In this paper, we study the well-know $g$-conjecture for rational homology spheres in a topological way. To do this, we construct a class of topological spaces with torus actions, which can be viewed as topological generalizations of toric varieties. Along this way we prove that after doing stellar subdivision operations at some middle dimensional faces of an arbitrary rational homology sphere, the $g$-conjecture is valid. Furthermore, we give topological proofs of several fundamental algebraic results about Buchsbaum complexes and simplicial manifolds. In this process, we also get a few interesting results in toric topology.
The fundamental Davis-Januszkiewicz construction of toric manifolds is reinterpreted in order to allow for generalization. Applications involve the simplicial wedge $J$-construction and Ayzenberg's recent identities arising from composed simplicial complexes.
Let $\mathbf{k}$ be the field $\mathbb{F}_p$ or the ring $\mathbb{Z}$. We study combinatorial and topological properties of the universal simplicial complexes $X(\mathbf{k}^n)$ and $K(\mathbf{k}^n)$ whose simplices are certain unimodular subsets of $\mathbf{k}^n$. As a main result we show that $X(\mathbf{k}^n)$, $K(\mathbf{k}^n)$ and the links of their simplicies are homotopy equivalent to a wedge of spheres specifying the exact number of spheres in the corresponding wedge decompositions. This is a generalisation of Davis and Januszkiewicz's result that $K(\mathbb{Z}^n)$ and $K(\mathbb{F}_2^n)…
A toric arrangement is a finite collection of codimension-$1$ subtori in a torus. These subtori stratify the ambient torus into faces of various dimensions. Let $f_i$ denote the number of $i$-dimensional faces; these so-called face numbers satisfy the Euler relation $\sum_i (-1)^i f_i = 0$. However not all tuples of natural numbers satisfying this relation arise as face numbers of some toric arrangement. In this paper we focus on toric arrangements in a $2$-dimensional torus and obtain a characterization of face numbers. In particular we show that the convex hull of these face numbers is a con…
We characterise integral Poincaré duality moment-angle complexes $\mathcal{Z}_{\mathcal{K}}$ in combinatorial terms of the Fan-Wang duality of the simplicial complex $\mathcal{K}$, and consequently in algebraic terms of the Gorenstein duality of the Stanley-Reisner ring $\mathbb{Z}[\mathcal{K}]$. We extend Poincaré duality results to certain polyhedral products using polyhedral join products of simplicial complexes.
We find families of graphs $G$ and subgraphs $H$ of $G$ such that the number of edge colorings of $G$ avoiding a monochromatic coloring of $H$ is determined by lattice point counts or a Hodge structure on the cohomology of a certain toric variety. In general, this gives a class of ``anti-Ramsey theory problems'' with a geometric structure. For example, we find one for Ramsey numbers of classes of such graphs. The key observation is that our previous result expressing simplicial chromatic polynomials in terms of $h$-vectors of auxiliary simplicial complexes can be reinterpreted as one on edge c…
A fundamental result of toric geometry is that there is a bijection between toric varieties and fans. More generally, it is known that some class of manifolds having well-behaved torus actions, called topological toric manifolds $M^{2n}$, can be classified in terms of combinatorial data containing simplicial complexes with $m$ vertices. We remark that topological toric manifolds are a generalization of smooth toric varieties. The number $m-n$ is known as the Picard number when $M^{2n}$ is a {compact smooth} toric variety. In this paper, we investigate the relationship between the topological t…
The simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and more generally, polyhedral products. In this paper we extend the analysis to include toric orbifolds. Our main results yield infinite families of toric orbifolds, derived from a given one, whose integral cohomology is free of torsion and is concentrated in even degrees, a property which might be termed \emph{integrally equivariantly formal}. In all cases, it …
In this paper we study the Buchstaber invariant of simplicial complexes, which comes from toric topology. With each simplicial complex $K$ on $m$ vertices we can associate a moment-angle complex $\mathcal Z_K$ with a canonical action of the compact torus $T^m$. Then $s(K)$ is the maximal dimension of a toric subgroup that acts freely on $\mathcal Z_K$. We develop the Buchstaber invariant theory from the viewpoint of the set of minimal non-simplices of $K$. It is easy to show that $s(K)=1$ if and only if any two and any three minimal non-simplices intersect. For $K=\partial P^*$, where $P$ is a…
Toric orbifolds are a generalization of simplicial projective toric varieties. In this paper, we show that there is a resolution of singularities of a toric orbifold. In a different category, the class of quasi-contact toric manifolds contains the class of good contact toric manifolds. We prove that a quasi-contact toric manifold is equivariantly a boundary. Moreover, we conclude that good contact toric manifolds and generalized lens spaces are equivariantly boundaries.
A small cover is a closed smooth manifold of dimension $n$ having a locally standard $\mathbb{Z}_2^n$-action whose orbit space is isomorphic to a simple polytope. A typical example of small covers is a real projective toric manifold (or, simply, a real toric manifold), that is, a real locus of projective toric manifold. In the paper, we classify small covers and real toric manifolds whose orbit space is isomorphic to the dual of the simplicial complex obtainable by a sequence of wedgings from a polygon, using a systematic combinatorial method finding toric spaces called puzzles.
Let $P_m(J)$ denote a simplicial complex obtainable from consecutive wedge operations from an $m$-gon. In this paper, we completely classify toric manifolds over $P_m(J)$ and prove that all of them are projective. As a consequence, we provide an infinite family of projective toric manifolds.
In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the complete case fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As special cases, we obtain new (or recover well-known) formulae for the Baum-Fulton-MacPherson Todd (or MacPherson-Chern) classes of toric varieties, as well as for the Thom-Milnor L-classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of …
Toric orbifolds are a topological generalization of projective toric varieties associated to simplicial fans. We introduce some sufficient conditions on the combinatorial data associated to a toric orbifold to ensure the existence of an invariant cell structure on it and call such a toric orbifold retractable. In this paper, our main goal is to study equivariant cohomology theories of retractable toric orbifolds. Our results extend the corresponding results on divisive weighted projective spaces.
We prove that the cohomology of the moduli space of morphisms of a fixed finite degree from a smooth projective curve $C$ of genus $g$ to a complete simplicial toric variety $\mathbb{P}(Σ)$, denoted by the rational polyhedral fan $Σ$, stabilizes. As an arithmetic consequence we obtain a resolution of the Batyrev-Manin conjecture for toric varieties over global function fields in all but finitely many characteristics.
In this paper, we consider homology groups induced by the exterior algebra generated by a simplicial compliment of a simplicial complex $K$. These homology groups are isomorphic to the Tor-groups $\mathrm{Tor}_{i, J}^{\mathbf{k}[m]}(\mathbf{k}(K),\mathbf{k})$ of the face ring $\mathbf{k}(K)$, which is very useful and much studied in toric topology. By using $\check{C}ech$ homology theory and Alexander duality theorem, we prove that these homology groups have dualities with the simplicial cohomology groups of the full subcomplexes of $K$. Then we give a new proof of Hochster's theorem.
We develop a framework to construct geometric representations of finite groups $G$ through the correspondence between real toric spaces $X^{\mathbb R}$ and simplicial complexes with characteristic matrices. We give a combinatorial description of the $G$-module structure of the homology of $X^{\mathbb R}$. As applications, we make explicit computations of the Weyl group representations on the homology of real toric varieties associated to the Weyl chambers of type $A$ and $B$, which show an interesting connection to the topology of posets. We also realize a certain kind of Foulkes representatio…
A real toric space is a topological space which admits a well-behaved $\mathbb{Z}_2^k$-action. Real moment-angle complexes and real toric varieties are typical examples of real toric spaces. A real toric space is determined by a pair of a simplicial complex $K$ and a characteristic matrix $Λ$. In this paper, we provide an explicit $R$-cohomology ring formula of a real toric space in terms of $K$ and $Λ$, where $R$ is a commutative ring with unity in which $2$ is a unit. Interestingly, it has a natural $(\mathbb{Z} \oplus \operatorname*{row} Λ)$-grading. As corollaries, we compute the cohomolog…
We investigate small covers and quasitoric over the duals of neighborly simplicial polytopes with small number of vertices in dimensions $4$, $5$, $6$ and $7$. In the most of the considered cases we obtain the complete classification of small covers. The lifting conjecture in all cases is verified to be true. The problem of cohomological rigidity for small covers is also studied and we have found a whole new series of weakly cohomologically rigid simple polytopes. New examples of manifolds provide the first known examples of quasitoric manifolds in higher dimensions whose orbit polytopes have …
A fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. The authors in their previous paper provided a new way to find all characteristic maps on a simplicial complex $K(J)$ obtainable by a sequence of wedgings from $K$. The main idea was that characteristic maps on $K$ theoretically determine all possible characteristic maps on a wedge of $K$. In this work, we further develop our previous work for classification of toric spaces. For a star-sh…
We introduce the notion of a topological toric manifold and a topological fan and show that there is a bijection between omnioriented topological toric manifolds and complete non-singular topological fans. A topological toric manifold is a topological analogue of a toric manifold and the family of topological toric manifolds is much larger than that of toric manifolds. A topological fan is a combinatorial object generalizing the notion of a simplicial fan in toric geometry. Prior to this paper, two topological analogues of a toric manifold have been introduced. One is a quasitoric manifold and…
After surveying higher K-theory of toric varieties, we present Totaro's old (c. 1997) unpublished result on expressing the corresponding homotopy theory via singular cohomology. It is a higher analog of the rational Chern character isomorphism for general toric schemes. In the special case of a projective simplicial toric scheme over a regular ring one obtains a rational isomorphism between the homotopy K-theory and the direct sum of m copies of the K-theory of the ground ring, m being the number of maximal cones in the underlying fan. Apart from its independent interest, in retrospect, Totaro…
The moment-angle complex Z_K is cell complex with a torus action constructed from a finite simplicial complex K. When this construction is applied to a triangulated sphere K or, in particular, to the boundary of a simplicial polytope, the result is a manifold. Moment-angle manifolds and complexes are central objects in toric topology, and currently are gaining much interest in homotopy theory, complex and symplectic geometry. The geometric aspects of the theory of moment-angle complexes are the main theme of this survey. We review constructions of non-Kahler complex-analytic structures on mome…
For a simplicial complex $K$ with $m$ vertices, there is a canonical $\mathbb Z_2^m$-space known as a real moment angle complex $\mathbb R \mathcal Z_K$. In this paper, we consider the quotient spaces $Y=\mathbb R \mathcal Z_K / \mathbb Z_2^{k}$, where $K$ is a pure shellable complex and $\mathbb Z_2^k \subset \mathbb Z_2^m$ is a maximal free action on $\mathbb R \mathcal Z_K$. A typical example of such spaces is a small cover, where a small cover is known as a topological analog of a real toric manifold. We compute the integral cohomology group of $Y$ by using the PL cell decomposition obtain…
The CW structure of certain spaces, such as effective orbifolds, can be too complicated for computational purposes. In this paper we use the concept of $\mathbf{q}$-CW complex structure on an orbifold, to detect torsion in its integral cohomology. The main result can be applied to well known classes of orbifolds or algebraic varieties having orbifold singularities, such as toric orbifolds, simplicial toric varieties, torus orbifolds and weighted Grassmannians.
