33 research outputs found

### 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

### 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

### 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

### 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