On independent GKM-graphs without nontrivial extensions

GKM-theory [gkm-98], [gu-za-01] provides with a useful method for computation of the equivariant cohomology ring for a wide class of manifolds equipped with a torus action (having only isolated fixed points) called GKM-manifolds. The main tool of this theory is a graph that is naturally asssociated with the orbit space of the equivariant $1$-skeleton for a GKM-manifold. The labels on the edges of this graph are given by the tangent weights of the fixed points associated with the torus action. By abstracting from the torus action one obtains the definition of a GKM-graph axiomatized in [gu-za-01]. Apart from the equivariant cohomology ring many other important geometric objects and topological invariants related to a GKM-manifold can be studied in terms of the respective GKM-graph, such as Betti numbers [gu-za-01], invariant almost complex structures [gu-ho-za-06], invariant symplectic and Kähler structures [go-ko-zo-20], etc.

A GKM-manifold $M^{2n}$ is called a GKM-manifold in $j$-general position if any $j$ tangent weights at any fixed point of this action on $M^{2n}$ are linearly independent. A $j$-independent GKM-graph is defined similarly. Properties of a GKM-action of a $T^{k}$-action on $M^{2n}$ in $j$-general position were studied in several papers (in a wider context of equivariantly formal smooth actions with isolated fixed points). The number $n-k$ is called the complexity of the $T^{k}$-action on $M^{2n}$. In [ma-pa-06] it was shown that the equivariant cohomology ring of a torus manifold with vanishing odd cohomology groups (that is, a $2n$-dimensional GKM-manifold of complexity $0$ which is automatically $n$-independent) is isomorphic to the Stanley-Reisner ring of the face poset of the GKM-graph. Also see [ma-ma-pa-07] for treatment of equivariant topology for torus manifolds from GKM-perspective. Furthermore, any GKM-manifold $M^{2n}$ of complexity $1$ with a GKM-action in general position (that is, in $(n-1)$-general position) has a description [ay-ma-so-22] of the equivariant cohomology ring in terms of face rings similar to [ma-pa-06]. The poset of faces $S_{M}$ in the orbit space $M/T$ was associated to a GKM-manifold $M$ with the $T$-action in [ay-ma-so-22]. We remark that in the case of complexity $0$ this object was studied in earlier papers [ma-pa-06] (for a locally standard action, so that $M/T$ is a manifold with corners) and in [ay-ma-19] (for a general case, so that $M/T$ is a homological cell complex). It was shown in [ay-ma-so-22] that for a GKM-manifold in $j$-general position the subposets $(S_{M})_{<s}$ and $(S_{M})_{r}$ are $\min\{\dim s-1,j+1\}$- and $\min\{r-1,j+1\}$-acyclic, respectively, for any $r>0$ and $s\in S_{M}$. This implies by [ma-pa-06, ay-ma-19] that the corresponding orbit space $M/T$ of $M$ is $(j+1)$-acyclic. This result was proved in [ma-pa-06] for $j=n$ ($n$-independent case) and in [ay-ma-19] for the case of arbitrary $j$. On the other hand, by dropping the condition of general position, an arbitrary $T^{n-1}$-action on a smooth manifold $M^{2n}$ (of complexity $1$) with isolated fixed points shows much more complicated behaviour of the orbit space homology [ay-ch-19]. Therefore, GKM-actions in $j$-general position form a particularly nice class of GKM-actions exhibiting many useful properties (depending on the value of $j$).

A natural question (called an extension problem in [ku-19]) is how to determine whether a given GKM-action extends to an effective GKM-action of a torus of greater dimension (so that the complexity of the new action is lower) on the same manifold, or not. This question leads to a search of a $k$-independent $(n,k)$-type GKM-action that is nonextendible for arbitrary $n\geq k\geq 2$. Homogeneous GKM-manifolds [gu-ho-za-06] supply with many interesting examples of GKM-graphs, including those of positive complexity. Recall that if the Euler characteristic of a homogeneous complex manifold $G/H$ is nonzero then it is a GKM-manifold by [gu-ho-za-06], where $G$ is a compact connected semisimple Lie group, $H$ is an arbitrary parabolic subgroup in $G$ and $T$ is any maximal torus of $H\subset G$. Two phenomena occur for homogeneous GKM-manifolds related to the search problem mentioned above. Firstly, S. Kuroki proved the following gap property for $G$ of type $A_{n}$ with the natural GKM $T$-action on $G/H$ as above (unpublished): if the respective GKM-graph is $j$-independent for some $j\geq 4$ then it is $n$-independent, where $n$ is the rank of $G$. This leads to a conjecture that any homogeneous GKM-manifold satisfies such a gap property. Secondly, the natural $T^{n-1}$-action on the Grassmanian $Gr_{k}(\mathbb{C}^{n})$ of $k$-planes in $\mathbb{C}^{n}$ is a GKM-action admitting no nonisomorphic extensions for any $k$. This can be shown by modifying the proof for $k=2$ from [ku-19] in a simple way. This result leads to another conjecture that most of the homogeneous GKM-manifolds are nonextendible with possibly a few exceptions determined by the condition that the respective automorphism group identity component $\operatorname{Aut}^{0}(G/H)$ is greater than $G$. (The description of $\operatorname{Aut}^{0}(G/H)$ is given in [ak-95], for instance.)

Therefore the extension problem seems to be a bit challenging. Instead, one can study a rather simplistic extension problem for arbitrary GKM-graphs by replacing a GKM-action of $T^{k}$ on $M^{2n}$ with an $(n,k)$-type $k$-independent GKM-graph $\Gamma$ (not necessarily coming from a GKM-action). An occurring new realization problem here is to determine whether a given GKM-graph $\Gamma$ is a GKM-graph of some GKM-manifold. We remark that some results on realization problem were obtained in dimensions $n=1,2$ in [ca-ga-ka] for GKM-graphs (in terms of conditions given by the ABBV localization formula), and for GKM-orbifolds in dimension $n=2$ in [da-ku-so-18, Remark 4.13].

In this paper, we give new examples of non-realizable and non-extendible actions of different complexity. The precise statement is given in the following main theorem of this paper (see Theorem LABEL:thm:example_).

The necessary tools for the proof of Theorem 1 are twofold. Firstly, in Section 2 we give a sufficient criterion (Corollary 2.12) of non-extendibility of any GKM-graph satisfying some condition formulated in terms of chords for a face in the GKM-graph. Secondly, in Section 3 for any $(j+1)$-independent GKM-manifold $M$ with the corresponding GKM-graph $\Gamma$, we compare (Proposition 3.9) some subposets in the face poset $S_{\Gamma}$ of a GKM-graph $\Gamma$ with the subposets in the face poset $S_{M}$ of $M$. The obtained comparison implies (by [ay-ma-so-22]) partial acyclicity for some subposets in $S_{\Gamma}$. In this case $(S_{\Gamma})_{\leq\Xi}^{op}$ turns out to be a simplicial poset for any face $\Xi$ of dimension not exceeding $j$ in $\Gamma$ (Proposition 3.14). The respective Euler characteristic of the order complex for the poset $(S_{\Gamma})_{<\Xi}^{op}$ is computed in terms of the corresponding $f$-numbers (Proposition 3.13).

In Section 4 we explicitly construct an example which is suitable for the proof of Theorem 1. Here is a brief outline of the construction. Firstly we introduce an infinite $(d+1)$-regular graph embedded into $\mathbb{R}^{d}$. We endow it with an axial function in order to obtain a torus graph $\Gamma(d,0)$. We add to it edges and define the axial function agreeing to that of $\Gamma(d,0)$ causing the increase of the complexity for a GKM-graph. The resulting $(d+1+r,d+1)$-type GKM graph $\Gamma(d,r)$ is $(d+1)$-independent. We remark that the definition of the axial function on $\Gamma(d,r)$ is obtained by applying the decision method of Tarski [ta-51] to some Vandermonde matrices (Lemma 4.9) and therefore it is implicit. The infinite GKM graph $\Gamma(d,r)$ is periodic (invariant) with respect to the subgroup $2^{r+1}\cdot\mathbb{Z}^{d}\subset\mathbb{Z}^{d}$ in the group of parallel translations in $\mathbb{R}^{d}$. The quotient $\Gamma_{d+1}^{d+1+r}:=\Gamma^{2^{r+1}}(d,r)$ of the GKM-graph $\Gamma(d,r)$ by the group $2^{r+1}\cdot\mathbb{Z}^{d}$ is shown to be a well-defined GKM-graph without multiple edges and loops.

In the concluding Section LABEL:sec:proof of this paper we prove that $\Gamma_{d+1}^{d+1+r}$ satisfies all conditions of Theorem 1 (where we put $n=d+1+r$, $k=n+1$). We apply the results on acyclicity of face subposets in GKM-manifolds (in the case of complexity $0$) from [ma-pa-06] in order to prove non-realizability of the constructed GKM-graph $\Gamma_{d+1}^{d+1+r}$ by studying the respective Euler characteristic (by the comparison results mentioned above). This argument relies on the explicit computation of face numbers in the torus graph $\Gamma^{a}(d,0)$ (Lemma LABEL:lm:eucomp). The nonextendibility is proved by using the method of chords mentioned above.

In this section we recall some definitions from GKM-theory (we follow the notation from [gu-za-01] and [ku-09]). We introduce an obstruction to have extensions for a given GKM-graph in terms of chords.

In this section we fix a GKM-graph $\Gamma$ with the corresponding connected $n$-valent graph $(V,E)=(V_{\Gamma},E_{\Gamma})$, axial function $\alpha\colon E\to\mathbb{Z}^{k}$ on $\Gamma$ and a connection $\nabla$ on it.

The next corollary is a principal tool for the proof of Theorem 1.

In this section we continue to recall some basic notions of GKM-theory and of the related [ma-ma-pa-07, ay-ma-so-22] posets $S_{M}$, $S_{\Gamma}$ of faces arising from the orbit space $M/T$ and from the GKM-graph $\Gamma$ of a given GKM-manifold $M$ with the $T$-action, respectively. We compare some specific simplicial subposets in $S_{M}$, $S_{\Gamma}$ under assumption of $j$-general position for $M$. After that we recall the P.Hall formula for the Euler characteristic of an order complex for a finite simplicial poset which is used later in the text.

Due to [ma-pa-06, Lemma 2.1] one can give the following definition of a GKM-manifold that is equivalent to the standard one (e.g. see [gu-za-01]).

Consider a GKM-action of $T=T^{k}$ on $M=M^{2n}$.

Recall that a topological space $X$ is called $j$-acyclic if $\tilde{H}^{i}(X)=0$ holds for any $i\leq j$, and acyclic, if $\tilde{H}^{*}(X)=0$ holds. Let $S_{M}$ be the poset of faces for a GKM-manifold $M$. Let $S_{\Gamma}$ be the poset of faces of nonnegative dimension (ordered by inclusion) of a GKM-graph $\Gamma$. We need the following particular case of a theorem from [ay-ma-so-22].

The claim of the following proposition is reminiscent to [ay-ma-so-22, Lemma 3.8] (although not quite the same).

Let $P$ be a finite poset [st-86]. Recall the following definitions.

In the following we need the following well-known Philip Hall’s theorem.

The computation of Euler characteristic for certain face subposets in $S_{\Gamma}$ for a $j$-complete GKM-graph $\Gamma$ is possible (by using Proposition 3.13) due to the following proposition.

In this section we give a detailed construction of the GKM-graph suitable for the proof of Theorem 1 and study some of its properties.

Notice that the graph $\Gamma$ is a $(d+1+r)$-valent connected graph with infinite set of vertices, and that $V_{\Gamma}=V_{\Gamma^{\prime}}$ holds (see Figure LABEL:fig:cross, LABEL:fig:edges).