and chromatic numbers,
but different Buchstaber invariants. To prove the first theorem we define
Buchstaber number as a generalized chromatic invariant. This approach allows to
guess the required example. The task then reduces to a finite enumeration of
possibilities which was done using GAP computational system. To prove the
second statement we use properties of Taylor resolutions of face rings.Comment: 19 pages, 2 figure
Moment-angle complexes from simplicial posets
We extend the construction of moment-angle complexes to simplicial posets by
associating a certain T^m-space Z_S to an arbitrary simplicial poset S on m
vertices. Face rings Z[S] of simplicial posets generalise those of simplicial
complexes, and give rise to new classes of Gorenstein and Cohen--Macaulay
rings. Our primary motivation is to study the face rings Z[S] by topological
methods. The space Z_S has many important topological properties of the
original moment-angle complex Z_K associated to a simplicial complex K. In
particular, we prove that the integral cohomology algebra of Z_S is isomorphic
to the Tor-algebra of the face ring Z[S]. This leads directly to a
generalisation of Hochster's theorem, expressing the algebraic Betti numbers of
the ring Z[S] in terms of the homology of full subposets in S. Finally, we
estimate the total amount of homology of Z_S from below by proving the toral
rank conjecture for the moment-angle complexes Z_S.Comment: 17 pages, 2 pictures; v2 revised, v3 minor correction
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