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This article is cited in 2 scientific papers (total in 2 papers) How is a graph not like a manifold? A. A. Ayzenberga, M. Masudaba, G. D. Solomadina a Faculty of Computer Science, National Research University Higher School of Economics, Moscow, Russia b Osaka City University Advanced Mathematical Institute, Osaka, Japan
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 6, DOI: https://doi.org/10.4213/sm9798 
Bibliographic databases:
    § 1. IntroductionToric topology studies actions of a compact torus $T^k$ on closed smooth manifolds $X^{2n}$ in terms of the related combinatorial structures. The classical examples are given by smooth toric varieties, which are classified by their simplicial fans, and their topological analogues — quasitoric manifolds classified by characteristic pairs. In both cases the torus $T^n$ acts on a manifold $X^{2n}$ with $H^{\mathrm{odd}}(X^{2n})=0$, and it turns out that the poset $S(X)$ of $T^n$-invariant submanifolds in $X^{2n}$ is a certain cell subdivision of a topological disc. The poset $S(X)$ has nice acyclicity properties: not only $S(X)$ is acyclic (which is obvious since it has the greatest element), but also its skeleta, the links of its simplices, and other naturally related objects are acyclic too. These acyclicity properties are intimately related to the Cohen-Macaulay property of Stanley-Reisner algebras, which are isomorphic to the equivariant cohomology of $X$. In this paper we study torus actions of arbitrary complexity. For a general action of $T=T^k$ on $X=X^{2n}$ having isolated fixed points, we consider the poset of face submanifolds $S(X)$. In [7] the local structure of $S(X)$ was studied: it was proved that $S(X)_{\geqslant s}$ is a geometric lattice for any element $s\in S(X)$, in particular, such ‘local’ poset is acyclic due to a result of Björner [8]. In this paper we concentrate on the global topological structure of $S(X)$; however, we restrict the class of actions under consideration. We assume that torus actions satisfy the following properties.
With these properties satisfied, we prove the following result. Theorem 1.1. Assume that an action of $T$ on $X$ is equivariantly formal and $j$-independent. Then the following acyclicity conditions hold. This theorem gives a necessary condition for a GKM graph to be a GKM graph of some (equivariantly formal) GKM manifold for $j$ large enough. Indeed, if $\Gamma$ is a $j$-independent GKM graph, then every $j-1$ edges adjacent to a vertex span a face, as follows, for example, from the results of [7]. Therefore, the poset $S(X)_{j-1}$ can be reconstructed only from the data of a GKM graph alone. If the poset is not $(j-2)$-acyclic, then the graph does not correspond to any equivariantly formal manifold. It should be noted that the problem of the reconstruction of a manifold with given tangent representations at the fixed points and, in particular, with a given equivariant 1-skeleton, is well known in equivariant topology; see [13]. The classical approach to the solution of this problem is via the Atiyah-Bott-Berline-Vergne formula (the instance of the localization formula in cohomology). Sometimes it is possible to reconstruct a manifold with torus action geometrically — by extending the action from lower dimensional strata to higher dimensions. This approach, however, does not guarantee that the resulting manifold is equivariantly formal, and that GKM theory itself is applicable. This is an important issue we want to address in our paper. As an application of Theorem 1.1, we describe the equivariant cohomology algebra for complexity $1$ actions in general position. Acyclicity arguments are applied to extend the study of complexity $1$ actions in general position which started in [5]. An action of $T^{n-1}$ on $X^{2n}$ is called an action in general position if it is $(n-1)$-independent. Theorem 1.2. Consider an equivariantly formal action of $T^{n-1}$ on $X^{2n}$ in general position. Assume that $n\geqslant 5$, $\pi_1(X)=1$ and the GKM graph of the action is bipartite. Then there exists a simplicial poset $S(\Gamma(X))$ and a regular linear element $\eta$ in the face ring $\mathbb{Q}[S(\Gamma(X))]$ such that as $H^*(BT;\mathbb{Q})$-algebras. The poset $S(\Gamma(X))$ is constructed from $S(X)$ by adding some extra elements and reversing the order. The poset $S(\Gamma(X))$ is Gorenstein*: this follows from the Gorenstein property of the algebra $H^*_T(X;\mathbb{Q})$, as explained in detail in Remark 6.9. In a certain sense Theorem 1.2 tells us that under certain assumptions complexity $1$ actions in general position behave much like the restrictions of $T^n$-actions on $X^{2n}$ to actions of generic subtori $T^{n-1}\subset T^n$, at least from the point of view of equivariant cohomology. This paper has the following structure. In § 2 we recall the necessary definitions: equivariant formality, face submanifolds and faces of a torus action. In § 3 we prove several statements about the homological properties of the orbit spaces of torus actions. Most of the arguments there follow the lines of [5]; however, we recall the key arguments. In § 4 we prove Theorem 1.1. In § 5 we provide all necessary definitions from the GKM theory and prove a combinatorial statement that for ${n\geqslant 5}$ the $n$-valent GKM graph of a complexity $1$ action in general position is bipartite if and only if it is $n$-colourable. This is a generalization of the result of Joswig [22], which asserts a similar fact for the edge skeleta of simple polytopes (hence can be viewed as a statement about GKM graphs of complexity $0$). The ability to properly colour a GKM graph implies that the graph determines a simplicial poset in a way similar to complexity $0$. In § 6 we recall the notion of the face ring of a simplicial poset and derive Theorem 1.2 from the GKM description of $H^*_T(X)$. § 2. Preliminaries2.1. Orbit-type filtrationsIn this section we define the faces of an action and list their main properties. Let a torus $T=T^k$ act on a topological space $X$, which is always assumed connected. Let $\operatorname{Sgr}(T)$ denote the set of all closed subgroups of $T$. For a point $x\in X$, $T_x\in \operatorname{Sgr}(T)$ denotes the stabilizer (stationary subgroup) of $x$, and $Tx\subset X$ is the orbit of $x$. In what follows we assume that $X$ is a $T$-CW-complex (see [1], Definition 1.1). In particular, this holds for smooth torus actions on smooth manifolds. Construction 2.1. For an action of $T$ on $X$ we define the fine subdivision of $X$ by orbit types: where $H$ is a closed subgroup of $T$, and $X^{(H)}=\{x\in T\mid T_x=H\}$. Moreover, for $H\in \operatorname{Sgr}(T)$ set Thus, $X^H$ is the set of $H$-fixed points of $X$. Each closed subgroup of a compact torus $T\cong T^k$ is isomorphic to a direct product of some torus (the continuous component) and a finite abelian group (the finite component). We say that a $T$-action on $X$ has connected stabilizers if all stabilizers $T_x$ are connected, that is, the finite components are trivial. When working with homology we follow the agreement that for actions with connected stabilizers the coefficients are taken in $\mathbb{Z}$ (or any field), but in the general case the coefficients are taken in $\mathbb{Q}$. Construction 2.2. For a $T$-action on a topological space $X$ consider the filtration where $X_i$ is the union of all orbits of the action having dimension $\leqslant i$. In other words, Remark 2.3. If $y\in Q$ is an orbit of the action, then the stabilizer subgroup $T_y$ is defined as the stabilizer $T_x$ for any representative $x\in y$. It is well defined because the torus is commutative. In what follows, when we speak about fixed points of an action, we abuse the notation by denoting a fixed point and its image in the orbit space by the same letter. 2.2. Smooth actionsDefinition 2.4. The lattice $N=\operatorname{Hom}(T^k,S^1)\cong \mathbb{Z}^k$ is called the weight lattice, and its dual lattice $N^*=\operatorname{Hom}(S^1,T^k)$ is called the lattice of one-dimensional subgroups. There are canonical isomorphisms
$$
\begin{equation*}
\operatorname{Hom}(T^k,S^1) \cong H^1(T^k;\mathbb Z)\cong H^2(BT^k;\mathbb Z)
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\operatorname{Hom}(S^1,T^k) \cong H_1(T^k;\mathbb Z)\cong H_2(BT^k;\mathbb Z).
\end{equation}
\tag{2.3}
$$
Let $X$ be a smooth closed connected orientable manifold, and let $T$ act smoothly and effectively on $X$. If $x\in X^T$ is a fixed point, then we have an induced representation of $T$ in the tangent space $\tau_xX$, called the tangent representation. Let $\alpha_{x,1},\dots,\alpha_{x,n}\in \operatorname{Hom}(T^k,S^1)\cong \mathbb{Z}^{k}$ be the weights of the tangent representation at $x$, which means by definition that
$$
\begin{equation*}
\tau_xX\cong V(\alpha_{x,1})\oplus\dots\oplus V(\alpha_{x,n})\oplus \mathbb R^{\dim X-2n},
\end{equation*}
\notag
$$
where $V(\alpha)$ is the standard one-dimensional complex representation given by ${tz=\alpha(t)\cdot z}$, $z\in \mathbb{C}$, and the action on $\mathbb{R}^{\dim X-2n}$ is trivial (see [20], Corollary I.2.1). It is assumed that all weight vectors $\alpha_{x,i}$ are nonzero, since otherwise the corresponding terms contribute to $\mathbb{R}^{\dim X-2n}$. If there is no $T$-invariant complex structure on $X$, then there is an ambiguity in the choice of the signs of the vectors $\alpha_i$. For the statements in this paper the choice of signs is inessential. We can also assume that the weight vectors $\alpha_{x,1},\dots,\alpha_{x,n}$ span linearly the weight lattice $\operatorname{Hom}(T^k,S^1)$ (otherwise there would exist an element $\lambda$ of the dual lattice $\operatorname{Hom}(S^1,T^k)$ such that $\langle\alpha_{x,1},\lambda\rangle=0$, which implies that the corresponding one-dimensional subgroup $\lambda$ lies in the noneffective kernel). This observation implies that, if the action has fixed points, then Each fixed point $x\in X^T$ has a neighborhood equivariantly diffeomorphic to the tangent representation $\tau_xX$. In particular, $x\in X^T$ is isolated if and only if $\dim X=2n$ and all tangent weights $\alpha_{x,1},\dots,\alpha_{x,n}$ are nonzero. Definition 2.5. Let $T$ act effectively on a smooth manifold $X$, and let the fixed point set $X^T$ be finite and nonempty. The nonnegative integer $\operatorname{compl} X=\frac{1}{2}\dim X-\dim T$ is called the complexity of the action. If the action is noneffective, then the symbol $\operatorname{compl} X$ denotes the complexity of the corresponding effective action: the action of the quotient by the noneffective kernel. Construction 2.6. For each closed subgroup $H\subset T$ the subset $X^H$ is a closed smooth submanifold in $X$. This submanifold is $T$-stable, as follows from the commutativity of $T$. A connected component of $X^H$ is called an invariant submanifold. Construction 2.7. For a smooth $T$-action on $X$ consider the canonical projection $p\colon X\to Q$ onto the orbit space and the filtration (2.2) on the orbit space. The closure of a connected component of $Q_i\setminus Q_{i-1}$ is called a face $F$ if it contains at least one fixed point. The number $i$ is called the rank of the face $F$, it is equal to the dimension of a generic $T$-orbit in $F$. Remark 2.8. In the case of locally standard action of $T=T^n$ on $X=X^{2n}$ the orbit space $Q=X/T$ is a nice manifold with corners, so the notion of faces is naturally defined. The preimages of the faces of $Q$ are all the invariant submanifolds of $X$. Only those faces of $Q$ (in the sense of manifold with corners) that have vertices are faces of $Q$ (in the sense of torus actions). Remark 2.9. In general, the notion of a face of an orbit space is determined by the action, so the face subdivision resembles an additional structure on the orbit space $Q$. Knowing the topology of $Q$ itself is not sufficient to define faces. Even in complexity 0, $Q$ is just a topological manifold with boundary, so one can only distinguish whether or not a point in $Q$ is a free orbit depending on whether it lies in the interior or on the boundary. There are examples in higher complexity when even free and non-free orbits cannot be distinguished in the orbit space [3]. However, we abuse the notation slightly by using the term ‘face of the orbit space’ as if it were defined intrinsically by the topology of $Q$. The following lemma is proved in [7]. Lemma 2.10. For a $T$-action on $X$, the full preimage $X_F=p^{-1}(F)$ of any face $F\subset Q$ is an invariant submanifold. In particular, it is a smooth closed submanifold of $X$. Definition 2.11. Let $F$ be a face of $Q=X/T$. Then the submanifold $X_F=p^{-1}(F)\subset X$ is called a face submanifold corresponding to $F$. Notice that the definition of a face implies that each face submanifold has necessarily a $T$-fixed point. Some details and formalism about the notions of faces and face submanifolds can be found in the recent preprint [7]. Construction 2.12. It is known (see [14], Theorem 5.11) that a smooth action of $T$ on a compact smooth manifold has only a finite number of possible stabilizers. This implies that there exists only a finite number of faces. The set of faces of $Q$ (equivalently, of face submanifolds of $X$) is partially ordered by inclusion and graded by ranks. We denote this poset by $S(X)$. Construction 2.13. Let $F$ be a face and $X_F$ be the corresponding face submanifold. The action of $T$ on $X_F$ has a noneffective kernel which we call a common stabilizer of the points in $F$. The number $\dim T/T_F$ equals the rank of $F$. The effective action of $T/T_F$ on $X_F$ satisfies the general assumption in Definition 2.5: its fixed-point set is nonempty (by the definition of a face) and finite (because it is a subset of $X^T$, and the latter is assumed to be finite). Therefore, the induced complexity $\operatorname{compl} X_F$ is well defined: Notice that each face $F$ of rank $r$ is a subset of the filtration term $Q_r$. It is tempting to say that $Q_r$ is the union of all faces of rank $r$. However, this can be false in general: an example when $Q_r$ is not the union of $r$-dimensional faces can be found in [7], Figure 1. Nevertheless, there is no such problem for equivariantly formal actions as explained in Remark 3.5 below. § 3. Acyclicity for independent actions3.1. Equivariant formalityThe definitions and statements of this subsection are well known and given for the convenience of the reader. For $T\cong T^k$ we have the universal principal $T$-bundle $ET\to BT$, ${BT\simeq (\mathbb{C}P^\infty)^k}$. The $T$-action on $X$ determines the Borel construction $X_T=X\times_T ET$, the Serre fibration $p\colon X_T\!\stackrel{X}{\to}\! BT$ and the equivariant cohomology ring $H^*_T(X;R)\!=\!H^*(X_T;R)$, which is a module over the polynomial ring $H^*(BT;R)\cong R[k]=R[v_1,\dots,v_k]$ (the module structure is induced by $p^*$). The fibration $p$ induces a Serre spectral sequence:
$$
\begin{equation}
E_2^{p,q}\cong H^p(BT^k;R)\otimes H^q(X;R)\Rightarrow H^{p+q}_T(X;R).
\end{equation}
\tag{3.1}
$$
Definition 3.1. The $T$-action on $X$ is called cohomologically equivariantly formal (over $R$) in the sense of Goresky-Kottwitz-MacPherson if the spectral sequence (3.1) collapses at $E_2$. We simply call such actions and spaces equivariantly formal. The definition of equivariant formality was given by Goresky-Kottwitz-MacPherson in [18] in the case of coefficients in $\mathbb{R}$. For convenience we provide a lemma proved in [17], which gives an equivalent reformulation of equivariant formality. Lemma 3.2 (see [17]). The following conditions are equivalent (all coefficients are either in $\mathbb{Z}$ or in a field). If the coefficients are in a field, then these conditions are equivalent to the freeness of the $H^*(BT)$-module $H^*_T(X)$. Over $\mathbb{Z}$, the freeness of a $H^*(BT)$-module is a strictly stronger property than equivariant formality (see [16]). We only consider actions with isolated fixed points. In this case the characterization of equivariant formality becomes easier. Lemma 3.3 (see [24], Lemma 2.1). Consider a smooth $T$-action on $X$ such that $X^T$ is finite and nonempty. Then the following conditions are equivalent If any of these conditions holds, then there is an isomorphism of graded $H^*(BT)$-modules $H^*_T(X)\cong H^*(BT)\otimes H^*(X)$. Another important statement asserts that equivariant formality is inherited by invariant submanifolds. Lemma 3.4 (see [24], Lemma 2.2). Let $T$ act on $X$ and $Y$ be an invariant submanifold (a connected component of $X^H$ for some $H\in \operatorname{Sgr}(T)$). Then condition $H^{\mathrm{odd}}(X)=0$ implies that $H^{\mathrm{odd}}(Y)=0$ and $Y^T\neq\varnothing$. Remark 3.5. Lemma 3.4 implies that the requirement for a face submanifold (or a face) to contain a fixed point is automatically satisfied for equivariantly formal actions with isolated fixed points. This, in turn, implies that the equivariant $r$-skeleton $Q_r$ is exactly the union of all faces $F$ of rank $r$ when we deal with equivariantly formal actions. 3.2. $j$-independent actionsDefinition 3.6. A $T$-action on a manifold $X$ is $j$-independent1 if, for any fixed point $x\in X^T$, any $\leqslant j$ of tangent weights $\alpha_{x,1},\dots,\alpha_{x,n}\in \operatorname{Hom}(T,S^1)\cong \mathbb{Z}^k$ are linearly independent over $\mathbb{Q}$. Remark 3.7. If $\operatorname{compl} X=0$, that is, $T=T^n$ acts on $X=X^{2n}$, then at each fixed point $x\in X^T$ we have $n$ weights $\alpha_{x,1},\dots,\alpha_{x,n}\in \operatorname{Hom}(T,S^1)\cong \mathbb{Z}^n$ which span $\mathbb{Q}^n$ linearly. Hence such actions are $n$-independent. It is also true that such an action is $\infty$-independent since any subset of the set of tangent weights is linearly independent. We only work with finite-dimensional manifolds, so this notation is just a matter of formalism; however, some statements about $j$-independent actions make perfect sense for $j=\infty$. Several technical statements about $j$-independent actions were proved in [5]. Lemma 3.8 (see [5], Lemma 3.1). Consider a $j$-independent action of $T$ on $X$, $j\geqslant 1$. Let $X_F$ be a face submanifold. Then the following hold: Construction 3.9. Let $F$ be a face of $Q$ for an action of $T$ on $X$. For simplicity we denote by $F_{-1}$ the union of the lower-rank subfaces of $F$. In the case of equivariantly formal actions the set $F_{-1}$ can equivalently be defined by Indeed, every $r$-dimensional orbit is contained in some invariant submanifold (hence a face submanifold in the equivariantly formal case) of rank $r$. In a similar way we define $(X_F)_{-1}$ as the union of the lower-rank face submanifolds. In the equivariantly formal case this subset coincides with $\{x\in X_F\mid \dim T_x>\operatorname{rk} F\}$. In the following propositions we assume that the coefficient ring $R$ is $\mathbb{Z}$ if all stabilizers are connected, or $\mathbb{Q}$ otherwise. Proposition 3.10 (see [24]). Consider an equivariantly formal $T$-action on $X$ of complexity $0$. Then, for each face $F$ (including $Q$ itself) the following statements hold true: In other words, each face $F$ is a homology cell and the filtration $\{Q_i\}$ is a homology cell complex. Combining Lemma 3.8, Proposition 3.10 and some inductive arguments we obtain the following statement. Proposition 3.11. Assume that the action of $T=T^k$ on $X=X^{2n}$ is equivariantly formal and $j$-independent, $j\geqslant 1$. Then the faces $F$ of the orbit space $Q=X/T$ have the following homological properties. Proof. Assertion 1 was proved in [5], Lemma 3.3. Assertion 2 was proved in [5], Lemma 3.2. Assertion 3 is Theorem 2 in [5]. Assertion 4 was not stated in [5] explicitly; however, its proof follows the same lines as the proof of assertion 3. We outline the main ideas.
Consider the cohomology spectral sequence associated with the filtration
$$
\begin{equation}
Q_0\subset Q_1\subset\dots\subset Q_{r-1}\subset Q_r\subset\dots\subset Q_k=Q,
\end{equation}
\tag{3.2}
$$
that is $E_1^{p,q}=H^{p+q}(Q_p,Q_{p-1})$. Notice that
$$
\begin{equation*}
H^*(Q_p,Q_{p-1})\cong \bigoplus_{F\mid \operatorname{rk} F=p}H^*(F,F_{-1}).
\end{equation*}
\notag
$$
Assertions 1 and 2 imply that $E_1$ vanishes, as shown in Figure 1.
The 0th row $(E^{p,0}_1,d^1)$ coincides with the differential complex
$$
\begin{equation}
\begin{aligned} \, \notag 0 &\to H^0_T(X_0)\stackrel{\delta_0}{\to} H^{1}_T(X_1,X_0)\stackrel{\delta_1}{\to}\dotsb \\ &\dotsb\stackrel{\delta_{k-2}}{\to}H^{k-1}_T(X_{k-1},X_{k-2})\stackrel{\delta_{k-1}}{\to}H^{k}_T(X,X_{k-1})\to 0, \end{aligned}
\end{equation}
\tag{3.3}
$$
which is the degree $0$ part of the nonaugmented version of the sequence of Atiyah, Bredon, Franz and Puppe
$$
\begin{equation}
\begin{aligned} \, \notag 0 &\to H^*_T(X)\stackrel{i^*}{\to} H^*_T(X_0)\stackrel{\delta_0}{\to} H^{*+1}_T(X_1,X_0)\stackrel{\delta_1}{\to}\dotsb \\ &\dotsb\stackrel{\delta_{k-2}}{\to}H^{*+k-1}_T(X_{k-1},X_{k-2})\stackrel{\delta_{k-1}}{\to}H^{*+k}_T(X,X_{k-1})\to 0 \end{aligned}
\end{equation}
\tag{3.4}
$$
(see details in [5]). The sequence (3.4) is exact for equivariantly formal actions according to [11] (for rational coefficients) and Franz and Puppe [17] (over the integers, presuming that the stabilizers are connected). Therefore, when passing from $E_1^{*,*}$ to $E_2^{*,*}=H(E_1^{*,*},d^1)$ the whole of the 0th row disappears except for $E_2^{0,0}\cong H^0_T(X)\cong R$. Thus, the second page has the form shown in Figure 2. This implies that $E_\infty^{p,q}=0$ for $0<p+q\leqslant j+1$, and therefore $\widetilde{H}_i(Q)=0$ for $i\leqslant j+1$, which proves assertion 3 of the proposition. To prove assertion 4 consider the spectral sequence associated with the filtration (3.2) cut at the $r$th term. In this case passing from $E_1$ to $E_2$ can result in the additional nonzero entry $E^{r,0}_2$ at the rightmost position of the $0$th row. If $r\geqslant j+2$, then we still have $E_\infty^{p,q}=0$ for $p+q\leqslant j+1$, and therefore $\widetilde{H}_i(Q_r)=0$ for $i\leqslant j+1$. Otherwise, if $r<j+2$, then vanishing in the spectral sequence only implies the $(r-1)$-acyclicity of $Q_r$. This completes the proof of assertion 4. The proposition is proved. Corollary 3.12. If a $T$-action on $X$ is equivariantly formal and $j$-independent, then each face $F\subset Q=X/T$ is $(j+1)$-acyclic. Proof. If $\operatorname{rk} F<j$, then $\operatorname{compl} X_F=0$ (by Lemma 3.8), so $F=X_F/T$ is acyclic by Proposition 3.10. If $\operatorname{rk} F\geqslant j$, then the induced action on $X_F$ is $j$-independent (again, by Lemma 3.8), so $F=X_F/T$ is $(j+1)$-acyclic by Proposition 3.11.
§ 4. Topology of face posetsRecall that $S(X)$ denotes the face poset of $Q$ (or the poset of face submanifolds in $X$) ordered by inclusion. Let $S(X)_r$ denote the subposet of all faces of rank $\leqslant r$. The symbol $|S|$ denotes the geometric realization of a poset $S$, that is, the geometric realization of the order complex $\operatorname{ord} S$ (the simplicial complex whose simplices are chains in $S$). The following is an almost immediate corollary to Proposition 3.11. Corollary 4.1. Assume that a $T$-action on $X$ is equivariantly formal and $j$-independent. Then the geometric realization $|S(X)_{r}|$ is $(r-1)$-acyclic for $r<j$. Proof. As noticed before, if $r<j$, then the orbit type filtration on $Q_r$ is a homological cell filtration with regular cells (essentially, because of Proposition 3.10). Standard arguments with the spectral sequences are used to prove the isomorphisms $H_*(|S(X)_{r}|)\cong H_*(Q_r)$ for regular homological cell complexes (see, for example, [24], Proposition 5.14, or [2], Proposition 2.7). The rest follows from assertion 4 of Proposition 3.11. Using slightly more complicated arguments we can prove a stronger statement. Theorem 4.2. Assume that a $T$-action on $X$ is equivariantly formal and $j$-independent. Then the geometrical realization $|S(X)_{r}|$ is $\min(r-1,j+1)$-acyclic for any $r$. To prove this, we recall several useful facts about homotopy colimits. Construction 4.3. Let $S$ be a finite poset and $\operatorname{cat}(S)$ the finite category whose objects are elements $s\in S$ and there is exactly one morphism $s_1\to s_2$ if ${s_1\leqslant s_2}$ (and no morphisms otherwise). An $S$-shaped topological diagram is a functor $D\colon\operatorname{cat}(S)\to \mathrm{Top}$ to the category of topological spaces. Two topological spaces can be associated with each topological diagram $D$: the colimit $\operatorname{colim}_S D$ and the homotopy colimit $\operatorname{hocolim}_SD$. Colimit is a synonym for the direct limit of a diagram in the category of topological spaces. Homotopy colimit is a modified version of colimit, well behaved under homotopy equivalences. An accessible exposition of homotopy colimits and their use in combinatorial topology can be found in [27]. There exists a constant diagram $\ast\colon\operatorname{cat}(S)\to\mathrm{Top}$ which maps each $s\in S$ to a point $\mathrm{pt}$. From the definitions it easily follows that $\operatorname{colim}_S\ast$ is a finite set of points corresponding to connected components of $|S|$, while $\operatorname{hocolim}_S\ast=|S|$. Construction 4.4. Let $T$ act on a smooth manifold $X$. Consider the diagram $D_Q\colon \operatorname{cat}(S(X))\to\mathrm{Top}$, which maps each face $F$ (as an abstract element of the face poset $S(X)$) to the face $F$ (as a topological space) with morphisms being the natural inclusions of faces. Since the poset $S(X)$ has the greatest element (the space $Q$ itself), the equality $\operatorname{colim}_{S(X)}D_Q=Q$ holds. Smooth toric actions always admit equivariant cell structures [21]. This implies that inclusions of subfaces $F_1\hookrightarrow F_2$ of the orbit space $Q=X/T$ admit cellular structures, hence they are cofibrations. Moreover, this argument shows that the diagram $D_Q$ is cofibrant. Therefore,
$$
\begin{equation}
\operatorname{hocolim}_{S(X)} D_Q\simeq \operatorname{colim}_{S(X)}D_Q.
\end{equation}
\tag{4.1}
$$
Let $t$ be a fixed nonnegative integer. Definition 4.5. A map $\psi\colon X\to Y$ of topological spaces is called a $t$-equivalence if the induced map $\psi_*\colon \pi_r(X,b)\to\pi_r(Y,\psi(b))$ is an isomorphism for all $r<t$ and is surjective for $r=t$, for all basepoints $b$. Lemma 4.6 (strong homotopy lemma; see [9], Lemma 2.8). Let $D_1$ and $D_2$ be $S$-shaped diagrams. Let $\alpha\colon D_1\to D_2$ be a map of diagrams such that for each $s\in S$ the map $\alpha_s\colon D_1(s)\to D_2(s)$ is a $t$-equivalence. Then the induced map from $\operatorname{hocolim}_SD_1$ to $\operatorname{hocolim}_SD_2$ is a $t$-equivalence. As usual, this statement has a homological version. Definition 4.7. A map $\psi\colon X\to Y$ is called a homological $t$-equivalence (over coefficient ring $R$) if the induced map is an isomorphism for all $r<t$ and surjective for $r=t$. Lemma 4.8. Let $D_1$ and $D_2$ be $S$-shaped diagrams. Let $\alpha\colon D_1\to D_2$ be a map of diagrams such that for each $s\in S$ the map $\alpha_s\colon D_1(s)\to D_2(s)$ is a homological $t$-equivalence. Then the induced map from $\operatorname{hocolim}_SD_1$ to $\operatorname{hocolim}_SD_2$ is a homological $t$-equivalence. Although the arguments used in [9] to prove Lemma 4.6 work for the homology version, we provide an alternative proof based on spectral sequences. Recall that any diagram of spaces (CW-complexes) induces a spectral sequence. Proposition 4.9 (see [15], Proposition 15.12). Let $D\colon I\to \mathrm{Top}$ be a diagram over a small category $I$ and $h_*(\,\cdot\,)$ a generalized homology theory. Then there is a spectral sequence The differentials have the form $d_r\colon E_{p,q}^r\to E^r_{p-r,q+r-1}$. Here $h_q(D)$ denotes the diagram of abelian groups obtained by applying the functor $h_q(\,\cdot\,)$ elementwise to the topological diagram $D$. The module $H_p(I;\mathcal{A})$ denotes the homology of a small category $I$ with coefficients in a functor $\mathcal{A}$, which can be defined by one of the equivalent constructions listed below.
For the equivalence of these constructions, we refer to [25]. Let us prove Lemma 4.8 using Proposition 4.9. Proof of Lemma 4.8. The diagram map $\alpha\colon D_1\to D_2$ induces the morphism of spectral sequences Since $\alpha\colon D_1(s)\to D_2(s)$ is a homology $t$-equivalence for each entry $s\in S$, the induced map $\alpha_*\colon E_{p,q}^r(D_1)\to E_{p,q}^r(D_2)$ is an isomorphism for $p+q<t$ or $({p+q=t}) \mathbin{\&} (q<t)$, while it is surjective for $(p,q)=(0,t)$. This can be proved using induction on $r$, the index of the page. Finally, this implies that $\alpha_*\colon H_{p+q}(\operatorname{hocolim}_ID_1)\to H_{p+q}(\operatorname{hocolim}_ID_2)$ is an isomorphism for $p+q<t$ and surjective for $p+q=t$. Hence $\alpha$ induces a $t$-equivalence of homotopy colimits. The lemma is proved. Now we prove Theorem 4.2. Proof of Theorem 4.2. Let $D_Q\colon\operatorname{cat}(S(X))_r\to\mathrm{Top}$ be the diagram of faces, described in Construction 4.4 and $\ast\colon \operatorname{cat}(S(X))_r\to\mathrm{Top}$ be the constant diagram (which maps every element to a single point). Then we have a natural morphism of diagrams $\alpha\colon D_Q\to\ast$. Each face $F\in S(X)$ is $(j+1)$-acyclic by Corollary 3.12. Therefore, $\alpha$ is a $(j+2)$-equivalence on each entry of the diagram. Then Lemma 4.8 states that the induced map
$$
\begin{equation*}
\operatorname{hocolim}_{S(X)_r}D_Q\to \operatorname{hocolim}_{S(X)_r}\ast
\end{equation*}
\notag
$$
is a homology $(j+2)$-equivalence. However, $\operatorname{hocolim}_{S(X)_r}D_Q\simeq \operatorname{colim}_{S(X)_r}D_Q$ since $D_Q$ is cofibrant. The colimit $\operatorname{colim}_{S(X)_r}D_Q$ is homeomorphic to the $r$-skeleton $Q_r$ by construction. The space $Q_r$ is $\min(r-1,j+1)$-acyclic by Proposition 3.11. Therefore, $\operatorname{hocolim}_{S(X)_r}\ast \cong |S(X)_r|$ is $\min(r-1,j+1)$-acyclic as well. The theorem is proved. These arguments prove assertion 1 of Theorem 1.1. Assertion 2 follows easily, since $S(X)_{\leqslant s}$ is naturally isomorphic to $S(Y)$ whenever $Y$ is a face submanifold corresponding to $s\in S(X)$. This finishes the proof Theorem 1.1. § 5. Actions of complexity $1$ in general positionIn the next two sections we prove Theorem 1.2. We apply a very particular case of acyclicity argument to describe the equivariant cohomology algebra of a manifold $X^{2n}$ with an equivariantly formal $(n-1)$-independent action of $T^{n-1}$ when $n\geqslant 5$. In this particular case the description boils down to the theory of Gorenstein face algebras, similar to the complexity $0$ actions studied in [24]. As in the case of complexity $0$, we use the theorem of Goresky, Kottwitz and MacPherson to describe equivariant cohomology. Let us recall the basics of GKM theory (see details in [18] and [23]). While GKM manifolds usually refer to complex algebraic varieties with an action of an algebraic torus, we deal with the topological version of the GKM theory. Definition 5.1. A $2n$-dimensional (orientable connected) compact manifold $X$ with an action of $T=T^k$ is called a GKM manifold (named after Goresky, Kottwitz and MacPherson) if the following conditions hold: The next proposition is often taken as a definition of a GKM manifold and is quite standard. Proposition 5.2. The one-dimensional equivariant skeleton $X_1$ of a GKM manifold is a union of $T$-invariant $2$-spheres. Each invariant $2$-sphere connects two fixed points. Corollary 5.3. The $1$-skeleton $Q_1=X_1/T$ is a graph on the vertex set $Q_0\cong X_0$ whose edges correspond to $2$-spheres between fixed points. Let $\operatorname{star}(p)$ denote the set of edges emanating from the vertex $p$ of a graph. If $e\in\operatorname{star}(p)$ is an edge emanating from a fixed point $p$ to a fixed point $q$, then this edge comes equipped with a weight $\alpha(pq)\in\operatorname{Hom}(T^k,T^1)$. This weight corresponds to the summand of the tangent representation $\tau_pX$ that is tangent to the invariant $2$-sphere corresponding to $e$. It easily follows that $\alpha(pq)=\pm\alpha(qp)$. Definition 5.4. A GKM graph $\Gamma$ is a finite $n$-valent regular graph $(V,E)$ equipped with a function $\alpha\colon E\to \operatorname{Hom}(T^k,T^1)$ which satisfies $\alpha(pq)=\pm\alpha(qp)$ for all edges $e=(pq)$. The function $\alpha$ is called an axial function. The numbers $k$ and $n$ in the definition are called the rank and dimension of a GKM graph $\Gamma$. Definition 5.5. A GKM graph $\Gamma$ with connection is a GKM graph equipped with additional data, a connection. A connection $\theta$ is a set of bijections $\theta_{(pq)}\colon \operatorname{star}(p)\to \operatorname{star}(q)$ for all edges $e=(pq)$ of a graph which satisfy the conditions Proposition 5.6 (see [10], Theorem 3.4). If $X$ is a GKM manifold then its $1$-skeleton $Q_1$ is a GKM graph. If, moreover, the action on $X$ is $3$-independent, then $Q_1$ is a GKM graph equipped with a canonical connection. Let $\Gamma(X)$ denote the GKM graph corresponding to a torus action on a manifold $X$. The definitions of $j$-independency and faces of this action inspire the following analogues for abstract GKM graphs. Definition 5.7. A GKM graph $\Gamma$ is called $j$-independent if for any vertex $p$ of $\Gamma$ the axial values of any $\leqslant j$ edges of $\operatorname{star}(p)$ are linearly independent over $\mathbb{Q}$. Proposition 5.6 (or, more precisely, its analogue for abstract GKM graphs) implies that if $j\geqslant 3$, then the connection $\theta$ on a $j$-independent GKM graph is uniquely determined. Definition 5.8 (see [19], Definition 1.4.2). Let $\Gamma$ be an abstract GKM graph with connection. A connected subgraph $\Gamma'\subset \Gamma$ is called a totally geodesic face of rank $r$ if it is a GKM graph of rank $r$ and for any edge $pq\in \Gamma'$ we have $\theta_{pq}(\operatorname{star}(p)\cap \Gamma')=\operatorname{star}(q)\cap \Gamma'$. In what follows we use the term face of a GKM graph instead ‘totally geodesic face’ for brevity. If a face has dimension $d$, then we call it a $d$-face. If a face of a graph has codimension $1$, then it is called a facet. Construction 5.9. If $Y$ is a face submanifold of a GKM manifold $X$, then the graph $\Gamma(Y)$ is naturally a face of the graph $\Gamma(X)$. We call such a face a geometric face of the GKM graph $\Gamma(X)$. Not every face of $\Gamma(X)$ is necessarily geometric. The simplest example is the full flag manifold $\mathrm{Fl}_3$: its GKM graph has three nongeometric totally geodesic faces (see [7], Figure 2). Let $S(\Gamma)$ denote the poset of all faces of $\Gamma$ ordered by inclusion. Although this poset does not coincide with the poset $S(X)$ of geometric faces in general, $S(X)$ can be reconstructed from the GKM graph $\Gamma(X)$ as described in [7]. Lemma 5.10. Consider a $j$-independent GKM action on a manifold $X$, $j\geqslant 3$, such that $\Gamma(X)$ is a $j$-independent GKM graph with connection. Then any $\leqslant j-1$ edges emanating from a common vertex span a unique face of $\Gamma(X)$. This face is geometric. The proof can be found, for instance, in [7], Proposition 5.7. Definition 5.11. An edge $e$ of $\Gamma$ emanating from a vertex of a face $H$ is called transversal to $H$ if $e$ is not an edge of $H$. Consider an $n$-valent $j$-independent abstract GKM graph $\Gamma$. If $j\geqslant 3$, then any two edges emanating from a common vertex determine a $2$-face $\Xi$. Notice that combinatorially a $2$-face is a cycle graph, so we have the monodromy map, the composition of the connection maps along the edges of the cycle. This monodromy acts on the set of edges incident to any given vertex of $\Xi$. Lemma 5.12. If a GKM graph is $j$-independent and $j\geqslant 4$, then the monodromy map along any $2$-face acts identically on transverse edges. Proof. Take an arbitrary edge $e$ transversal to $\Xi$ at a vertex $p$. This transversal edge, together with the two edges of $\Xi$ emanating from $p$ determines a $3$-face since $j\geqslant 4$. Therefore, if we translate $e$ along $\Xi$, then it returns back to $e$ (since it stays inside a $3$-face). Since $e$ is arbitrary, this proves the lemma. The condition $j\geqslant 4$ in Lemma 5.12 cannot be weakened. The monodromy map $\mu_F$ is not necessarily trivial when $j=3$, as evidenced by the following examples. Example 5.13. Consider the natural torus action of $T^3$ on the complex Grassmann manifold $\mathrm{Gr}_{4,2}$ of $2$-planes in $\mathbb{C}^4$. This is a complexity $1$ action in general position. Its GKM graph is shown in Figure 3, a. This graph is embedded in $\mathbb{R}^3$ as a skeleton of an octahedron, and the values of the axial function correspond to the actual geometrical directions of edges in $\mathbb{R}^3$. A triangular face of an octahedron corresponds to a face submanifold $\mathbb{C}P^2$ inside $\mathrm{Gr}_{4,2}$; it is a $2$-face in the GKM sense. There are precisely two transversal edges to a face in each vertex. It can be seen that the monodromy along a triangular face transposes the transversal edges. Example 5.14. There is a canonical action of $T^3$ on the quaternionic projective plane $\mathbb{H}P^2$. This is a complexity $1$ action in general position similarly to the previous example. The detailed analysis of the faces of this action was done in [4]. The GKM graph is shown schematically in Figure 3, b; it has three vertices, any two of which are connected by a pair of edges. Again, the monodromy along any triangular face permutes its transversal edges. It is also true that the monodromy along any biangle permutes the transversal edges. Lemma 5.12 asserts that the monodromy is trivial on transversal edges for highly independent actions. The monodromy along a $2$-face can, however, be nonidentical on the edges of this face. The next statement is straightforward from the properties of a connection. Lemma 5.15. The monodromy along a $2$-face is identical on the edges of this face if and only if this face is a cycle of even length. Let us introduce several more combinatorial definitions. Definition 5.16. A GKM graph $\Gamma$ is called bipartite if it is bipartite as an unlabelled graph. A graph $\Gamma$ is called even if every $2$-face of $\Gamma$ is a cycle of even length. A graph is bipartite if and only if its vertices can be properly coloured with two colours. Bipartiteness of a graph obviously implies that the graph is even. The converse is also true under the assumption that every closed path in $\Gamma$ is a composition of $2$-faces as an element of $\pi_1(\Gamma)$. Definition 5.17. An $n$-valent GKM graph $\Gamma$ is called balanced if there exists a colouring of its edges with $n$ colours such that If $\Gamma$ is balanced, then every (totally geodesic) face is balanced too. Therefore, the condition of being balanced implies evenness. The aim of the remaining part of this section is to prove the converse statement for complexity $1$ actions in general position. Proposition 5.18. Let $X=X^{2n}$ be a simply-connected GKM manifold with smooth effective action of $T=T^{n-1}$ of complexity $1$ in general position, and let $n\geqslant 5$. If the GKM graph $\Gamma(X)$ is even, then it is balanced. Proof. The graph $\Gamma(X)$ is an $(n-1)$-independent GKM graph of dimension $n$ and rank $n-1$. Since $n\geqslant 5$, we are in a position to apply Lemma 5.12: an edge $e$ is preserved by the monodromy along any $2$-face transversal to $e$. On the other hand, since the graph is even, Lemma 5.15 applies as well, so that the monodromy along a $2$-face preserves $e$ if $e$ lies in this face. Therefore, the monodromy is trivial along all closed paths in the subgroup of $\pi_1(\Gamma(X))$ generated by the $2$-faces of $\Gamma(X)$.
Let us prove that the $2$-faces generate $\pi_1(\Gamma(X))$. Consider the orbit space $Q$ of $X$ and the equivariant skeleta $Q_0\subset Q_1\subset Q_2$. Here $Q_1$ is homeomorphic to a graph $\Gamma(X)$, and $Q_2$ is a homology cell complex according to Proposition 3.11. However, in dimensions 1 and 2 every homology cell is an actual cell. The independence assumption on the action implies that $Q_2$ is $1$-acyclic. Combining this with simple connectedness gives $\pi_1(Q_2)=1$. Therefore, all closed paths in $Q_1=\Gamma(X)$ are generated by the boundaries of $2$-faces. This argument shows that monodromy acts trivially. We can colour $\operatorname{star}(p)$ with $n$ colours arbitrarily and then use monodromy to transfer this colouring consistently to all other vertices. This procedure determines a proper colouring. The proposition is proved. Definition 5.19. A GKM graph $\Gamma$ is called a graph with facets if for any vertex $p$ and any edge $e\in\operatorname{star}(p)$ there exists a facet of $\Gamma$ spanned by the edges $\operatorname{star}(p)\setminus\{e\}$. Proof. If $\Gamma$ has dimension $n$ and a proper colouring $c\colon E_\Gamma\to[n]=\{1,\dots,n\}$, then the facets are the connected components of the subgraphs $\Gamma_i=c^{-1}([n]\setminus\{i\})$ for $i\in[n]$. The lemma is proved. Examples 5.13 and 5.14 show that GKM graphs $\Gamma(\mathrm{Gr}_{4,2})$ and $\Gamma(\mathbb{H}P^2)$ do not have facets. By contrast, combining Proposition 5.18 and Lemma 5.20 we obtain the following Corollary 5.21. Under the assumptions of Theorem 1.2 the GKM graph $\Gamma(X)$ is a graph with facets. It follows easily that in the case described above all faces of $\Gamma(X)$ are either geometric (faces of dimension $\leqslant n-2$ and the whole graph) or are facets provided by Corollary 5.21. In this case $S(\Gamma(X))$ is a dually simplicial poset in the following sense: every upper order ideal $S(\Gamma(X))_{\geqslant s}$ is a Boolean lattice. Let $S(\Gamma(X))^*$ denote this poset with reversed order; this is a simplicial poset. Remark 5.22. We expect that much weaker assumptions are required to guarantee that a GKM graph $\Gamma(X)$ of an action of $T^{n-1}$ on $X^{2n}$ in general position has facets. In the first version of this paper we claimed this for $n\geqslant 5$, without assuming the graph to be bipartite, but we found a gap in the original proof which we were unable to fix. However, we do not know any examples of graphs of complexity 1 in general position that have no facets. § 6. Cohomology and face ringsLet us recall the basic theorem used to describe the equivariant cohomology ring of a GKM manifold. Theorem 6.1 (GKM theorem (after Goresky, Kottwitz and MacPherson)). Let $X$ be a GKM manifold and $\Gamma(X)$ its GKM graph with vertex set $V=X^T$, edge set $E$ and axial function $\alpha$. Consider the $H^*(BT)$-algebra where the value $\alpha(pq)$ of the axial function is considered as an element of $H^2(BT)$. Then there is a canonical isomorphism of graded $H^*(BT)$-algebras: The GKM theorem provides an explicit description of $H^*(BT)$ as a subring of the direct sum $\bigoplus_{p\in X^T}H^*(BT)$. An additional work is required if one needs an expression for $H^*(BT)$ in terms of generators and relations. The classical cases are smooth toric varieties and quasitoric manifolds: it is possible to describe their equivariant cohomology rings as Stanley-Reisner algebras. More generally, equivariant cohomology rings of equivariantly formal torus manifolds were described as the face rings of their face posets in [24]. Here we adopt some ideas of that work for the case of actions of complexity $1$ in general position under the assumption that their GKM graphs have facets. In what follows cohomology rings are taken with coefficients in $R=\mathbb{Q}$. In this section we study an equivariantly formal complexity $1$ action of $T=T^{n-1}$ on $X=X^{2n}$ in general position and assume that $\Gamma(X)$ has facets. Therefore, $S(\Gamma(X))^*$ is a simplicial poset as explained in the previous section. For a simplicial poset there is a well-known notion of face ring, a generalization of Stanley-Reisner ring. Definition 6.2. Consider the face ring $R[\Gamma(X)]$ of the simplicial poset $S(\Gamma(X))^*$, that is the quotient ring where the ideal $\mathcal{I}$ is generated by the relations where $E$ runs over all connected components of the intersection $F\cap H$ and ${F\vee H}$ denotes the minimal face of $S(\Gamma(X))^*$ which contains both $F$ and $H$. The ring $R[\Gamma(X)]$ is a graded commutative ring with grading $\deg v_F=2\operatorname{codim} F=2(n- \dim F)$. Notice that the element $F\vee H$ is well defined and unique if $F\cap H\neq \varnothing$. Otherwise, if $F\cap H=\varnothing$, then the sum over an empty set of indices is set to be zero, so there is no need to define a unique element $F\vee H$. Elements of degree $2$ of $R[\Gamma(X)]$ are called linear. The component $R[\Gamma(X)]_2$ is generated by the $v_F$, where each $F$ is a facet. Theorem 6.3. Let $\Gamma(X)$ be as described above. Then there exists a nonzero linear form $\eta\in R[\Gamma(X)]_2$ such that $H^*_T(X)$ is isomorphic to $R[\Gamma(X)]/(\eta)$. The proof follows in many aspects the lines of a similar result on actions of complexity $0$ which is given in [24]. The essential tool of the proof is the notion of the Thom class of a face of a GKM graph, which is recalled below. A technical remark is needed for the definition. We impose an omniorientation on a GKM manifold (which means that we orient all of its faces) and also an omniorientation on GKM graphs. The latter means that all edges of $\Gamma$ are assumed to be directed, and the values of the axial function are sensitive to the change of direction, that is $\alpha(pq)=-\alpha(qp)$. Construction 6.4. Let $F$ be a (totally geodesic) face of an omnioriented GKM graph $\Gamma$ with connection on the vertex set $V$. Consider the following element of $\bigoplus_{p\in V}H^*(BT)$, called the Thom class of $F$: The value at a vertex $p$ of $F$ is the product of the weights at $p$ transversal to the face $F$. The element $\tau_F$ is homogeneous of degree $2(n-\dim F)$, twice the codimension of $F$. From the properties of the connection on $\Gamma$ it easily follows that $\tau_F$ belongs to the submodule $H_T^*(\Gamma)\subset \bigoplus_{p\in V}H^*(BT)$. From this general construction and Theorem 6.1 it follows that whenever $X$ is a GKM manifold and $F$ is a face of its GKM graph, its Thom class $\tau_F$ is a well-defined element of $H^{2(n-\dim F)}_T(X)$. Thom class makes sense even if $F$ is not geometric. However, if $F$ is a GKM graph of a face submanifold $Y\subset X$, then $\tau_F$ is the equivariant Poincaré dual of the submanifold $Y\subset X$. In other words, $\tau_F$ is the image of the identity element under the equivariant Gysin homomorphism $H^0_T(Y)\to H^{2n-2\dim F}(X)$. This fact is easily proved since localizing the element $\tau_F$ to a fixed point $p\in X^T=V$ gives either the Euler class of the normal space to $Y\subset X$ at $p$ (if ${p\in Y}$) or zero (if ${p\notin Y}$). Returning to complexity $1$ actions in general position with facets one can notice that for all faces $F$ of $\Gamma(X)$ the elements $\tau_F$ satisfy similar polynomial relations to those in Definition 6.2 of the face ring. Lemma 6.5. For a complexity $1$ action in general position with facets, the Thom classes of faces of $\Gamma(X)$ satisfy the relations The proof follows easily by localizing the relation to each fixed point $p$; see [24], Lemma 6.3. The assignment $v_F\mapsto \tau_F$ defines therefore a homomorphism Proof. The same argument as in Proposition 7.4 in [24] shows that $H^*_T(X)$ is generated by the elements $\tau_F$ as a module over $H^*(BT)$, and $H^2(BT)$ is generated over $R$ by the $\tau_G$, where the $G$ are facets. This implies the lemma. Proposition 6.7. Assume that $X$ is a GKM manifold of dimension $2n$ such that the action has complexity $1$ in general position and $\Gamma(X)$ has facets. Then there exists a nontrivial linear form $\eta\in R[\Gamma(X)]$ such that $\varphi(\eta)=0$ and $\varphi$ induces an isomorphism Proof. We consider the commutative diagram where $\psi$ is induced by $v_F\to \tau_F(p)$ for each $p\in X^T$. The vertical map $r$ is injective since the action is equivariantly formal (see, for example, the GKM model given by Theorem 6.1). The vertical map $s$ is also injective: this follows from the fact that the face algebra is an algebra with straightening law (see, for example, [12], Theorem 3.5.6, and the remark after it).
Notice that each term $R[\Gamma(X)]/(v_H\mid p\notin H)$ in the expression on the left is isomorphic to the polynomial algebra in $n$ generators (see the explanation below). Since $\dim T=n-1<n$, the term $H^*(BT)$ on the right is a polynomial algebra in ${n-1}$ generators. It follows that the map $\psi$ is surjective but not injective, even in degree $2$. Therefore, as the above diagram is commutative, $\ker\varphi$ contains a nontrivial linear form $\eta=\sum_{i=1}^m c_i\tau_i$, where $c_i\in R$ and the $\tau_i$ are the Thom classes of facets of $\Gamma(X)$. Then $r(\varphi(\eta))=0$, and so $\sum_{i=1}^mc_i\tau_{i}(p)=0$ for any $p\in X^T$. Here $\tau_{i}(p)\neq0$ if and only if $p$ is a vertex of the facet corresponding to $\tau_i$. Since the $\tau_{i}(p)$ span $H^2(BT)$, which is of rank ${n-1}$, the coefficient vector $(c_1,\dots,c_m)$ is uniquely determined up to proportionality. This shows that the epimorphism
$$
\begin{equation*}
\overline{\varphi}\colon R[\Gamma(X)]/(\eta)\to H^*_T(X),
\end{equation*}
\notag
$$
induced from $\varphi$ is an isomorphism on degree $2$.
Note that any $c_i$ is nonzero. Indeed, for any fixed point $p\in X^T$ we have $n$ tangent weights $\alpha_{p,1},\dots,\alpha_{p,n}\in \operatorname{Hom}(T,T^1)\cong \mathbb{Z}^{n-1}$ attached to this point. A unique (up to proportionality) linear relation $\sum_{j\in [n]}c'_j\alpha_{p,j}=0$ holds for these vectors, and the coefficients $c'_j$ are nonzero since every $n-1$ weights are linearly independent; see details in [3]. Each $c'_j$ is equal to the number $c_i$ corresponding to the facet transversal to the weight $\alpha_{p,j}$ at $p$. By moding out the ideals generated by $\eta$ in the commutative diagram (6.2) we obtain the commutative diagram where $\overline{s}$ is injective since so is $s$.Indeed, since $\eta=\sum_{i=1}^mc_i\tau_i$ and $\tau_i(p)\neq0$ if and only if the corresponding facet contains $p$, we have
$$
\begin{equation}
R[\Gamma(X)]/(\eta)/(v_H\mid p\notin H)=R[\tau_i\mid i\in I(p)]/\biggl(\sum_{i\in I(p)} c_i\tau_i\biggr),
\end{equation}
\tag{6.4}
$$
where $I(p)=\{i\mid \tau_i(p)\neq0\}$. Since $|I(p)|=n$ and the coefficients $c_i$ are nonzero, the ring in (6.4) is isomorphic to $H^*(BT)$. This and the surjectivity of $\overline{\psi}$ imply that $\overline{\psi}$ is an isomorphism, proving the claim. Since both $\overline{s}$ and $\overline{\psi}$ are injective, the commutativity of the above diagram shows that the epimorphism $\overline{\varphi}$ is indeed injective on any degree, proving the proposition. Proposition 6.7 proves Theorem 6.3. Combining it with Corollary 5.21 in the previous section, we obtain the proof of Theorem 1.2. Remark 6.9. Since $X$ is equivariantly formal, the equivariant cohomology algebra $H^*_T(X)\cong R[\Gamma(X)]/(\eta)$ is a free module over $H^*(BT)$, which is a subalgebra in $H^*_T(X)$ generated freely by some linear forms $\theta_1,\dots,\theta_{n-1}$. Hence $\theta_1,\dots,\theta_{n-1}$ is a regular sequence in $R[\Gamma(X)]/(\eta)$. Notice that $\eta$ is a regular element of $R[\Gamma(X)]$ since its localization to each fixed point $p\in X^T$ is nonzero. Therefore, the face ring $R[\Gamma(X)]$ has a regular sequence $\eta,\widetilde{\theta}_1,\dots,\widetilde{\theta}_{n-1}$ of length $n$, where $\widetilde{\theta}_i$ is a lift of $\theta_i$ to $R[\Gamma(X)]$. Since the quotient $R[\Gamma(X)]/(\eta,\widetilde{\theta}_1,\dots,\widetilde{\theta}_{n-1})\cong H^*(X)$ is finite dimensional, this is a maximal regular sequence. Hence the face ring $R[\Gamma(X)]$ is Cohen-Macaulay. Moreover, since the quotient is a Poincaré duality algebra, the ring $R[\Gamma(X)]$ is Gorenstein. Therefore, the simplicial poset $S(\Gamma(X))^*$ is Gorenstein (see [26]). It is also a Gorenstein* poset since the top-degree component of $R[\Gamma(X)]/(\eta,\widetilde{\theta}_1,\dots,\widetilde{\theta}_{n-1})\cong H^*(X)$ has degree $2n$. This implies that the geometric realizations of $S(\Gamma(X))^*$ and all of its links are homology spheres. We conclude the paper with an observation which relates the cohomology of $X$ with that of its face submanifolds. Proposition 6.7 implies that if $Y$ is a face submanifold of $X$ such that any intersection of faces of $\Gamma(X)$ with $\Gamma(Y)$ is connected unless empty, then the restriction map is surjective when $\Gamma(X)$ has facets. This is not true in general.Example 6.10. Consider $X=\mathrm{Gr}_{4,2}$ as a continuation of Example 5.13. Let $Y$ be a face submanifold of $\mathrm{Gr}_{4,2}$ diffeomorphic to $\mathbb{C}P^1\times \mathbb{C}P^1$. It corresponds to an equatorial square cycle of an octahedron shown in Figure 3, a. The restriction map $\iota^*\colon H^2_T(\mathrm{Gr}_{4,2})\to H^2_T(\mathbb{C}P^1\times \mathbb{C}P^1)$ is not surjective: the rank of the target module is obviously larger than the rank of the source. However, in this example surjectivity holds in higher degrees. Proposition 6.11. Let $X$ be a GKM manifold of complexity $1$ in general position. If any intersection of geometric faces of $\Gamma(X)$ with $\Gamma(Y)$ is connected unless empty, then $\iota^*$ in (6.5) is surjective in degrees ${\geqslant 4}$. Proof. We think of $H^*_T(X)$ and $H^*_T(Y)$ as the cohomology of the GKM graphs $\Gamma(X)$ and $\Gamma(Y)$. For any face $G$ of codimension ${\geqslant 2}$ of $\Gamma(Y)$ there exists a face $F$ of $\Gamma(X)$ such that $F$ and $\Gamma(Y)$ intersect transversally in the vertices of $G$. Since $F$ also has codimension ${\geqslant 2}$ in $X$, it is a geometric face. By assumption $G=F\cap \Gamma(Y)$ since there are no other connected components in the intersection. Hence ${\iota^*(\tau_F)=\tau_G}$, proving the proposition. AcknowledgementsThe authors thank Fedor Pavutnitskiy from whom we knew about the works of Quillen on spectral sequences of homotopy colimits. The last-named author would like to thank Shintarô Kuroki for many fruitful discussions of GKM theory. We are also grateful to Ivan Limonchenko, who pointed out an inaccuracy in the proof of the second theorem that appeared in the first version of this text. We thank the anonymous referees for many useful remarks, which helped us to improve the exposition.
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