From convex polytopes to toric varieties: the constructions of Davis and Januszkiewicz One of the most useful tools in the study of convex polytopes is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on the combinatorics of the polytopes. This construction requires that the polytope is rational which is a real restriction when the polytope is general (neither simple nor simplicial). Often we would like to consider general polytopes and even polyhedral spheres (and more general objects) where the toric variety construction does not work.
I am aware of very general constructions by M. Davis, and T. Januszkiewicz,
(one relevant paper might be Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991) and several subsequent papers). Perhaps these constructions allow you to start with arbitrary polyhedral spheres and perhaps even in greater generality.
I ask about an explanation of the scope of these constructions, and, in simple terms as possible, how does the construction go?
 A: The DJ construction works with a simplicial complex $K$ and a subtorus $W\leq\prod_{v\in V}S^1$ (where $V$ is the set of vertices of $K$).  People tend to be interested in the case where $|K|$ is homeomorphic to a sphere, but that isn't really central to the theory.  However, it is important that we have a simplicial complex rather than something with more general polyhedral structure.  It is also important that we have a subtorus, which gives a sublattice $\pi_1(W)\leq\prod_{v\in V}\mathbb{Z}$, which is integral/rational information.  I don't think that the DJ approach will help you get away from the rational case.
I like to formulate the construction this way.  Suppose we have a set $X$ and a subset $Y$.  Given a point $x\in\prod_{v\in V}X$, we put $\text{supp}(x)=\{v:x_v\not\in Y\}$ and $K.(X,Y)=\{x:\text{supp}(x) \text{ is a simplex}\}$.  The space $K.(D^2,S^1)$ is a kind of moment-angle complex, and $K.(D^2,S^1)/W$ is the space considered by Davis and Januskiewicz; it has an action of the torus $T=\left(\prod_{v\in V}S^1\right)/W$.  Generally we assume that $W$ acts freely on $K.(D^2,S^1)$.  There is a fairly obvious complexification map $K.(D^2,S^1)/W\to K.(\mathbb{C},\mathbb{C}^\times)/W_{\mathbb{C}}$.  Under certain conditions relating the position of $W$ to the simplices of $K$, one can check that $K$ gives rise to a fan, that the complexification map is a homeomorphism, and that both $K.(D^2,S^1)/W$ and $K.(\mathbb{C},\mathbb{C}^\times)/W_{\mathbb{C}}$ can be identified with the toric variety associated to that fan.
A: Dear Gil, in addition to Dan's answer let me mention that the construction of toric varieties à la Cox has been generalized to arbitrary convex polytopes in Geometric spaces from arbitrary convex polytope, Int. J. Math., 23, (2012) (the simple case had been treated in a previous paper joint with E. Prato).
A: Dear Gil, we have a nonrational construction with F. Battaglia in the case of simplicial fans here:     arXiv:1108.1637, where we use foliated compact manifolds instead of toric varieties. In this setting, Stanley's proof of (the necessary part of) the g-conjecture carries over.
A: The construction of Davis and Januskiewicz can be realized as an equivariant colimit. 
Let $P$ be a simple polytope of dimension $n$ and let $G$ be either the mod 2 torus ${\mathbb{Z}}_2^n$ or the usual torus ${\mathbb{T}}^n$. A characteristic function on $P$ corresponds to a order-reserving map $\chi:{\mathrm{Face}} \, P\to {\mathrm{Sub}}_{\mathbf{Grp}} G$ from the face poset of $P$ to the poset of subgroups of $G$ such that


*

*The image of $\chi$ lands in the unimodular subgroups of G.

*$\chi$ is graded, in the sense that ${\mathrm{codim}} \,F = {\mathrm{rank}} \, \chi F$. 


There is a functor $-\times G/-:{\mathrm{Face}} P\times {\mathrm{Sub}}_{\mathbf{Grp}} G\to {\mathbf{Top}}G$ from the poset product to the category of $G$-spaces that carries $(F,H)$ to the $G$-space $F\times G/H$. Here $G$ acts on the second factor of the product naturally: 
$$(x,Hg)g':=(x,Hgg')$$ 
Pick out a certain subposet $Q$ of ${\mathrm{Face}} P\times {\mathrm{Sub}}_{\mathbf{Grp}} G$ by requiring that $(F,G/H)$ is in $Q$ if and only if $H$ is a unimodular subgroup of $\chi F$. The real or complex quasitoric manifold $M(\chi)$ over $P$ with characteristic function $\chi$ is the colimit of the composite $$Q\hookrightarrow {\mathrm{Face}} P\times {\mathrm{Sub}}_{\mathbf{Grp}} G\xrightarrow{-\times G/-} {\mathbf{Top}}_G$$
Without reference to morphisms and wrting $H\prec \chi F$ to mean that $H$ is a unimodular subgroup of $\chi F$, we can write $$M(\chi)={\mathrm{colim}}_{H \prec \chi F} F\times G/H$$
Replacing the face poset by a general poset and taking a general topological group $G$ (or Lie group), one can construct $G$-spaces via such an equivariant decomposition. The defect of such a generalization is that it is unclear whether the resulting $G$-space has a manifold, variety or Kahler structure. 
Gil, does a polyhedral sphere have a naturally associated poset over which we could construct $G$-spaces with interesting combinatorial invariants?
A: In the definition of moment-angle manifold in Davis-Januszkiewicz's 1991 paper, the orbit space is a simple convex polytope $P$ which is contractible. We can replace $P$ by an arbitrary smooth nice manifold with corners $Q$ and define the generalized moment-angle manifold $\mathcal{Z}_Q$ in the similar way as usual moment-angle manifold (see https://arxiv.org/abs/2011.10366). Let $F_1,\cdots,F_m$ be all the facets of $Q$ and $\lambda: \{ F_1,\cdots,F_m \} \rightarrow
    \mathbb{Z}^m$ be a map such that $\{\lambda(F_1),\cdots,\lambda(F_m)\}$ is a unimodular basis of $\mathbb{Z}^m\subset \mathbb{R}^m=T_e(S^1)^m$.
\begin{equation}
       \mathcal{Z}_{Q} = Q\times (S^1)^m / \sim
     \end{equation}
where $(x,g) \sim (x',g')$ if and only if $x=x'$ and $g^{-1}g' \in \mathbb{T}^{\lambda}_x$
where $\mathbb{T}^{\lambda}_x$ is the subtorus of $(S^1)^m$ determined by
the linear subspace of $\mathbb{R}^m$ spanned by the set $\{ \lambda(F_j) \, |\, x\in F_j \}$.
The free quotient of $\mathcal{Z}_Q$ under the action of some subtorus in $(S^1)^m$ of rank $m-n$ gives analogues of toric manifolds in this setting (where $n$ is the dimension of $Q$). Such a space can also be defined from a (non-degenerate) characteristic function on the facets of $Q$. The equivariant cohomology ring of $\mathcal{Z}_Q$ with $\mathbf{k}$-coefficients is isomorphic to a ring $\mathbf{k}\langle Q\rangle$ (called the topology face ring of $Q$) that is determined not only by the face poset of $Q$ but also by the $\mathbf{k}$-cohomology rings of all the faces of $Q$. The definition of topology face ring is a direct generalization of the face ring of a simple convex polytope.
Furthermore, we can replace $Q$ by an arbitrary finite CW-complex $X$ with a panel structure $\mathcal{P}$ and define moment-angle complex $(D^2,S^1)^{(X,\mathcal{P})}$ and do the similar calculations for its cohomology and equivariant cohomology (see https://arxiv.org/abs/2103.04281). The definition of panel structure
is due to M. Davis's 1983 paper "Groups generated by reflections and aspherical manifolds not covered by
Euclidean space" in Ann. of Math.
Roughly speaking, a panel structure on $X$ defines "abstract faces" on $X$ which allows us to do the similar construction as $\mathcal{Z}_Q$. But it is more convenient to think of
$(D^2,S^1)^{(X,\mathcal{P})}$ as the colimit of a diagram of CW-complexes of the form
$ f\times \underset{j\in I_f}{\prod} D^2_{(j)} \times
 \underset{j\in [m]\backslash I_f}{\prod} S^1_{(j)}$ where $f$ ranges over all
the "abstract faces" of $(X,\mathcal{P})$ and $I_f$ denotes all indices of the panels that contain $f$. Note that "panels" plays the role of facets here.
In general, $(D^2,S^1)^{(X,\mathcal{P})}$ may not be a manifold. The free quotient of $(D^2,S^1)^{(X,\mathcal{P})}$ under some torus action can also be considered as a far-reaching generalization of manifolds with locally standard torus actions. In addition, the topology face ring of the panel structure $(X,\mathcal{P})$ also makes perfect sense, which is isomorphic to the equivariant cohomology ring of $(D^2,S^1)^{(X,\mathcal{P})}$.
In particular, when $X$ is the cone of the barycentric subdivision of a simplicial complex $K$ (with a canonical panel structure $\mathcal{P}_K$), the topological face ring of $(X,\mathcal{P}_K)$ is nothing but the face ring (Stanley-Reisner ring) of $K$ (see Section 5 of that paper).
