Topological model for h"-vectors of simplicial manifolds

Any manifold with boundary gives rise to a Poincare duality algebra in a natural way. Given a simplicial poset $S$ whose geometric realization is a closed orientable homology manifold, and a characteristic function, we construct a manifold with boundary such that graded components of its Poincare duality algebra have dimensions $h_k"(S)$. This gives a clear topological evidence for two well-known facts about simplicial manifolds: the nonnegativity of $h"$-numbers (Novik--Swartz theorem) and the symmetry $h"_k=h"_{n-k}$ (generalized Dehn--Sommerville relations).Comment: 8 page

Locally standard torus actions and h'-vectors of simplicial posets

We consider the orbit type filtration on a manifold $X$ with locally standard action of a compact torus and the corresponding homological spectral sequence $(E_X)^r_{*,*}$. If all proper faces of the orbit space $Q=X/T$ are acyclic, and the free part of the action is trivial, this spectral sequence can be described in full. The ranks of diagonal terms are equal to the $h'$-numbers of the Buchsbaum simplicial poset $S_Q$ dual to $Q$. Betti numbers of $X$ depend only on the orbit space $Q$ but not on the characteristic function. If $X$ is a slightly different object, namely the model space $X=(P\times T^n)/\sim$ where $P$ is a cone over Buchsbaum simplicial poset $S$, we prove that $\dim (E_X)^{\infty}_{p,p} = h''_p(S)$. This gives a topological evidence for the fact that $h''$-numbers of Buchsbaum simplicial posets are nonnegative.Comment: 21 pages, 3 figures + 1 inline figur

Torus action on quaternionic projective plane and related spaces

For an action of a compact torus $T$ on a smooth compact manifold~$X$ with isolated fixed points the number $\frac{1}{2}\dim X-\dim T$ is called the complexity of the action. In this paper we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that $\mathbb{H}P^2/T^3\cong S^5$ and $S^6/T^2\cong S^4$, for the homogeneous spaces $\mathbb{H}P^2=Sp(3)/(Sp(2)\times Sp(1))$ and $S^6=G_2/SU(3)$. Here the maximal tori of the corresponding Lie groups $Sp(3)$ and $G_2$ act on the homogeneous spaces by the left multiplication. Next we consider the quaternionic analogues of smooth toric surfaces: they give a class of 8-dimensional manifolds with the action of $T^3$, generalizing $\mathbb{H}P^2$. We prove that their orbit spaces are homeomorphic to $S^5$ as well. We link this result to Kuiper--Massey theorem and some of its generalizations.Comment: 22 pages, 6 figure

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

Homology cycles in manifolds with locally standard torus actions

Let $X$ be a $2n$-manifold with a locally standard action of a compact torus $T^n$. If the free part of action is trivial and proper faces of the orbit space $Q$ are acyclic, then there are three types of homology classes in $X$: (1) classes of face submanifolds; (2) $k$-dimensional classes of $Q$ swept by actions of subtori of dimensions $<k$; (3) relative $k$-classes of $Q$ modulo $\partial Q$ swept by actions of subtori of dimensions $\geqslant k$. The submodule of $H_*(X)$ spanned by face classes is an ideal in $H_*(X)$ with respect to the intersection product. It is isomorphic to $(\mathbb{Z}[S_Q]/\Theta)/W$, where $\mathbb{Z}[S_Q]$ is the face ring of the Buchsbaum simplicial poset $S_Q$ dual to $Q$; $\Theta$ is the linear system of parameters determined by the characteristic function; and $W$ is a certain submodule, lying in the socle of $\mathbb{Z}[S_Q]/\Theta$. Intersections of homology classes different from face submanifolds are described in terms of intersections on $Q$ and $T^n$.Comment: 25 pages, 3 figures. Minor correction in Lemma 3.3 and a calculations of Subsection 7.

Dimensions of multi-fan algebras

Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data. The procedure relies on the theory of volume polynomials and multi-fans. This construction includes many important examples, such as cohomology of toric varieties and quasitoric manifolds, and Gorenstein algebras of triangulated homology manifolds, introduced by Novik and Swartz. In all these examples the dimensions of graded components of such duality algebras do not depend on the vector coloring. It was conjectured that the same holds for any simplicial cycle. We disprove this conjecture by showing that the colors of singular points of the cycle may affect the dimensions. However, the colors of smooth points are irrelevant. By using bistellar moves we show that the number of different dimension vectors arising on a given 3-dimensional pseudomanifold with isolated singularities is a topological invariant. This invariant is trivial on manifolds, but nontrivial in general.Comment: 18 pages, 5 labeled figures + 2 unlabeled figure