https://doi.org/10.1007/s00209-021-02853-0
Mathematische Zeitschrift
Non-negatively curved GKM orbifolds
Oliver Goertsches1·Michael Wiemeler2
Received: 26 January 2021 / Accepted: 20 July 2021 / Published online: 2 September 2021
© The Author(s) 2021
Abstract
In this paper we study non-negatively curved and rationally elliptic GKM4 manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational coho- mology 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 man- ifolds. This characterisation was originally proved in Wiemeler (J Lond Math Soc 91(3):
667–692, 2015) and was used there to obtain a classification of non-negatively curved torus manifolds.
Keywords Non-negative curvature·Rationally elliptic orbifolds·GKM orbifolds
1 Introduction
The classification of Riemannian manifolds with positive or non-negative sectional curvature is one of the most prominent open problems in differential geometry. Many authors have investigated it under the additional assumption of an isometric Lie group action, e.g. [1,8, 9,11–13,16,19,29,32,33]. In this paper we continue our investigation [17] of isometric torus actions of GKM type on Riemannian manifolds with sectional curvature bounded from below.
The GKM condition—introduced in [14]—requires that the one-skeletonM1of the action, i.e. the union of all orbits of dimension less than or equal to one, is of a particularly simple type.
Namely, it is required thatM1is a union of two-dimensional spheres, such that theT-action restricts to a cohomogeneity one action on each two-sphere. The orbit space= M1/T is then ann-valent graph, where 2n=dimM. The isotropy representations at the fixed points induce a labeling of the edges of the graph, as explained in Sect.2.
MW was supported by DFG-Grants HA 3160/6-1 and HA 3160/11-1 and SFB 878.
B
Oliver Goertschesgoertsch@mathematik.uni-marburg.de Michael Wiemeler
wiemelerm@uni-muenster.de
1 Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein-Straße, 35032 Marburg, Germany
2 Mathematisches Institut, WWU Münster Einsteinstraße 62, 48149 Münster, Germany
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 [14], see Theorem2.8below.
Similarly to the GKM condition, we say that an action is GKMk if for all 0 ≤k <k the union of the orbits of dimension at most k is a union of 2k-dimensional invariant submanifolds. Their GKM graphs are then thek-dimensional faces of the GKM graph of M.
In [17] we showed that a positively curved Riemannian manifold admitting an isometric GKM3 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 4-dimensional positively curvedT2-manifolds [16], 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 [19,29], 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 O be a orientable GKM4orbifold with an invariant metric of non-negative curvature. Then the GKM graph of O is finitely covered by the vertex-edge graph of a finite product
ini×
imi.
In the above theorem and later on,ndenotes ann-dimensional simplex andmthe orbit space of the linear effective action of them-dimensional torus onS2m.
The stronger GKM4condition implies that the GKM graph ofOhas three-dimensional faces. The restriction on the two-dimensional faces of the graph imply that the combinatorial types of the three-dimensional faces is also very restricted. They are all combinatorially equivalent to one of the following:I3, 3, 3, 2×I, 2×I.
Using these restrictions, we show that the combinatorial type of a neighborhood of a vertex in the GKM graphof a non-negatively curved GKM4 manifold is the same as that of a neighborhood of a vertex in the vertex edge graph˜ of a finite product
ini ×
imi. Extending this local result to all of˜ then yields the covering described in Theorem1.1.
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 O of dimension2n be as in the previous theorem, then the total Betti number b(O)=
ibi(O)is either smaller than or equal to2n−2·3or equal to2n. The latter case appears if and only if the GKM graph of O is combinatorially equivalent to the vertex-edge graph of In.
Note that the upper bound on the total Betti number is sharp. Therefore we are verifying a conjecture of Gromov [15] in the special case of non-negatively curved GKM4-manifolds.
He conjectured that for general non-negatively curved manifolds of dimensiondthe total Betti number is bounded from above by 2d.
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 GKM4graphs with underlying graph equal to the vertex-edge graph of
ini ×
imi extend — in the sense of Kuroki [23] — to
GKMngraphs, i.e. to GKM graphs of torus orbifolds over
ini×
imi. This reasoning then leads to
Theorem 1.3 If O is a GKM4 orbifold such that the GKM graph of O is the vertex-edge graph of a product
ini ×
imi, then the rational cohomology of O 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 groupGof the covering˜ →from Theorem1.1acts on the model torus orbifoldO˜associated to the extended GKM graph˜ as in Theorem1.3. The quotientO˜/Gis a GKM4orbifold realising the GKM graph. Hence we get
Theorem 1.4 Let O be a non-negatively curved GKM4 orbifold. Then there is a non- negatively curved torus orbifoldO and an isometric action of a finite group G on˜ O such˜ that
H∗(O;Q)∼=H∗(O/G˜ ;Q).
Moreover, if O is a manifold thenO 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 Sect.7. We show that in the situation of Theorem1.1this implies that the covering ofis trivial, and that the covering graph is the vertex edge graph of
ini. Moreover, the torus manifold corresponding to the extended GKMn graph will also admit an invariant almost complex structure. Torus manifolds over
ini admitting an invariant almost complex structure were classified in [6]. They are all diffeomorphic to so-called generalised Bott manifolds. These manifoldsXare total spaces of iteratedCPni-bundles
X=Xk→Xk−1→ · · · →X1 →X0= {pt},
where eachXiis the total space of the projectivisation of a Whitney sum ofni+1 complex line bundles overXi−1. Their cohomology rings can be easily determined from the Chern classes of the involved line bundles. Indeed, if P(E)is the projectivisation of a complex vector bundleEof dimensionnover a base spaceB, then we have
H∗(P(E);Z)∼=H∗(B)[x]/(f(x)), wherexhas degree two and f(x)=n
i=0ci(E)xn+1−i. Hereci(E)denotes thei-th Chern class ofE. By iterating this formula one gets the cohomology rings of all generalised Bott manifolds. By the above discussion we get
Theorem 1.5 Let M be a non-negatively curved GKM4manifold which admits an invariant almost complex structure. Then the rational cohomology ring of M is isomorphic to the rational cohomology ring of a generalised Bott manifold.
In [32] 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
ini×
imi. 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 conceptual proof of this result.
The Bott conjecture asks if any simply connected non-negatively curved manifold is rationally elliptic. Therefore it is natural to consider the question if the above theorems also hold for rationally elliptic GKM4manifolds or orbifolds.
By [11], 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 GKM4 orbifolds instead of non-negatively curved ones.
The remaining sections of this paper are structured as follows. In Sect.2we gather back- ground material about GKM manifolds and orbifolds as well as on torus manifolds and orbifolds. Then in Sect.3we prove Theorem1.1.
In Sect.4we show that a GKM4graph with underlying graph the vertex edge graph of a product
ini×
imi always extends to a GKMngraph withn=
ini+
imi. This is then used in Sect.5to prove Theorems1.3and1.4.
In Sect.6we give an example of a non-negatively curved GKM4manifold which does not have the same rational cohomology as a torus manifold. This shows that the groupGfrom Theorem1.4cannot be omitted.
In Sect.7we consider GKM manifolds with invariant almost complex structure and prove Theorem1.5. Moreover, in the last Sect.8we 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.
We would like to thank the anonymous referee for comments which helped to improve the presentation of this paper.
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 T on a smooth, compact, orientable manifoldMof dimension 2nwith finite fixed point set, such that the one-skeleton
M1= {p∈M|dimT p≤1}
of the action is a union of invariant 2-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 2-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 algebratofT and are well-defined up to sign.
We need to abstract from group actions and consider the occurring graphs detached from any geometric situation, as in [18] or [3].
For a graphwe denote its set of vertices byV()and its set of edges by E(). We consider only graphs with finite vertex and edge set, but we allow multiple edges between vertices. Edges are oriented; fore∈E()we denote byi(e)its initial vertex and byt(e)its terminal vertex. The edgee, with opposite orientation, is denotede. For a vertex¯ v∈V() we denote the set of edges starting atvbyEv.
Definition 2.1 Letbe a graph. Then aconnectionon is a collection of bijective maps
∇e:Ei(e)→Et(e), fore∈E(), such that 1. ∇e(e)= ¯eand
2. ∇e¯= ∇e−1, for alle∈E().
Definition 2.2 Letk ≥ 2. AGKMk graph(GKM graphfork = 2)(, α)consists of an n-valent connected graphand a mapα:E()→H2(BTm)/{±1}such that
1. Ife1, . . . ,ekare edges ofwhich meet in a vertexvofthen theα(eˆ i),i =1, . . . ,k, are linearly independent. Hereα(eˆ i)∈ H2(BTm)denotes a lift ofα(ei). Note that this property is independent of the choice ofα(eˆ i).
2. There exists a connection∇onsuch that for any two edgese1,e2which meet in a vertex vthere arep,q∈Qsuch that
ˆ
α(∇e1(e2))= pα(ˆ e2)+qα(ˆ e1). (2.1) Note here that p,q are determined up to sign by α. Moreover, if we fix a sign for
ˆ
α(∇e1(e2)),α(eˆ 2), andα(eˆ 1), thenp,qare uniquely determined.
3. For each edgeewe haveα(e)=α(¯e).
Definition2.2is slightly more general than usual, as pandqare allowed to be rational numbers. The reason will become clear below, when we consider orbifolds. The construction of the graph described above always results in GKM graphs:
Proposition 2.3 For any action of a compact torus T on a smooth, compact, orientable manifold M of dimension2n with finite fixed point set, and whose one-skeleton M1is the union of invariant2-spheres, the graph associated to the action in the way prescribed above admits a connection for which Eq.(2.1)holds, with p= ±1and q an integer. In particular the graph is a GKM graph.
Moreover, if the weights at every vertex are3-independent, i.e. if any three weights at every vertex are linearly independent, then the connection is unique.
Proof Let N be one of the invariant 2-spheres andTN the principal isotropy group of the T-action onN. Moreover, letx1,x2be the twoT-fixed points inN. Then we have twoT- representationsTx1M⊗CandTx2M⊗Con the complexified tangent spaces at these fixed points. Let
TxiM⊗C=
j
Wi j Wi j = {v∈TxiM⊗C; tv=αi j(t)vfor allt∈T}, be the decomposition ofTxiM⊗Cinto weight spaces. Here theαi jare some homomorphisms T → S1. By the GKM condition each Wi j has complex dimension one. Moreover, the derivatives of theαi jagree –up to sign– with the weights of the edges starting inxi.
Since theTN-action onN is trivial, it follows from the proof of Proposition 2.2 of [30]
thatT M⊗C|NsplitsTN-equivariantly as T M⊗C|N ∼=
k i=1
Vi⊗Ei,
where the Vi are complex irreducible TN-representations and the Ei are complex vector bundles with trivialTN-action. Therefore the isomorphism type of theTN-representation
on TxM ⊗C is independent of x ∈ N. In particular, there is an isomorphism of TN- representations
Tx1M⊗C∼=Tx2M⊗C.
So the homomorphismsα1j andα2j agree after restriction toTN(and reordering).
BecauseTN has codimension one inT there is a homomorphismα :T → S1 –unique up to complex conjugation onS1 ⊂ C– such that kerα = TN. By the definition ofTN, thisα–or its conjugateα¯– induces theT-action onTxiN ∼=C,i = 1,2. Moreover, every homomorphismT →S1which is trivial onTN factors throughα.
We now apply this toα1j· ¯α2j, where·and¯denote multiplication and complex conjugation inS1, respectively. So we get some factorisation
α1j· ¯α2j =αkj (2.2)
Forming the derivative of this expression leads to equation (2.1) with p = ±1 andq =
±kj ∈ Z. Moreover, in case the weights atx2are 3-independent, for givenα1j there is at most oneα2j for which equation (2.2) holds. So our claim follows.
Remark 2.4 An alternative proof of this proposition was given in [18, p. 6]; there, the fact thatq∈Zfollowed from the Atiyah–Bott–Berline–Vergne localization theorem.
Remark 2.5 For GKM3-graphs the connection∇in Condition 2 of2.2is unique.
Remark 2.6 AT-invariant almost complex structure on a manifoldMallows to speak about weights of the isotropy representation that are well-defined elements oft∗, 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 liftαˆ :E()→H2(BTm)ofαsuch that (2.1) holds withp=1 and qan integer andα(e)ˆ = − ˆα(e), for all edges¯ e.
The relevance of this type of actions is grounded in the fact that for manifoldsMwith van- ishing odd-dimensional (rational) cohomology, the (equivariant) cohomology is determined by the associated GKM graph. We define
Definition 2.7 We say that an action of a compact torusT on a smooth, compact, orientable manifoldMisof type GKMk(simplyGKMfork=2) ifHodd(M)=0, the fixed point set is finite, and for every fixed point anyk weights of the isotropy representation are linearly independent.
Theorem 2.8 [14, Theorem 7.2]For a T -action of GKM type with fixed points p1, . . . ,pr, the inclusion MT →M induces an injection
HT∗(M)→HT∗(MT)= r
i=1
H∗(BT) whose image is given by the set of tuples(fi)∈r
i=1H∗(BT)such that fi−fj=αβwith someβ∈H∗(BT)whenever piand pjare joined by an edge with labelα.
Obviously, the image in particular depends only on the labelled graph, so it is sensible to use the notationHT∗()for theH∗(BT)-subalgebra of
v∈V()H∗(BT)defined in the theorem above.
It is well-known that the vanishing of the odd rational cohomology implies that the canon- ical mapHT∗(M)→H∗(M)is surjective. In particular, the rational cohomology ring ofM 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 [18, Sect. 1.2]. In this paper we consider general (orientable) orbifoldsO, which are given by orbifold atlases on a topological space, consisting of good local charts(U˜p, p) for any point p∈ O. For the precise definition, and all basics on orbifolds and Lie group actions we use Sect. 2 of [11] as a general reference. Other sources of background material on orbifolds are [20–22,25,34]. But note that the notation differs slightly from source to source.
We consider an action of a torus on a compact, orientable orbifoldO, in the sense of [11, Definition 2.10]. In [11] it is shown that orbits, as well as components of fixed point setsOH, whereH ⊂Tis a connected Lie subgroup, are strong suborbifolds ofO. Moreover, for any p ∈ O, with good local chartπ :(U˜p, p) →Up there is an extensionT˜p ofTp byp, acting onU˜p. TheT˜p-action fixes the single pointp˜in the preimageπ−1(p). We thus obtain a well-defined isotropy representation ofT˜ponTpU˜p. Its restriction to the identity component T˜po ofT˜p has well-defined weightsα, which we consider, via the projectionT˜po →Tp, as elements int∗p/{±1}.
With this definition of weights, Definition2.7applies to torus actions on orbifolds equally well, and we can speak about torus actions on orbifolds of type GKMk. This leads to the following definition.
Definition 2.9 An action of a compact torusT on a smooth, compact, orientable orbifoldO isof type GKMk(simplyGKMfork=2) ifHodd(O)=0, the fixed point set is finite, and for every fixed point anykweights of the isotropy representation are linearly independent.
To any such action we can associate ann-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 [11, Lemma 3.9]. For the labelling, we rescale the weights as follows: for a weightαat a fixed point p, the intersection ofRαwith the integer lattice int∗is isomorphic toZ. We letαbe a generator of this group, andkbe the number of components of the principal isotropy group of theT-action on the 2-sphere to which the weight space ofαis tangent. We defineβ=kα.
We remark that the factorkis irrelevant for what follows: we include it in order for the GKM graph to encode the full isotropy groups. Consideringβas a homomorphismT →S1, its kernel is precisely the principal isotropy group of the corresponding 2-sphere.
In order to construct a connection on, we now restrict to actions of type GKM3. Letvbe a vertex of the graph, corresponding to the fixed pointp∈O, andean edge withi(e)=p, with labelα∈t∗/{±1}. Letqbe the fixed point corresponding tot(e). For any other edgeeat v, with weightβ, we consider the connected subgroupH ⊂Twith Lie algebra kerα∩kerβ.
By the GKM3-condition and the slice theorem for actions on orbifolds [11, Theorem 2.18]
the connected component ofOHis a four-dimensional strong suborbifold ofO. It containsq, and there exists an edge f withi(f)=t(e), with weightγ, such that kerγ∩kerα=h. We define a connection onby∇ee:= f. By construction,γis a (rational) linear combination ofαandβ, so that Equation (2.1) holds.
Also Theorem2.8holds true for GKM actions on orbifolds. This was (for torus orbifolds) already observed in [11, 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 [4,5,7,11,27].
We start with a general construction of such manifolds and orbifolds. Ann-dimensional manifold with cornersPis called nice if at each vertex ofPthere meet exactlynfacets ofP, that is, exactlyncodimension-one faces. The faces ofPordered by inclusion form a poset P(P), the so-called face poset ofP. We also denote byF=F(P)the set of facets ofPand letm= |F|.
Now letPbe a nice manifold with corners with only contractible faces, and assume that there is a mapλ:F→Zn such that for every vertexvofP,
λ(F1), . . . , λ(Fn)
are linearly independent, where theFiare the facets ofPwhich meet inv.
Then we can construct a torus orbifold, i.e. an orientable 2n-dimensional orbifoldOwith an action of then-dimensional torusTn =Rn/ZnwithOTn = ∅, such that the orbit space of theTn action onOis homeomorphic toP. The orbifoldOis defined as
O=(P×Rn/Zn)/∼,
where(x, v)∼(x, v)if and only ifx =xandv−vis contained in theR-span of all the λ(F)withx ∈F. HereTn acts on the second factor ofO. Note that replacing theλ(F)by nonzero multiples does not change the associated orbifoldO.
If for every vertexvtheλ(Fi)of the facetsFi which meet atvform a basis ofZn, then O is a manifold and theTn-action is locally standard, i.e., locally modelled on effective n-dimensional complex representations ofTn.
The preimages of the facets ofPunder the orbit map are invariant suborbifolds of codi- mension two inO. Therefore their equivariant Poincaré dualsv1, . . . , vm ∈HT2(O;Q)are defined. Moreover, these Poincaré duals form a basis ofHT2(O;Q)because the faces of P are contractible (see [27] and [28]).
The contractibility of the faces ofPalso implies that the rational cohomology of a torus orbifoldOas above is concentrated in even degrees.
Equivalently to giving the labelsλof the facets, one can also defineOby a labelingαˆof the edges ofOin such a way that for edgese1, . . . ,enmeeting a vertex,α(eˆ 1), . . . ,α(eˆ n)is the basis ofQn∗dual toλ(F1), . . . , λ(Fn)∈Zn⊗Q. In this way the vertex-edge graph ofP becomes a so-called torus graph (see [26]), i.e. a GKM graph of a torus manifold or orbifold.
Similar to the definition of the torus orbifold O associated to the pair(P, λ)one can associate a moment angle manifold of dimensionn+mtoP. This goes as follows.
Denote the facets ofPbyF1, . . . ,Fmand for eachi =1, . . . ,mletSi1be a copy of the circle group. Then define
ZP =(P×( m i=1
Si1))/∼,
where(x,t)∼(x,t)if and only ifx =xand tt−1∈
i;x∈Fi
Si1⊂ m i=1
Si1. There is an action ofTm=m
i=1S1i onZPinduced by multiplication on the second factor.
Moreover the torus orbifold O from above is the quotient of the action of the kernel of
a homomorphismϕ : Tm → Tn = Rn/Zn. Hereϕ is defined by the condition that its restriction toSi1induces an isomorphismSi1→Rλ(Fi)/(Zn∩Rλ(Fi)).
Note that ifPis a product of simplicesni and quotientsmi =S2mi/Tmi, thenZPis a product of spheres. We can equip this product with the product metric of the round spheres.
If we do so,Tmis identified with a maximal torus of the isometry group ofZP.
Example 2.10 At the end of this section we give examples of torus orbifolds. LetP=2be a triangle. Then we haven=2 andm=3. Denote byF1,F2,F3the facets ofP. Moreover, let
λ(F1)=(1,0), λ(F2)=(0,1), λ(F3)=(α, β)
withα, β ∈Z− {0}. Then by the above discussion the pair(P, λ)defines a torus orbifold O. Note thatOis a torus manifold if and only if|α| = |β| =1.
The moment angle manifoldZPassociated toPisS5 ⊂C3with a linearT3 =R3/Z3- action. The mapZP→Oconstructed above is the orbit map for the action of the subtorus ofT3whose Lie-algebra is generated by(−α,−β,1). ThereforeOis a so-called weighted projective space.
3 Coverings of GKM graphs
In this section we construct a covering of a GKMkgraph,k≥4, with small three-dimensional faces by the vertex edge graph of a product
ini ×
imi. We start with the definition of what we mean by the faces of a graph.
Definition 3.1 Letbe a graph with a connection∇. Then anl-dimensional faceofis a connectedl-valent∇-invariant subgraph of.
Lemma 3.2 Letbe a GKMk-graph, where k≥3. Then for any vertexvofand any edges e1, . . .elthat meet atv, where1≤l≤k−1, there exists a unique l-dimensional face of that contains e1, . . . ,el.
Proof First recall that by Remark2.5there is a unique connection onwhich is compatible with the weights. Therefore it makes sense to speak about the faces of a GKMk-graph,k≥3.
LetV be the (l-dimensional) span of theα(ei)in H2(BTm). Consider the subgraph of that consists of those edges whose labeling is contained inV, and let˜ be its connected component ofv. We claim that this subgraph is∇-invariant andl-valent.
Asis GKMk, withk>l, the only edges atvcontained in˜ aree1, . . . ,el. Moreover, wheneverwis anl-valent vertex of˜ande∈E()withi(e)=w, then alsot(e)isl-valent.
In fact, (2.1) shows that for any edgeeatwinE()˜ , also∇eeis an edge ofE()˜ , and the GKMkproperty ofshows that there is no further edge att(e)contained in˜. A simply connected spaceXis called rationally elliptic if it has finite dimensional rational homotopy and finite dimensional rational cohomology, i.e.,
i≥2
dimπi(X)⊗Q<∞ and
i≥2
dimHi(X)⊗Q<∞. (3.1) In this case the minimal Sullivan model ofX is elliptic. Moreover, for simply connected spaces the other implication also holds. However, there are non-simply connected spaces whose minimal Sullivan model is elliptic which do not satisfy the conditions (3.1).
Lemma 3.3 Let O be a GKM3manifold or orbifold which admits an invariant metric of non- negative curvature or whose minimal Sullivan model is elliptic. Then each two-dimensional face of the GKM graph of O has at most four vertices.
Proof The proof is essentially the same as the proof of Lemma 4.2 in [32]. In the non- negatively curved case it was first discussed in [12]. It has been translated to the orbifold setting in [11]. Here we repeat it for the sake of completeness.
The two-dimensional faces of the GKM graph ofOare GKM graphs of four-dimensional invariant totally geodesic suborbifolds ofO. These submanifolds are fixed point components of codimension-two subtori of T. They are non-negatively curved ifO is non-negatively curved. LetObe one of these suborbifolds. ThenO/Tis a two-dimensional non-negatively curved Alexandrov space with totally geodesic boundary, such that the points in(O)T correspond to corners of the orbit space, i.e. points on the boundary whose space of directions has diameterπ/2. Leta0, . . . ,ak−1be these corners such that fori ∈Z/kZ,aiis connected toai+1 by a totally geodesic arc which is contained in the boundary. Fori = 0 choose geodesicsγifroma0toai, such thatγ1andγk−1are part of the boundary.
Then by Toponogov’s Theorem the sum of angles in each of the triangles spanned by γi, γi+1 and the part of the boundary betweenai andai+1is at leastπ. Summing over all these triangles we get the inequality
π(k−2)≤ π 2k.
Hence the claim follows in this case.
Now assume that the minimal Sullivan model ofOis elliptic. Then, by [2], the minimal model(V,d) ofOis also elliptic. Sinceχ(OT) = χ(O), the number of vertices in the GKM graph of Ois equal to the Euler characteristic of O. Moreover, since H∗(O) is evenly graded, Hodd(O) = 0 by localization in equivariant cohomology. In particular, H1(O)=0.
Hence, by [10, Theorem 32.6 (ii)] and [10, Proposition 12.2], we have 4≥2 dimV2=2b2(O).
Therefore we haveχ(O)≤4 and there are at most four vertices in the GKM graph of
O.
Lemma 3.4 Let O be a orientable GKM4orbifold such that all two-dimensional faces of the GKM graph of O have at most four vertices. Then each three-dimensional face of the GKM graph of M has one of the following combinatorial types:3,3,2×I ,2×I , I3. Moreover, the face structure of the three-dimensional faces induced by the connection is the natural one.
Proof First note that the three-dimensional faces of the GKM graph ofO are GKM graphs of six-dimensional torus orbifoldsN1, . . . ,Nk.
Note that since O is orientable, all Ni (and therefore all the orbit spaces Ni/T) are orientable orbifolds without (and with, respectively) boundary. Moreover, because the coho- mology ofO(and therefore that of theNi) is concentrated in even degrees, the orbit space Ni/T and all its faces are acyclic over the rationals. Because two-dimensional orbifolds are homeomorphic to surfaces it follows from the classification of surfaces that the facets, i.e.
2-dimensional faces, ofNi/Tare homeomorphic to discsD2. Moreover, by Lemma3.2, any two edges of the GKM graph ofNimeeting in a vertex span a unique 2-dimensional face of Ni/T.
There are the following cases:
Fig. 1 These graphs do not occur
1. There is a facetF1 ofNi/T, which has two vertices. And, there is another facet F2 of Ni/T, which intersects withF1in an edge and has
(a) two vertices, or (b) three vertices, or (c) four vertices.
2. There is a facetF1 ofNi/T, which has three vertices. And, there is another facetF2 of Ni/T, which intersects withF1in an edge and has
(a) three vertices, or (b) four vertices.
3. All facets ofNi/Thave four vertices.
In case 1.(a), it is clear thatP=Ni/T is combinatorially equivalent to3. Moreover, in case 1.(c), using the 3-valence of the graph ofNione easily sees thatPis of type2×I.
Next assume thatF2is of type2, that is, we are in case 1.(b). Letv0be the vertex ofF2
which does not belong toF1andv1andv2be the other vertices. Then one sees by considering the faces which meet atv1 that the two edgesv0v1 andv0v2 belong to two different faces which both contain all three vertices, as in the first graph in Fig.1. But any two edges atv0
must span a unique face, a contradiction.
Alternatively, the following argument also leads to a contradiction. By the 3-valence of the graph there is a third edgeestarting fromv0. Since it is contained in a two-dimensional face ofNi/T, the end point ofemust bev1orv2(or must be connected via one edge to one ofv1, v2). But this cannot happen because of the 3-valence of the graph (atv1andv2).
Hence the case 1.(b) does not occur.
We now assume thatF1is of type2. That is, we are in one of the cases 2.(a) or 2.(b).
If all other facets ofPwhich have an edge withF1in common are also of type2, then Pmust be of type3. If all faces which have an edge withF1in common are of typeI2, thenPis of type2×I.
Therefore we have to exclude the case that there is a faceF2of type2and a faceF3of typeI2such thatF1∩F2∩F3is a vertex, as in the second graph in Fig.1.
In this case the third faceF4which has an edge withF1in common must be of typeI2. Using the 3-valence of the graph one now gets a contradiction in a similar way as in case 1.b).
Fig. 2 Small three-dimensional faces
In the remaining case 3) let F1be one of the two-dimensional faces ofNi/T. Then by Lemma3.2there are four not necessarily pairwise distinct two-dimensional facesF2, . . . ,F5
ofNi/T which have non-trivial intersection with F1. If all the intersections Fi ∩Fj,i = 1, . . . ,5, are connected or empty, thenF2, . . . ,F5are pairwise distinct and it is clear thatP is of typeI3.
Therefore assume thatF1∩F2has two components. These two components must then both be edges ofP.
Letv1, . . . , v4be the vertices ofF1such thatviandvi+1are connected by an edgevivi+1
for alli∈Z/4Z.
Assume thatv1v2 andv3v4are contained in the intersection of F1andF2. Ifv1 andv4
were connected by an edge inF2, then there would be a facet of type2inP, contradicting our assumption. Thereforev1andv3are connected by an edge inF2(and similarlyv2andv4).
SoF1∪F2is homeomorphic to a Moebius strip, contradicting our orientability assumption
onO.
Definition 3.5 Letbe a connected graph with a connection∇. We say thatis agraph with small three-dimensional facesif the following conditions hold true:
1. For any x ∈ V() and distinct edgese1,e2,e3 meeting atx, there exits a unique 3- dimensional face ofcontaininge1,e2ande3.
2. The conclusion of Lemma3.4holds true, i.e., any three-dimensional face ofhas the combinatorial type of3, 3,2×I,2×IorI3(see Fig.2) and these faces have the natural face structure.
Lemmas3.2and3.4thus say that the GKM graph of a connected nonnegatively curved GKM4orbifold is a graph with small three-dimensional faces.
Definition 3.6 Letx ∈V(), whereis a graph with small three-dimensional faces. We call a subgraph⊂alocal factoratxifcontainsx, is∇-invariant, has the combinatorial type ofkork,∇induces the natural face structure onandis maximal with these properties.
Lemma 3.7 Letbe a graph with small three-dimensional faces. For two edges e,eema- nating from x we have:
1. e and ebelong to the same local factor of typekif and only if e and espan a triangle.
2. e and ebelong to the same local factor of typekif and only if e and espan a biangle.
3. e and edo not belong to the same local factor if and only e and espan a square.
Proof By definition, all two dimensional faces ofhave at most four vertices. Therefore the lemma follows from the definition of the local factors, because two-dimensional faces ofk
andkare triangles and biangles, respectively.
Lemma 3.8 Letbe a graph with small three-dimensional faces. Then the local factors at x ∈V()are partitioning Ex =G1 · · · Gnx in such a way that each Gi contains the edges which span a local factor at x.
Proof We have to show that belonging to the same local factor is an equivalence relation on Ex. We only have to show transitivity.
Lete,e,e∈Exbe such thateandebelong to one local factorσiandeandebelong to another local factor. Then, by Lemma3.7,eandeandeandespan a triangle or a biangle, respectively.
Consider the three-dimensional faceFofspanned bye,e,e. It has one of the combi- natorial types described in Definition3.5. Since none of the faces spanned bye,eande,e are squares, it follows thatFhas the combinatorial type of3or3. Hence, we have shown
transitivity and the claim follows.
Lemma 3.9 Let be a graph with small three-dimensional faces and let e be an oriented edge of. Then the connection∇e : Ei(e) →Et(e)preserves the partitions Ei(e)= G1
· · · Gni(e)and Et(e)=G1 · · · Gnt(e). Moreover, the combinatorial types of the local factors spanned by Giand∇e(Gi)are the same.
Proof Lete,ebe other edges ofemanating fromi(e). By Lemma3.7, we have to show that the two-dimensional faces spanned by any choice of two edgese1,e2 frome,e,e, respectively, has the same combinatorial types as the two-dimensional face spanned by
∇e(e1),∇e(e2).
To do so, we consider the three-dimensional face ofspanned bye,e,e. It contains the two-dimensional faces spanned bye,e ande,e, respectively. Moreover, these two- dimensional faces are also spanned bye,∇e(e)ande,∇e(e), respectively.
The three-dimensional face also contains the face spanned byeande. By an inspection of the list of three-dimensional faces in Definition3.5, it follows that the two-dimensional faces spanned bye,eand∇e(e),∇e(e), respectively, have the same combinatorial types.
Hence the lemma is proved.
Lemma 3.10 Letbe a graph with small three-dimensional faces. Let e,eand f be edges inwith the same initial point.
1. If e and espan a biangle, then∇ef = ∇ef .
2. If e and espan a square, then∇e1∇ef = ∇e1∇ef , where e1,e1 are the edges opposite to e and e, respectively, in the square spanned by e and e.
Proof To see the first claim, one has to show that ife,espan a biangle2and f is an edge with the same initial point aseandethen
∇ef = ∇ef.
To see this, first assume thateand f span a biangle. Then by Definition3.5,e,e, f span a face ofwhich is combinatorially equivalent to3. Hence,∇ef = ¯f = ∇ef follows.
Next assume thateand f span a square. Then, by Definition3.5,e,e, and f span a face with the combinatorial type of2×I. Therefore∇ef = g = ∇ef, wheregis the third edge att(e)=t(e). Hence the claim follows in this case.
By Definition3.5, the case thate and f 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 ofks andks. 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 aren-valent, for somen≥1; forn=2 a covering ofn-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, x ∈ V(), and σ1, . . . , σk the local factors at x. Consider the product graph ˜ := k
i=1σi, equipped with its natural connection ˜∇. Let x0 ∈ ˜be a base point, and f : E()˜ x0 → E()x a bijection sending the edges of a local factor to the edges of a local factor. Then there exists a unique coveringπ: ˜→extending f that is compatible with the connections, i.e., which satisfies∇π(e)◦π=π◦ ˜∇efor all edges e∈E().˜
Proof The compatibility condition∇π(e)◦π=π◦ ˜∇eshows that ife,eare edges meeting at some vertexv, andπ(e)andπ(e)are given, thenπ(∇ee)is uniquely determined. This implies the uniqueness ofπ.
We have to show the existence ofπ. Note that for any pathγ in˜, say fromx0toy, the connection ˜∇on˜ induces a bijection
˜∇γ :E()˜ x0 →E()˜ y ˜∇γ = ˜∇em ◦ · · · ◦ ˜∇e1,
whereγ =e1∗ · · · ∗em. Similarly, for paths infromx toythere is a bijection
∇γ :E()x →E()y
induced by∇.
Forn≥0, we let˜nbe the subgraph of˜ whose vertices are those that have distance at mostntox0, and whose edges are all the edges ofconnecting two such vertices. We prove by induction that we can construct a map of graphsπ: ˜n →extending f that satisfies
∇π(γ )◦π(e)=π◦ ˜∇γe
for all shortest pathsγin˜nstarting atx0, and all edgeseof˜nwithi(e)=i(γ ). Forn=0 there is nothing to do. Forn =1, the existence ofπis guaranteed by the fact that f sends local factors to local factors.
We assume thatπ: ˜n−1→is already constructed, and wish to constructπ: ˜n →. Letebe an edge of˜nwhich is not an edge of˜n−1, but whose initial vertexi(e)is a vertex of˜n−1, and choose a shortest pathγ fromx0 toi(e). Note thatγ is a path in˜n−1. We want to defineπ(e)= ∇π(γ )π(˜∇γ−1e); in order to do so we have to show that this definition is independent of the choice ofγ.
Ifx,yare vertices of, then the shortest paths between˜ xandyare of the following form:
ei1∗ · · · ∗eil,
