For several important classes of manifolds acted on by the torus, the information about the action can be encoded combinatorially by a regular n-valent graph with vector labels on its edges, which we refer to as the torus graph. By analogy with the GKM-graphs, we introduce the notion of equivariant cohomology of a torus graph, and show that it is isomorphic to the face ring of the associated simplicial poset. This extends a series of previous results on the equivariant cohomology of torus manifolds. As a primary combinatorial application, we show that a simplicial poset is Cohen-Macaulay if its face ring is Cohen-Macaulay. This completes the algebraic characterisation of Cohen-Macaulay posets initiated by Stanley. We also study blow-ups of torus graphs and manifolds from both the algebraic and the topological points of view.
As is well known, h-vectors of simple (or simplicial) convex polytopes are characterized. In fact, those h-vectors must satisfy Dehn-Sommerville equations and some other inequalities. Simple convex polytopes determine Gorenstein* simplicial posets and h-vectors are defined for simplicial posets. It is known that h-vectors of Gorenstein* simplicial posets must satisfy Dehn-Sommerville equations and that every component in the h-vectors must be non-negative. In this paper we will show that h-vectors of Gorenstein* simplicial posets must satisfy one more subtle condition conjectured by R. Stanley…
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 iso…
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…
We introduce the notion of nonevasive reduction, and show that for any monotone poset map $ϕ:P\to P$, the simplicial complex $Δ(P)$ {\tt NE}-reduces to $Δ(Q)$, for any $Q\supseteq{\text{\rm Fix}}ϕ$. As a corollary, we prove that for any order-preserving map $ϕ:P\to P$ satisfying $ϕ(x)\geq x$, for any $x\in P$, the simplicial complex $Δ(P)$ collapses to $Δ(ϕ(P))$. We also obtain a generalization of Crapo's closure theorem.
We show that hypernetworks can be regarded as posets which, in their turn, have a natural interpretation as simplicial complexes and, as such, are endowed with an intrinsic notion of curvature, namely the Forman Ricci curvature, that strongly correlates with the Euler characteristic of the simplicial complex. This approach, inspired by the work of E. Bloch, allows us to canonically associate a simplicial complex structure to a hypernetwork, directed or undirected. In particular, this greatly simplifying the geometric Persistent Homology method we previously proposed.
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).
This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples are an Euler formula for a class of `continuous convex polytopes' (conjectured by Kalai and Wigderson), a duality result for a class of `continuous matroids', a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of Gian-Carlo Rota), an exposition of t…
We review a class of problems on the borders of topology of torus actions, commutative homological algebra and combinatorial geometry, which is currently being investigated by Victor Buchstaber and the author. The text builds on the lectures delivered on the transformation group courses in Osaka City University and Universitat Autonoma de Barcelona. We start with discussing several well-known results and problems on combinatorial geometry of polytopes and simplicial complexes, and then move gradually towards investigating the combinatorial structures associated with spaces acted on by the toru…
In this paper we study general torus actions on manifolds with isolated fixed points from combinatorial point of view. The main object of study is the poset of face submanifolds of such actions. We introduce the notion of a locally geometric poset -- the graded poset locally modelled by geometric lattices, and prove that for any torus action, the poset of its faces is locally geometric. Next we discuss the relations between posets of faces and GKM-theory. In particular, we define the face poset of an abstract GKM-graph and show how to reconstruct the face poset of a manifold from its GKM-graph…
Let $X$ be 2n-dimensional compact manifold with a locally standard action of a compact torus. The orbit space $X/T$ is a manifold with corners. Suppose that all proper faces of $X/T$ are acyclic. In the paper we study the homological spectral sequence $E^*_{*,*}\Rightarrow H_*(X)$ corresponding to the filtration of $X$ by orbit types. When the free part of the action is not twisted, we describe the whole spectral sequence in terms of homology and combinatorial structure of $X/T$. In this case we describe the kernel and the cokernel of the natural map $k[X/T]/(l.s.o.p.) \to H_*(X)$, where $k[X/…
We present a method to compute integral cohomology of posets. This toolbox is applicable as soon as the sub-posets under each object possess certain structure. This is the case for simplicial complexes and simplex-like posets. The method is based on homological algebra arguments in the category of functors and on a spectral sequence built upon the poset. We show its relation to discrete Morse theory. As application we give an alternative proof of Webb's conjecture for saturated fusion systems and we compute the cohomology of Coxeter complexes for finite and infinite Coxeter groups.
We consider a sheaf of exterior algebras on a simplicial poset $S$ and introduce a notion of homological characteristic function. Two natural objects are associated with these data: a graded sheaf $\mathcal{I}$ and a graded cosheaf $\widehatΠ$. When $S$ is a homology manifold, we prove the isomorphism $H^{n-1-p}(S;\mathcal{I})\cong H_{p}(S;\widehatΠ)$ which can be considered as an extension of the Poincare duality. In general, there is a spectral sequence $E^2_{p,q}\cong H^{n-1-p}(S;\mathcal{U}_{n-1+q}\otimes \mathcal{I})\Rightarrow H_{p+q}(S;\widehatΠ)$, where $\mathcal{U}_*$ is the local hom…
A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orientation data. It may be considered as a far-reaching generalisation of toric manifolds from algebraic geometry. The orbit space of a torus manifold has a rich combinatorial structure, e.g., it is a manifold with corners provided that the action is locally standard. Here we investigate relationships between the cohomological properties of torus manifolds and the combinatorics of their orbit quotients. We show that the cohomology ring of a torus manifold is…
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 inter…
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 th…
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 …
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.
We introduce new methods for understanding the topology of $\Hom$ complexes (spaces of homomorphisms between two graphs), mostly in the context of group actions on graphs and posets. We view $\Hom(T,-)$ and $\Hom(-,G)$ as functors from graphs to posets, and introduce a functor $(-)^1$ from posets to graphs obtained by taking atoms as vertices. Our main structural results establish useful interpretations of the equivariant homotopy type of $\Hom$ complexes in terms of spaces of equivariant poset maps and $Γ$-twisted products of spaces. When $P = F(X)$ is the face poset of a simplicial complex $…
A foundation is laid for a theory of combinatorial groupoids, allowing us to use concepts like ``holonomy'', ``parallel transport'', ``bundles'', ``combinatorial curvature'' etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes and other combinatorial objects. A new, holonomy-type invariant for cubical complexes is introduced, leading to a combinatorial ``Theorema Egregium'' for cubical complexes non-embeddable into cubical lattices. Parallel transport of Hom-complexes and maps is used as a tool for extending Babson-Kozlov-Lovasz graph coloring results to more gen…
The {\it torus manifolds} have been defined and studied by M. Masuda and T. Panov (arXiv:math.AT/0306100) who in particular describe its cohomology ring structure. In this note we shall describe the topological $K$-ring of a class of torus manifolds (those for which the orbit space under the action of the compact torus is a {\it homology polytope} whose {\it nerve} is a {shellable} simplicial complex) in terms of generators and relations. Since these torus manifolds include the class of quasi-toric manifolds this is a generalisation of earlier results due to the author and P. Sankaran (arXiv: …
Let the compact torus $T^{n-1}$ act on a smooth compact manifold $X^{2n}$ effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space $X^{2n}/T^{n-1}$ if the action is cohomologically equivariantly formal (which essentially means that $H^{odd}(X^{2n};\mathbb{Z})=0$). It happens that homology of the orbit space can be arbitrary in degrees $3$ and higher. For any finite simplicial complex $L$ we construct an equivariantly formal manifold $X^{2n}$ such that $X^{2n}/T^{n-1}$ is homotopy equivalent to $Σ^3L$. The constructed manifold $X^{2n}$ i…
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…
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 prove that any mapping torus of a closed 3-manifold has zero simplicial volume. When the fiber is a prime 3-manifold, classification results can be applied to show vanishing of the simplicial volume, however the case of reducible fibers is by far more subtle. We thus analyse the possible self-homeomorphisms of reducible 3-manifolds, and use this analysis to produce an explicit representative of the fundamental class of the corresponding mapping tori. To this end, we introduce a new technique for understanding self-homeomorphisms of connected sums in arbitrary dimensions on the level of clas…
We study torsion in the integral cohomology of a certain family of $2n$-dimensional orbifolds $X$ with actions of the $n$-dimensional compact torus. Compact simplicial toric varieties are in our family. For a prime number $p$, we find a necessary condition for the integral cohomology of $X$ to have no $p$-torsion. Then we prove that the necessary condition is sufficient in some cases. We also give an example of $X$ which shows that the necessary condition is not sufficient in general.
