Non-negatively curved GKM orbifolds

# Non-negatively curved GKM orbifolds

Oliver Goertsches and Michael Wiemeler MW was supported by DFG-Grants HA 3160/6-1 and HA 3160/11-1 and SFB 878.
###### Abstract

In this paper we study non-negatively curved and rationally elliptic GKM manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifolds.

Moreover, we give a simplified proof of a characterisation of products of simplices among orbit spaces of locally standard torus manifolds. This characterisation was originally proved in [Wie15] and was used there to obtain a classification of non-negatively curved torus manifolds.

## 1 Introduction

By the Chang–Skjelbred Lemma [CS74], the equivariant cohomology of an action of a torus on a rational cohomology manifold (e.g., a orientable manifold or a orientable orbifold) with cohomology concentrated in even degrees can be computed from the equivariant cohomology of the one-skeleton of the action, that is from the union of all the orbits of dimension less than or equal to one.

The GKM condition — introduced in [GKM98] — requires that is is of a particularly simple type. Namely, it is required that is a union of two-dimensional spheres, such that the -action restricts to a cohomogeneity one action on each two-sphere. The orbit space is then an -valent graph, where . The isotropy representations in the fixed points induce a labeling of the edges of the graph, as explained in Section 2. From this labelled graph one can compute the equivariant and non-equivariant rational cohomology rings of a GKM manifold or orbifold. This is made explicit in the GKM Theorem [GKM98], see Theorem 2.6 below.

Similarly to the GKM condition, we say that an action is GKM if for all the union of the orbits of dimension at most is a union of -dimensional invariant submanifolds. Their GKM graphs are then the -dimensional faces of the GKM graph of .

In this paper we continue our investigation of isometric torus actions of GKM type on Riemannian manifolds with sectional curvature bounded from below. In [GW15] we showed that a positively curved Riemannian manifold admitting an isometric GKM torus action has the same real cohomology ring as a compact rank one symmetric space. The assumption of positive curvature forces, by the classification of -dimensional positively curved -manifolds [GS94], the two-dimensional faces of the GKM graph to be just biangles or triangles – this condition turned out to be a severe enough restriction to classify all occurring graphs.

Considering the same setting for non-negatively instead of positively curved manifolds, we observe that now also quadrangles appear as two-dimensional faces [HK89], [SY94], which increases the possibilities for the GKM graphs greatly. Still, we are able to show the following theorem on the structure of the GKM graph (without the labelling):

###### Theorem 1.1.

Let be a GKM orbifold with an invariant metric of non-negative curvature. Then the GKM graph of is finitely covered by the vertex-edge graph of a finite product .

In the above theorem and later on, denotes an -dimensional simplex and the orbit space of the linear effective action of the -dimensional torus on .

Since the number of the vertices in the GKM graph is equal to the total Betti number of the orbifold we get the following gap phenomenon:

###### Corollary 1.2.

Let of dimension be as in the previous theorem, then the total Betti number is either smaller or equal to or equal to . The latter case appears if and only if the GKM graph of is combinatorially equivalent to the vertex-edge graph of .

Note that the upper bound on the total Betti number is sharp and better than the upper bound conjectured by Gromov in [Gro81] for general non-negatively curved manifolds of dimension .

By the GKM Theorem, the GKM graph determines the rational cohomology of a GKM orbifold. Therefore, if we can show that all GKM graphs appearing in the above theorem can be realised as GKM graphs of certain model GKM orbifolds, any non-negatively curved GKM orbifold will have rational cohomology isomorphic to the rational cohomology of one of the model orbifolds.

To construct the models, we have to show that GKM graphs with underlying graph equal to the vertex-edge graph of extend — in the sense of Kuroki [Kur15] — to GKM graphs, i.e. to GKM graphs of torus orbifolds over . This reasoning then leads to

###### Theorem 1.3.

If is a GKM orbifold such that the GKM graph of is the vertex-edge graph of a product , then the rational cohomology of is isomorphic to the rational cohomology of a non-negatively curved torus orbifold.

To get the models in the general case, we show that the deck transformation group of the covering from Theorem 1.1 acts on the model torus orbifold associated to the extended GKM graph as in Theorem 1.3. The quotient is a GKM orbifold realising the GKM graph . Hence we get

###### Theorem 1.4.

Let be a non-negatively curved GKM orbifold. Then there is a non-negatively curved torus orbifold and an isometric action of a finite group on such that

 H∗(O;Q)≅H∗(~O/G;Q).

Moreover, if is a manifold then is a simply connected manifold.

In the literature it is often assumed that a GKM manifold has an invariant almost complex structure. This assumption results in the fact that in this case the weights of the GKM graph have preferred signs. We consider this special case in Section 7. We show that in the situation of Theorem 1.1 this implies that the covering of is trivial, and that the covering graph is the vertex edge graph of . Moreover, the torus manifold corresponding to the extended GKM graph will also admit an invariant almost complex structure. Torus manifolds over admitting an invariant almost complex structure were classified in [CMS10a]. They are all diffeomorphic to so-called generalised Bott manifolds. Hence we get

###### Theorem 1.5.

Let be a non-negatively curved GKM manifold which admits an invariant almost complex structure. Then the rational cohomology ring of is isomorphic to the rational cohomology ring of a generalised Bott manifold.

Note here that the structure of the cohomology ring of a generalised Bott manifold is relatively simple (see for example Section 3 in [CMS10b]).

In [Wie15] a classification of non-negatively curved simply connected torus manifolds was given. A crucial step in the proof was to show that the orbit space of such a torus manifold is combinatorially equivalent to a product . The proof of this intermediate result was very long and highly technical. With the methods of the paper at hand we can give a short conceptional proof of this result.

The Bott conjecture asks if any simply connected non-negatively curved manifold or orbifold is rationally elliptic. Therefore it is natural to consider the question if the above theorems also hold for rationally elliptic GKM orbifolds. By [GGKRW14], the two-dimensional faces of the corresponding GKM graphs have at most four vertices. Moreover, since our arguments are purely graph-theoretic we conclude that all the above theorems also hold for rationally elliptic GKM orbifolds instead of non-negatively curved ones.

The remaining sections of this paper are structured as follows. In Section 2 we gather background material about GKM manifolds and orbifolds as well as on torus manifolds and orbifolds. Then in Section 3 we prove Theorem 1.1.

In Section 4 we show that a GKM graph with underlying graph the vertex edge graph of a product always extends to a GKM graph with . This is then used in Section 5 to prove Theorems 1.3 and 1.4.

In Section 6 we give an example of a non-negatively curved GKM manifold which does not have the same cohomology as a torus manifold.

In Section 7 we consider GKM manifolds with invariant almost complex structure and prove Theorem 1.5. Moreover, in the last Section 8 we give a short proof of the “big lemma” which is used in the classification of non-negatively curved torus manifolds.

Throughout, cohomology will be taken with rational coefficients.

## 2 Preliminaries

### 2.1 GKM manifolds

We begin with a review of GKM theory for manifolds; below we will describe the changes that are necessary to treat orbifolds as well.

Consider an effective action of a compact torus on a smooth, compact, orientable manifold of dimension with finite fixed point set, such that the one-skeleton

 M1={p∈M∣dimTp≤1}

of the action is a union of invariant -spheres. Given that the fixed point set is finite, the second condition is equivalent to the condition that for every fixed point, the weights of the isotropy representation are pairwise linearly independent. To such an action one associates its GKM graph: its vertices are given by the fixed points of the action; to any invariant -sphere – which contains exactly two fixed points – one associates an edge connecting the corresponding vertices. Finally, any edge is labeled with the weight of the isotropy representation in any of the two fixed points which corresponds to the two-dimensional submodule given by the tangent space of this sphere. These labels are linear forms on the Lie algebra of well-defined up to sign.

We need to abstract from group actions and consider the occurring graphs detached from any geometric situation, as in [GZ01] or [BGH02].

For a graph we denote its set of vertices by and its set of edges by . We consider only graphs with finite vertex and edge set, but we allow multiple edges between vertices. Edges are oriented; for we denote by its initial vertex and by its terminal vertex. The edge , with opposite orientation, is denoted . For a vertex we denote the set of edges starting at by .

###### Definition 2.1.

Let be a graph. Then a connection on is a collection of maps , for , such that

1. and

2. , for all .

###### Definition 2.2.

Let . A GKM graph (GKM graph for ) consists of an -valent connected graph , a map and a connection on , such that

1. If are edges of which meet in a vertex of then the , , are linearly independent. Here denotes a lift of . Note that this property is independent of the choice of .

2. For any two edges which meet in a vertex there are such that

 ^α(∇e1(e2))=p^α(e2)+q^α(e1). (2.1)

Note here that are determined up to sign by . Moreover, if we fix a sign for , , , then are uniquely determined.

3. For each edge we have .

The construction of the graph described above always results in GKM graphs:

###### Proposition 2.3.

For any action of a compact torus on a smooth, compact, orientable manifold of dimension with finite fixed point set, and whose one-skeleton is the union of invariant -spheres, the graph associated to the action in the way prescribed above canonically admits a connection for which Equation (2.1) holds, with and an integer. In particular the graph is a GKM graph.

Definition 2.2 is slightly more general than usual, as and are allowed to be rational numbers. The reason will become clear below, when we consider orbifolds.

###### Remark 2.4.

A -invariant almost complex structure on a manifold allows to speak about weights of the isotropy representation that are well-defined elements of , not only up to sign. On the level of graphs we say that a GKM graph admits an invariant almost complex structure if there is a lift of such that (2.1) holds with and an integer and , for all edges .

The relevance of this type of actions is founded in the fact that for manifolds with vanishing odd-dimensional (rational) cohomology, the (equivariant) cohomology is determined by the associated GKM graph. We define

###### Definition 2.5.

We say that an action of a compact torus on a smooth, compact, orientable manifold is of type GKM (simply GKM for ) if , the fixed point set is finite, and for every fixed point any weights of the isotropy representation are linearly independent.

###### Theorem 2.6 ([Gkm98, Theorem 7.2]).

For a -action of GKM type with fixed points , the inclusion induces an injection

 H∗T(M)→H∗T(MT)=r⨁i=1S(t∗)

whose image is given by the set of tuples such that whenever and are joined by an edge with label .

Obviously, the image in particular depends only on the labelled graph , so it is sensible to use the notation for the -subalgebra of defined in the theorem above.

It is well-known that the vanishing of the odd rational cohomology implies that the canonical map is surjective. In particular, the rational cohomology ring of is determined by the GKM graph of the action.

### 2.2 GKM orbifolds

The fact that GKM-theory works equally well for torus actions on orbifolds was already remarked in [GZ01, Section 1.2]. However, they considered only orbifolds that arise as global quotients of locally free Lie group actions. In this paper we consider general orbifolds , which are given by orbifold atlases on a topological space, consisting of good local charts for any point . For the precise definition, and all basics on orbifolds and Lie group actions we use Section 2 of [GGKRW14] as a general reference.

We consider an action of a torus on a compact, orientable orbifold , in the sense of [GGKRW14, Definition 2.10]. In [GGKRW14] it is shown that orbits, as well as components of fixed point sets , where is a connected Lie subgroup, are strong suborbifolds of . Moreover, for any , with good local chart there is an extension of by , acting on . The -action fixes the single point in the preimage . We thus obtain a well-defined isotropy representation of on . Its restriction to the identity component of has well-defined weights , which we consider, via the projection , as elements in .

With this definition of weights, Definition 2.5 applies to torus actions on orbifolds equally well, and we can speak about torus actions on orbifolds of type GKM. To any such action we can associate an -valent graph as in the case of manifolds, because any non-trivial torus action on a two-dimensional compact, orientable orbifold with a fixed point has exactly two fixed points, see [GGKRW14, Lemma 3.9]. For the labelling, we rescale the weights as follows: for a weight at a fixed point , the intersection of with the integer lattice in is isomorphic to . We let be a generator of this group, and be the number of components of the principal isotropy group of the -action on the -sphere to which the weight space of is tangent. We define .

We remark that the factor is irrelevant for what follows: we include it in order for the GKM graph to encode the full isotropy groups. Considering as a homomorphism , its kernel is precisely the principal isotropy group of the corresponding -sphere.

In order to construct a connection on , we now restrict to actions of type GKM. Let be a vertex of the graph, corresponding to the fixed point , and an edge with , with label . Let be the fixed point corresponding to . For any other edge at , with weight , we consider the connected subgroup with Lie algebra . By the GKM-condition and the slice theorem for actions on orbifolds [GGKRW14, Theorem 2.18] the connected component of is a four-dimensional strong suborbifold of . It contains , and there exists an edge with , with weight , such that . We define a connection on by . By construction, is a (rational) linear combination of and , so that Equation (2.1) holds.

Also Theorem 2.6 holds true for GKM actions on orbifolds. This was (for torus orbifolds) already observed in [GGKRW14, Theorem 4.2].

### 2.3 Torus manifolds and orbifolds

Here we gather the facts we need to know about torus manifolds and torus orbifolds. General references for the constructions used here are [BP02], [BP15], [MP06], [DJ91], [GGKRW14].

We start with a general construction of such manifolds and orbifolds. An -dimensional manifold with corners is called nice if at each vertex of there meet exactly facets of , that is exactly codimension-one faces. The faces of ordered by inclusion form a poset , the so-called face poset of . We also denote by the set of facets of .

Now let be a nice manifold with corners with only contractible faces, and assume that there is a map such that for every vertex of ,

 λ(F1),…,λ(Fn)

are linearly independent, where the are the facets of which meet in .

Then we can construct a torus orbifold, i.e. an orientable -dimensional orbifold with an action of the -dimensional torus with , such that the orbit space of the action on is homeomorphic to . The orbifold is defined as

 O=(P×Rn/Zn)/∼,

where if and only if and is contained in the -span of all the with . Here acts on the second factor of . Note that replacing the by nonzero multiples does not change the associated orbifold .

If for every vertex the of the facets which meet at form a basis of , then is a manifold and the -action is locally standard, i.e., locally modelled on effective -dimensional complex representations of .

The preimages of the facets of under the orbit map are invariant suborbifolds of codimension two in . Therefore their equivariant Poincaré duals are defined. Moreover, these Poincaré duals form a basis of because the faces of are contractible (see [MP06] and [PS10]).

The contractibility of the faces of also implies that the rational cohomology of a torus orbifold as above is concentrated in even degrees.

Equivalently to giving the labels of the facets, one can also define by a labeling of the edges of in such a way that for edges meeting a vertex, is the basis of dual to . In this way the vertex-edge graph of becomes a so-called torus graph (see [MMP07]), i.e. a GKM graph of a torus manifold or orbifold.

Similar to the definition of the torus orbifold associated to the pair one can associate a moment angular manifold to . This goes as follows.

Denote the facets of by and for each let be a copy of the circle group. Then define

 ZP=(P×(m∏i=1S1i))/∼,

where if and only if and

 t′t−1∈∏i;x∈FiS1i⊂m∏i=1S1i.

There is an action of on induced by multiplication on the second factor. Moreover the torus orbifold from above is the quotient of the action of the kernel of a homomorphism . Here is defined by the condition that its restriction to induces an isomorphism .

Note that if is a product of simplices and quotients , then is a product of spheres. We can equip this product with the product metric of the round spheres. If we do so, becomes identified with a maximal torus of the isometry group of .

## 3 Coverings of GKM graphs

In this section we construct a covering of a GKM graph with small three-dimensional faces by the vertex edge graph of a product . We start with the definition of what we mean by the faces of a graph.

###### Definition 3.1.

Let be a graph with a connection . Then an -dimensional face of is a connected -valent -invariant subgraph of .

###### Lemma 3.2.

Let be a GKM-graph, where . Then for any vertex of and any edges that meet at , where , there exists a unique -dimensional face of that contains .

###### Proof.

Let be the (-dimensional) span of the in . Consider the subgraph of that consists of those edges whose labeling is contained in , and let be its connected component of . We claim that this subgraph is -invariant and -valent.

As is GKM, with , the only edges at contained in are . Moreover, whenever is an -valent vertex of and with , then also is -valent. In fact, (2.1) shows that for any edge at in , also is an edge of , and the GKM property of shows that there is no further edge at contained in . ∎

###### Lemma 3.3.

Let be a GKM manifold or orbifold which admits an invariant metric of non-negative curvature or is rationally elliptic. Then each two-dimensional face of the GKM graph of has at most four vertices.

###### Proof.

The two dimensional faces of the GKM graph of are GKM graphs of four-dimensional invariant totally geodesic suborbifolds of . Hence the claim follows in the same way as in the proof of Proposition 5.1 (e) in [GGKRW14]. ∎

###### Lemma 3.4.

Let be a GKM orbifold with an invariant metric of non-negative curvature. Then each three-dimensional face of the GKM graph of has one of the following combinatorial types: , , , , .

###### Proof.

As in the proof of the previous lemma one sees that the three-dimensional faces of the GKM graph of are GKM graphs of six dimensional torus manifolds . such that all two-dimensional faces of the orbit spaces have at most four vertices.

By considering all the possible cases for the orbit spaces of one gets the conclusion. To do so assume first that a two-dimensional face of is combinatorially equivalent to . Let be another face which has an edge with in common. If is also of type then is of type . Moreover, if is of type , then, using the -valence of the graph of one easily sees that is of type .

Next assume that is of type . Then let be the vertex of which does not belong to and and be the other vertices. Then one sees by considering the faces which meet at that the two edges and belong to two different faces which both contain all three vertices, as in the first graph in Figure 1. But any two edges at must span a unique face, a contradiction. Hence this case does not appear.

We now assume that is of type . If all other faces of which have an edge with in common are also of type , then must be of type . If all faces which have an edge with in common are of type , then is of type .

Therefore we have to exclude the case that there is a face of type and a face of type such that is a vertex, as in the second graph in Figure 1.

In this case the third face which has an edge with in common must be of type . Using the -valence of the graph one gets now a contradiction, because the vertex of which is not contained in or is also contained in . Moreover two of the three edges which meet at this vertex are contained in the intersection of and .

In the remaining case that all two-dimensional faces of are equivalent to , is of type . ∎

###### Definition 3.5.

Let be a connected graph with a connection . We say that is a graph with small three-dimensional faces if the following conditions hold true:

1. For any and distinct edges meeting at , there exits a unique -dimensional face of containing and .

2. The conclusion of Lemma 3.4 holds true, i.e., any three-dimensional face of has the combinatorial type of , , or (see Figure 2).

###### Definition 3.6.

Let , where is a graph with small three-dimensional faces. We call a subgraph a maximal simplex at if contains , has the combinatorial type of or and is maximal with these two properties.

Lemma 3.4 thus says that the GKM graph of a connected nonnegatively curved GKM orbifold is a graph with small three-dimensional faces.

###### Lemma 3.7.

Let be a graph with small three-dimensional faces. For two edges emanating from we have:

1. and belong to the same maximal simplex of type if and only if and span a triangle.

2. and belong to the same maximal simplex of type if and only if and span a biangle.

3. and do not belong to the same maximal simplex if and only and span a square.

###### Lemma 3.8.

Let be a graph with small three-dimensional faces. Then the maximal simplices at are partitioning in such a way that each contains the edges which span a maximal simplex at .

###### Proof.

We have to show that belonging to the same maximal simplex is an equivalence relation on . We only have to show transitivity.

Let such that and belong to one maximal simplex and and belong to another maximal simplex.

Then, by Lemma 3.7, and and and span a triangle or a biangle, respectively.

Consider the three-dimensional face of spanned by . It has one of the combinatorial types described in Definition 3.5. Since none of the faces spanned by and are squares, it follows that has the combinatorial type of or . Hence, we have shown transitivity and the claim follows. ∎

###### Lemma 3.9.

Let be a graph with small three-dimensional faces and let be an oriented edge of . Then the connection preserves the partitions and . Moreover, the combinatorial types of the maximal simplices spanned by and are the same.

###### Proof.

Let be other edges of emanating from . By Lemma 3.7, we have to show that the two-dimensional faces spanned by and , respectively, have the same combinatorial types.

To do so, we consider the three-dimensional face of spanned by . It contains the two-dimensional faces spanned by and , respectively. Moreover, these two-dimensional faces are also spanned by and , respectively. Hence the claim follows from the list of combinatorial types of three-dimensional faces given in Definition 3.5. ∎

###### Lemma 3.10.

Let be a graph with small three-dimensional faces. Let and be edges in with the same initial point.

1. If and span a biangle, then .

2. If and span a square, then , where are the edges opposite to and , respectively, in the square spanned by and .

###### Proof.

To see the first claim, one has to show that if span a biangle and is an edge with the same initial point as and then

 ∇ef=∇e′f.

To see this first assume that and span a biangle, then by Definition 3.5, , , span a face of which is combinatorially equivalent to . Hence follows.

Next assume that and span a square, then, by Definition 3.5, , , span a face with the combinatorial type of . Hence the claim follows in this case.

By Definition 3.5, the case that and span a triangle does not occur. So the claim follows.

The second claim follows in a similar way, again by considering the three-dimensional faces of . ∎

The following theorem states that any graph with small three-dimensional faces is covered by a product of simplices. Here by a covering of a graph by another graph we mean the following: We consider graphs as one-dimensional CW-complexes, and coverings should be cellular. Note that the graphs we consider are -valent, for some ; for a covering of -valent graphs is automatically cellular, because in this case the neighbourhoods of points in the interior of a one-cell and the neighbourhoods of the vertices are not homeomorphic.

###### Theorem 3.11.

Let be a graph with small three-dimensional faces, , and the maximal simplices at . Consider the product graph , equipped with its natural connection . Let be a base point, and a bijection sending the edges of a maximal simplex to the edges of a maximal simplex. Then there exists a unique covering extending that is compatible with the connections, i.e., which satisfies for all edges .

###### Proof.

The compatibility condition shows that if are edges meeting at some vertex , and and are given, then is uniquely determined. This implies the uniqueness of .

We have to show the existence of . Note that for any path in , say from to , the connections and on and induce bijections

 ∇π(γ):E(Γ)x→E(Γ)π(y)

and

 ~∇γ:E(~Γ)x0→E(~Γ)y.

For , we let be the subgraph of whose vertices are those that have distance at most to , and whose edges are all the edges of connecting two such vertices. We prove by induction that we can construct a map of graphs extending that satisfies

 ∇π(γ)∘π(e)=π∘~∇γe

for all shortest paths in starting at , and all edges of with . For there is nothing to do.

We assume that is already constructed, and wish to construct . Let be an edge of which is not an edge of , but whose initial vertex is a vertex of , and choose a shortest path from to . Note that is a path in . We want to define ; in order to do so we have to show that this definition is independent of the choice of .

If are vertices of , then the shortest paths between and are of the following form:

 ei1∗⋯∗eil,

where

• is an edge tangent to

• the are unique up to ordering and if

• If , then is unique

• If , , then by Lemma 3.10, does not depend on the choice of edge in .

By Lemma 3.10, we have

 ~∇eij∘~∇eij+1=~∇e′ij+1∘~∇e′ij,

and

 ∇π(eij)∘∇π(eij+1)=∇π(e′ij+1)∘∇e′ij,

where and are the opposite edges to and , respectively, in the square spanned by and . Hence and do not depend on the chosen shortest path from to .

Next, we show that for two such edges with initial point in and same end point , which is not a vertex of , we have . To do so let and be two minimising curves from to and , respectively. Then

 γ ∗e γ′ ∗e′

are minimising curves from to . Therefore they coincide up to ordering of the edges (replacing edges by parallel edges) and possible disambiguity with multiple edges.

At first assume that ; then and span a biangle. Moreover we can, by induction hypothesis, assume that . Hence it follows from Lemma 3.9 that and have the same end points.

Next assume that . Then we may assume

 γ =γ1∗~e′ γ′ =γ1∗~e,

where is some minimising path and , are parallel to and respectively. Hence and span a square at the end point of . Hence it follows from Lemma 3.9 that also and span a square in . Therefore it follows that and have the same end points.

Finally we have to consider edges of such that both and are not vertices in . In this case there are shortest paths from to , as well as to , of the following form:

 γ1 ∗e1 γ1 ∗f1,

where the edges in the two paths satisfy the same relations as above and and are tangent to the same factor. This is because if two vertices of are connected by an edge, then they only differ in one coordinate. We have to show that the two possible definitions and for the image of are compatible.

Since the length of the two paths are the same and they are minimising, it follows that and span a triangle, with as third edge.

Then we have, using the induction hypothesis,

 ∇π(γ1∗e1)π(~∇−1γ1∗e1e) =∇π(e1)∇π(γ1)π(~∇−1γ1~∇−1e1(e)) =∇π(e1)∇π(γ1)π(~∇−1γ1f1)=∇π(e1)π(f1),

and analogously,

 ∇π(γ1∗f1)π(~∇−1γ1∗f1¯e)=∇π(f1)π(e1).

Thus we are done if we can show that and span a triangle in . But this follows from Lemma 3.9.

We have thus shown that is a well-defined map of graphs. Next, we confirm that is compatible with the connections and .

By construction has the following properties:

• maps two-dimensional faces of of a given type () to a face of of the same type.

• If is an edge of which is part of a shortest path to the base point, then we have

 ∇π(e)∘π=π∘~∇e,

i.e. is compatible with and .

Hence, it only remains to be shown that is also compatible with the connection of those edges which are not part of a minimising path to the base point. These edges are tangent to factors , and are opposite to the closest point to . Then together with two edges which connect with the initial and end point of form a triangle . Note that and are part of minimising paths to . Therefore for an edge starting at the same point as and not tangent to we have:

 π(~∇ee′)=π(~∇f2∘~∇¯f1e′)=∇π(f2)∘∇π(¯f1)∘π(e′)=∇π(e)∘π(e′),

because form a triangle in .

Moreover, by the same reason, we have

 π(~∇e¯f1)=π(¯f1)=∇π(e)π(¯f2).

Hence is compatible with the connections. Finally it follows that is a covering. ∎

This theorem directly implies Theorem 1.1:

###### Proof of Theorem 1.1.

The GKM graph of a GKM orbifold with an invariant metric of nonnegative curvature is a graph with small three-dimensional faces. By Theorem 3.11, any such graph is finitely covered by the vertex-edge graph of a finite product of simplices. ∎

###### Definition 3.12.

Let be a graph with small three-dimensional faces, , and a covering as in Theorem 3.11. Then a deck transformation of is an automorphism such that and for all edges .

Clearly, the deck transformations of form a group.

###### Proposition 3.13.

The covering is Galois, i.e., the deck transformation group of acts simply transitively on the fibers of .

###### Proof.

Let such that . The covering induces a bijection

 f:E(~Γ)xπ⟶E(Γ)zπ−1⟶E(~Γ)y.

We first claim that respects the combinatorial types of the maximal simplices at and . In fact, this property is clear for . At the vertex , the covering necessarily maps a maximal simplex attached to to a simplex of the same type. As the combinatorial structure of the simplices at is, by Lemma 3.9, the same as that of the simplices at , and hence also the same as that of the simplices at , it follows that has to respect the combinatorial structures of the maximal simplices at as well.

Thus Theorem 3.11, applied to , implies that extends uniquely to an automorphism respecting the natural connection of . By the uniqueness statement of Theorem 3.11, the maps and are identical, hence is a deck transformation.

This shows that the deck transformation group acts transitively on the fibers of . The uniqueness statement of Theorem 3.11 implies that the action of the deck transformation group on the fibers is also free. ∎

Now let be the GKM graph of a nonnegatively curved GKM manifold, and as above. Using , we pull back the labeling of ; in this way becomes a GKM graph. By construction, the deck transformation group of leaves invariant the labeling of .

## 4 Extending GKM graphs

We say that a GKM graph is effective if at each vertex , we have that generate , where are the edges emanating from .

Note that every GKM graph reduces to an effective GKM graph.

###### Definition 4.1.

Let . We consider two effective GKM graphs with the same underlying graph with connection, and , for an - respectively -dimensional torus respectively . If there exists a linear map , such that , then we say that is an extension of .

Recall that is only well-defined up to signs, i.e., is a map . The composition is thus also well-defined only up to signs.

Because a linear map sends linearly dependent vectors to linearly dependent vectors, any extension of a GKM graph is again GKM.

Let be an extension of an effective GKM graph in the sense of Definition 4.1. If are two edges that meet in a vertex , then by (2.1)

 ^β(∇e1(e2)) =q′^β(e1)+p′^β(e2), ^β(∇e1(e2)) =q^β(e1)+p^β(e2),

for some . If we have chosen the signs of in such a way that and , then we see by applying to the first equation and by using the -independence of the weights that and .

From this argument we get a necessary and sufficient condition for extending an -valent GKM graph to a GKM graph. This goes as follows.

At first choose a basis of and a base point of . Let be the edges with initial point . We want to define . Hence must be defined by the equations , for some choices of signs . As the weights on the edges emanating from are defined, we next want to define the weights for the edges emanating from a vertex connected to by one edge, say .

As in the above computation for these edges , the numbers such that

 ^α(∇e1(ej))=q^α(e1)+p^α(ej)

(for an arbitrary choice of sign of ) also have to satisfy

 ^β(∇e1(ej))=q^β(e1)+p^β(ej),

where the sign of is chosen such that