An Orlik-Raymond type classification of simply connected
six-dimensional torus manifolds with vanishing odd degree cohomology
Shintarô Kuroki
Dedicated to Professor Mikiya Masuda on his 60th birthday.

Abstract. The aim of this paper is to classify simply connected 6-dimensional torus manifolds
with vanishing odd degree cohomology. It is shown that there is a one-to-one correspondence
between equivariant diﬀeomorphism types of these manifolds and 3-valent labelled graphs, called
torus graphs introduced by Maeda-Masuda-Panov. Using this correspondence and combinatorialarguments, we prove that a simply connected 6-dimensional torus manifold with H odd (M ) = 0
is equivariantly diﬀeomorphic to the 6-dimensional sphere S 6 or an equivariant connected sumof copies of 6-dimensional quasitoric manifolds or S 4 -bundles over S 2 .

1. Introduction
Let M be a 2n-dimensional closed, connected, oriented manifold with an eﬀective n-dimensional(i.e., half-dimensional) torus T n -action. We call M (or (M, T )) a torus manifold if M T ̸= ∅ (see[5]), where M T is the set of ﬁxed points. A toric manifold (i.e., a non-singular, complete toricvariety viewed as a complex analytic space) with the restricted T n -action is a typical example oftorus manifolds. Recall that a toric manifold is a complex (C∗ )n -manifold with the dense orbit (see[19], [3]), and T n is the maximal compact subgroup of (C∗ )n . A fundamental result of toric geometry tells us that there is a one-to-one correspondence between toric manifolds and combinatorial
objects called fans. Thus, topological (more precisely, geometric) invariants of toric manifolds canbe described in terms of combinatorial invarinats of fans, such as equivariant cohomology rings,
equivariant characteristic classes and other topological invariants.
In 2003, Hattori-Masuda introduced a torus manifold as the topological generalization of atoric manifold in [5]. They also introduced the combinatorial objects, called multi-fans (see [15],[5]), and computed topological invariants (such as equivariant characteristic classes or Todd genus
for unitary torus manifolds) in terms of multi-fans. However, unlike toric geometry, a multi-fan
does not contain enough information to determine some topological invariants of torus manifolds(e.g. equivariant cohomology). So, in 2007, Maeda-Masuda-Panov introduced another combinatorial objects, called torus graphs motivated by GKM graphs introduced by Guillemin-Zara in [4].
The combinatorial information of torus graphs can completely determine the equivariant cohomology rings of torus manifolds with vanishing odd degree cohomology, i.e., H odd (M ; Z) = 0 (inthis paper, we only consider the integer coeﬃcient), see [17], [14] (and also see Section 3 in thispaper about torus graphs). However, in general, there is no one-to-one correspondence between
torus manifolds with H odd (M ) = 0 and torus graphs.
2000 Mathematics Subject Classification. Principal: 57S25, Secondly:94C15.
Key words and phrases. Torus manifold and Torus graph and GKM graph and Equivariant connected sum.
The author was partially supported by Grant-in-Aid for Scientiﬁc Research (S) 24224002, Japan Society for
Promotion of Science, and the JSPS Strategic Young Researcher Overseas Visits Program for Accelerating Brain
Circulation ”Deepening and Evolution of Mathematics and Physics, Building of International Network Hub basedon OCAMI”.
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So, we are naturally led to ask the following two questions: (1) which subclasses of torus
manifolds are completely determined by combinatorial objects (like multi-fans or torus graphs);(2) if we ﬁnd such a subclass of torus manifolds, how we can classify such torus manifolds. Several
mathematicians have answered to the 1st question; for example, Davis-Januszkiewicz [2] for the
subclass called quasitoric manifolds (see [1] or Section 4.3 in this paper), Ishida-Fukukawa-Masuda[6] for the subclass called topological toric manifolds, and Wiemeler for the class of simply connected 6-dimensional torus manifolds with H odd (M ) = 0 in [21] (see Theorem 2.7). The aim of
this paper is to answer to the 2nd question for the class of simply connected 6-dimensional torus
manifolds with H odd (M ) = 0 by using torus graphs.
Let us brieﬂy recall the classiﬁcation results for torus manifolds with lower dimensions. If
T 1 acts on a compact 2-dimensional manifold M , then M is the 2-dimensional sphere S 2 , the2-dimensional real projectiove space RP 2 or the 2-dimensional torus T 2 . Because M T ̸= ∅ and
M is oriented, M must be equivariantly diﬀeomorphic to S 2 with T 1 -action (also see [9]). Whendim M = 4. By Orlik-Raymond’s theorem in [20], we have the following fact:
Theorem 1.1 (Orlik-Raymond). Let M be a 4-dimensional simply connected torus manifold.
Then, M is equivariantly diﬀeomorphic to the 4-sphere S 4 or an equivariant connected sum of2
copies of complex projective spaces CP 2 , CP (reversed orientation) or Hirzebruch surfaces Hk .
Here, in Theorem 1.1, a Hirzebruch surface Hk (k ∈ Z) is a manifold which is deﬁned by the
projectivization of the complex 2-dimensional vector bundle γ ⊗k ⊕ ϵ over CP 1 , where γ is thetautological and ϵ is the trivial complex line bundles over CP 1 .
In this paper, we prove that an Orlik-Raymond type theorem in Theorem 1.1 also holds
for simply connected 6-dimensional torus manifolds with H odd (M ) = 0. Before we state ourmain results, we introduce the result for non-simply connected torus manifolds. As one of theconsequences of Masuda-Panov’s theorem (see Theorem 2.2 in Section 2.2), we have the following
proposition (also see [21]):
Proposition 1.2. Let W be a 6-dimensional torus manifold with H odd (W ) = 0 (might not
be simply connected). Then, there are a simply connected, 6-dimensional torus manifold M with
H odd (M ) = 0 and a homology 3-sphere hS 3 such that
∼ M #T (hS 3 × T 3 )W =up to equivariantly diﬀeomorphism.
Here, in Proposition 1.2, the product manifold hS 3 × T 3 is the product of hS 3 and the 3dimensional torus T 3 with the free T 3 -action on the 2nd factor, and the symbol #T represents
the equivariant gluing along two free orbits of M and hS 3 × T 3 .
Because the fundamental groups π1 (W ) and π1 (hS 3 ) are isomorphic in Proposition 1.2, W
is simply connected if and only if hS 3 is simply connected, i.e., the standard sphere. Our main
theorem is a classiﬁcation of simply connected torus manifolds appeared in Proposition 1.2.
Theorem 1.3. Let M be a simply connected 6-dimensional torus manifold with H odd (M ) = 0.
Then, M is equivariantly diﬀeomorphic to the 6-sphere S 6 or obtained by an equivariant connected
sum of copies of 6-dimensional quasitoric manifolds or S 4 -bundles over S 2 equipped with thestructure of a torus manifold.
This type of classiﬁcation, i.e., classiﬁcation by equivariant connected sum, may be regarded
as the 6-dimensional analogue of Orlik-Raymond’s classiﬁcation in Theorem 1.1. So, in this paper,
we call this theorem an Orlik-Raymond type classiﬁcation (also see the papers [18] and [10]).
Remark 1.4. In the paper [7], Izmestiev proves an Orlik-Raymond type classiﬁcation for someclass of 3-dimensional small covers (i.e., the real analogue of quasitoric manifolds, see Section 4.2),
called a linear model.
The organization of this paper is as follows. In Section 2, we recall the basic facts about
torus manifolds. In Section 3, we recall a torus graph. In particular, Corollary 3.5 is the key fact
to prove Theorem 1.3. In Section 4, we introduce the torus graphs of S 6 , quasitoric manifolds
2and S 4 -bundles over S 2 . These torus graphs will be the basic graphs to classify simply connected6-dimensional torus manifolds with H odd (M ) = 0. In Section 5, we introduce the “oriented” torus
graphs and translate the equivariant connected sum around ﬁxed points of torus manifolds to theconnected sum around vertices of oriented torus graphs. In Section 6 and 7, we prove Theorem 1.3.
The brief outline of the proof is as follows. Due to Corollary 3.5, there is a one-to-one corresponds
between 6-dimensional simply connected torus manifolds with H odd (M ) = 0 and 3-valent torus
graphs. Therefore, in order to prove Theorem 1.3, it is enough to prove that an (oriented) torus
graph can be decomposed into basic torus graphs in Section 4 by the connected sum. We prove
this by using combinatorial arguments.
2. Orbit spaces of torus manifolds
In this section, we recall some basic facts about torus manifolds (see [15] or [5] for details).
2.1. Torus manifolds. A 2n-dimensional torus manifold M is said to be locally standard ifevery point in M has a T -invariant open neighborhood U which is weakly equivariantly homeomorphic to an open subset ΩU ⊂ Cn invariant under the standard T n -action on Cn , where twogroup actions (U, T ) and (ΩU , T ) are said to be weakly equivariantly homeomorphic if there is anequivariant homeomorphism from U to ΩU up to an automorphism on T n (see e.g. [12, Section
2.1] for details).
Let Mi , i = 1, . . . , m, be a codimension-two torus submanifold in a 2n-dimensional torus
manifold M which is ﬁxed by some circle subgroup Ti in T . Such Mi is a (2n − 2)-dimensional
torus manifold with T /Ti -action, called a characteristic submanifold. Because a torus manifold
M is compact, the cardinality of all characteristic submanifolds in M is ﬁnite. If M is locallystandard, each characteristic submanifold is also locally standard.
An omniorientation O of M is a choice of orientation for the torus manifold M as well as foreach characteristic submanifold. If there are just m characteristic submanifolds in M , there areexactly 2m+1 omniorientations (see [1], [5]). If M has a T -invariant almost complex structure J (inthis case, M is automatically locally standard), then there exists the canonical omniorientation OJ
determined by J. We call the torus manifold M with a ﬁxed omniorientation O an omnioriented
torus manifold and denote it by (M, O).
2.2. Orbit spaces of locally standard torus manifolds. The orbit space M/T of a locallystandard torus manifold M naturally admits the structure of a “topological” manifold with corners.
We next recall the basic facts of a topological manifold with corners (cf. the deﬁnition of a smoothmanifold with corners in [13]) and introduce the structure on M/T .
The following notations will be often used:[n] = {0, 1, . . . , n},and
Rn+ = {(x1 , . . . , xn ) ∈ Rn | xi ≥ 0, i = 1, . . . , n}.
Let Qn be an n-dimensional topological manifold with boundary. A chart with corners for Qn isa pair (V, ψV ), where V is an open subset of Qn andψV : V → Rn+
is homeomorphic from V to a (relatively) open subset ΩV ⊂ Rn+ . Two charts with corners−1(V, ψV ), (W, ψW ) are said to be (topologically) compatible if the composition of functions ψV ◦ψW:ψW (V ∩W ) → ψV (V ∩W ) is a strata-preserving homeomorphism. This implies that if ψW (p) ∈ Rn+
contains exactly k zero coordinates then ψV (p) ∈ Rn+ also contains exactly k zero coordinates for0 ≤ k ≤ n. We call the collection of compatible charts with corners {(V, ψV )} whose domains
cover Qn an atlas. Then, its maximal atlas is called a structure with corners of Qn . A topologicalmanifold with boundary together with a structure with corners is called a (topological) manifoldwith corners. Let p ∈ Qn be a point of an n-dimensional manifold with corners Qn . A chart (V, ψV )with corners such that p ∈ V deﬁnes the number d(p) ∈ [n] by the number of zero-coordinates ofψV (p) ∈ Rn+ . By the compatibility of charts, this number is independent on the choice of a chart3with corners which contains p. Therefore, the map d : Qn → [n] is well-deﬁned. The number d(p)is called a depth of p. We call the closure of a connected component of d−1 (k) (0 ≤ k ≤ n) a
codimension-k face. In particular, the codimension-0 face is Qn itself. Moreover, codimension-1,(n − 1) and n faces are called facet, edge and vertex, respectively. The set of all edges and vertices
is called a one-skeleton of Qn (or a graph of Qn ). By restricting the structure with corners on Qn
to faces, we may regard each codimension-k face as an (n − k)-dimensional (sub)manifold withcorners.
Definition 2.1 (Manifold with faces). An n-dimensional manifold with corners Q is said tobe a manifold with faces (or a nice manifold with corners) if Q satisﬁes the following conditions:(1) for every k ∈ [n], there exists a codimension-k face;(2) for each codimension-k face H, there are exactly k facets F1 , . . . , Fk such that H is aconnected component of ∩ki=1 Fi ; moreover, H∩F ̸= H for any facet F ̸= Fi (i = 1, . . . , k).
Let (M, T ) be a torus manifold. When (M, T ) is locally standard, by using the diﬀerentiable
slice theorem, the orbit space M/T has the structure of an n-dimensional manifold with faces. On
the other hand, when M satisﬁes H odd (M ) = 0, its orbit space M/T satisﬁes a stronger conditionby the following Masuda-Panov theorem (see [17, Lemma 2.1] and [17, Theorem 2]):
Theorem 2.2 (Masuda-Panov). Let M be a 2n-dimensional torus manifold. Then, the following two conditions are equivalent:(1) H odd (M ) = 0;
(2) the T -action on M is locally standard and its orbit space M/T has the structure of ann-dimensional face acyclic manifold with corners.
Here, in Theorem 2.2, an n-dimensional face acyclic manifold with corners Q is an n-dimensional
manifold with faces such that all faces F (include Q) of Q are acyclic, i.e., H∗ (F ) ≃ H0 (F ) ≃ Z.
For example, if Q is a simply connected 3-dimensional face acyclic manifold with corners, then it
is easy to check that the boundary of Q is homeomorphic to the 2-sphere S 2 . Moreover, in thiscase, we can also check that Q itself is homeomorphic to the 3-dimensional disk D3 . Therefore,
as one of the conclusions of Theorem 2.2, we have the following corollary:
Corollary 2.3. Let M be a simply connected 6-dimensional torus manifold with H odd (M ) =0. Then, its orbit space M/T is homeomorphic to the 3-dimensional disk.
By the deﬁnition of a manifold with faces Q, we can deﬁne a simplicial poset (partially orderedset) P(Q), called a face poset of Q (see [16]), by the set of faces in Q with the empty-set ∅ orderedby inclusion, where the empty set ∅ is the smallest element by this order, say ≼. We often denotethe face poset structure of Q as (P(Q), ≼). Let Q1 and Q2 be n-dimensional manifolds with faces.
We call Q1 and Q2 are combinatorially equivalent if their face posets (P(Q1 ), ≼1 ) and (P(Q2 ), ≼2 )are isomorphic as a poset (i.e., there is an order-preserving bijection between them). We denotethem by Q1 ≈c Q2 . By deﬁnition of weakly equivariant homeomorphism, if two locally standard
torus manifolds M1 and M2 are weakly equivariantly homeomorphic then M1 /T ≈c M2 /T .
2.3. Characteristic functions. Let M be a 2n-dimensional locally standard torus manifold.
By the argument demonstrated in Section 2.2, the orbit map π : M → M/T = Q may be regarded
as the projection onto some manifold with faces Q. Let F(Q) = {F1 , . . . , Fm } ⊂ P(Q) be the setof all facets in Q. By the deﬁnition of facet Fi ∈ F (Q), its preimage π −1 (Fi ) is a characteristic
submanifold Mi . Then, there exists the circle subgroup Ti (⊂ T ) ﬁxing Mi = π −1 (Fi ) (recall that
dim Mi = 2n − 2). Recall that Ti is determined by a primitive element in tZ ≃ Zn (the lattice of
Lie algebra of T ). Therefore, by taking this primitive element (up to sign) in tZ , we can deﬁne thefollowing map:
λ : F(Q) → tZ /{±1}where tZ /{±1} represents the quotient of tZ by signs. We call λ a characteristic function.
Now the choice of omniorientation O of M determines the sign of λ as follows. Fix anomniorientation O of M . Namely, we ﬁx the orientation of the tangent bundle of M (resp. Mi ),say τ (resp. τi ). Restricting τ to the submanifold Mi , say τ |Mi , we obtain the T n -equivariant
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decomposition τ |Mi ≃ τi ⊕ νi , where νi is the Ti -equivariant normal bundle of Mi . Therefore,
because we ﬁx the orientation of τ |Mi (induced from the orientation of τ ) and that of τi , we may
choose an orientation of νi such that the orientation of τ |Mi coincides with that of τi ⊕ νi (thus,
we may regard νi as the complex line bundle over Mi ). Because Ti acts on νi , we may choose
an orientation of Ti such that the Ti -action preserves the orientation of νi . This orientation of Ti
determines the sign of λ(Fi ) for i = 1, . . . , m. By this way, we have the following function:
λO : F(Q) → tZ .
This is called an omnioriented characteristic function (of (M, O)), in this paper.
Remark 2.4. The characteristic function deﬁned in [21] may be regarded as the characteristic
function λ as above. On the other hand, the characteristic function deﬁned in [2] may be regarded
as the characteristic function λO as above by taking an appropriate omniorientation (also see [1,Section 5.2]).
Let p ∈ M T . We deﬁne the subset Ip ⊂ [m] as follows:
Ip = {i ∈ [m] | p ∈ Mi }.
By the diﬀerentiable slice theorem around p ∈ M T , we have that its cardinality |Ip | = n for everyp ∈ M T . Put Ip = {i1 , . . . , in }. Because the T -action on M is eﬀective, {λ(Fi1 ), . . . , λ(Fin )} spanst∗Z /{±1}, i.e., the determinant of the induced (n × n)-matrix(λ(Fi1 ) · · · λ(Fin ))satisﬁes that
(2.1)

det (λ(Fi1 ) · · · λ(Fin )) = ±1.

Similarly, we have
(2.2)

det (λO (Fi1 ) · · · λO (Fin )) = ±1.

for each n facets such that ∩nj=1 Fij = {p} for some vertex p ∈ Q (called the facets around a vertex).
Motivated by the observations as above, we may abstractly deﬁne the characteristic function
on a manifold with faces as follows (see [1], [2] for simple polytopes and [17], [21] for manifoldwith faces):
Definition 2.5. Let Q be an n-dimensional manifold with faces and F(Q) be the set of its
facets. Let tZ be the lattice of Lie algebra of T n and tZ /{±1} be its quotient by {±1}. Then, afunction λ : F(Q) → tZ /{±1} is said to be a characteristic function if λ satisﬁes the relation (2.1)
for the facets around every vertex, and a function λO : F(Q) → tZ is said to be an omnioriented
characteristic function if λO satisﬁes the relation (2.2) for the facets around every vertex.
We denote an n-dimensional manifold with faces Q with its characteristic function λ (resp.
omnioriented characteristic function λO ) by (Q, λ) (resp. (Q, λO )).
Let Q1 and Q2 be manifolds with faces and λ1 (resp. λO1 ) and λ2 (resp. λO2 ) be their (resp.
omnioriented) characteristic functions, respectively. Assume that Q1 ≈c Q2 and it is induced by
the bijective map fe : P(Q1 ) → P(Q2 ). Denote its restriction onto the set of facets as
f = fe|F (Q1 ) : F(Q1 ) → F (Q2 ).
Then, we call that (Q1 , λ1 ) and (Q2 , λ2 ) are combinatorially equivalent if the following diagram
commutes:λ1

F(Q1 )
f/ tZ /{±1}Id
F(Q2 )λ25/ tZ /{±1}Similarly, (Q1 , λO1 ) and (Q2 , λO2 ) are combinatorially equivalent if the following diagram commutes:
λO1/ tZ
F(Q1 )f
F(Q2 )

λO2



Id

/ tZ

Note that the characteristic function λ can be obtained by ignoring sings from the omnioriented
characteristic function λO ; we call such λ an induced characteristic function from λO . On the other
hand, by choosing a sign for each facet, we can obtain an omnioriented characteristic function λO
from the characteristic function λ; we call such λO an induced oriented characteristic function
from λ. Therefore, we have the following lemma:
Lemma 2.6. If (Q1 , λO1 ) and (Q2 , λO2 ) are combinatorially equivalent, then their induced(Q1 , λ1 ) and (Q2 , λ2 ) are also combinatorially equivalent.
If (Q1 , λ1 ) and (Q2 , λ2 ) are combinatorially equivalent, then there are induced omnioriented
characteristic functions λO1 and λO2 such that (Q1 , λO1 ) and (Q2 , λO2 ) are combinatorially equivalent.
Now we may introduce one of the key facts to prove our main theorem (see [21, Theorem 1.3,
Theorem 6.1]):
Theorem 2.7 (Wiemeler). Let M1 and M2 be a 6-dimensional simply connected torus manifold with H odd (M ) = 0 and (Q1 , λ1 ) and (Q2 , λ2 ) be their orbit spaces with characteristic functions.
Then, the following three statements are equivalent:(1) (Q1 , λ1 ) and (Q2 , λ2 ) are combinatorially equivalent;(2) M1 and M2 are equivariantly homeomorphic;(3) M1 and M2 are equivariantly diﬀeomorphic.
Therefore, by Corollary 2.3 and Theorem 2.7, in order to classify all 6-dimensional simply connected torus manifolds with H odd (M ) = 0, it is enough to classify all (Q, λ)’s up to combinatorial
equivalent, where Q is a 3-dimensional disk equipped with the structure of a manifold with faces.
3. Torus graph induced from manifold with faces
Let (M, O) be an omnioriented locally standard 2n-dimensional torus manifold and (Q, λO )be its orbit space with an omnioriented characteristic function. From the one-skeleton of (Q, λO ),we can deﬁne a labelled graph, called a torus graph. One of the key arguments to prove the main
theorem is to classify all possible torus graphs (see Section 7). To do that, in this section, werecall a torus graph deﬁned by Maeda-Masuda-Panov [14].
Let Γ be the graph of Q. Let V (Γ) be its vertices and E(Γ) be its oriented edges, i.e., we
distinguish two edges pq and qp. For p ∈ V (Γ), we denote the set of outgoing edges from p as
Ep (Γ). Because Q is an n-dimensional manifold with faces, |Ep (Γ)| = n and each edge e ∈ E(Γ)
n−1
is a connected component of ∩i=1
Fi , for some F1 , F2 , . . . , Fn−1 ∈ F (Q). Moreover, for p ∈ V (Γ)
which is one of two vertices on e, there is another facet Fn ∈ F(Q) such that {p} is a connectedcomponent of ∩ni=1 Fi . In other words, Fn may be regarded as a normal facet of e ∈ E(Γ) onp ∈ V (Γ). Put λO (Fi ) = ai ∈ tZ ≃ Zn . Then, there exists unique α ∈ t∗Z such that
(3.1)⟨α, ai ⟩ = 0 for i = 1, . . . , n − 1 and⟨α, an ⟩ = +1,∗

where ⟨, ⟩ represents the pairing of t and t. Therefore, by this way, we can deﬁne a map A : E(Γ) →t∗Z from the omnioriented characteristic function λO . This map A is called an axial function on
Γ. We call the labelled graph (Γ, A) a torus graph induced from (Q, λO ) (or equivalently (M, O)).
We denote such a torus graph as Γ(Q, λO ) (or (ΓM , AM )). We can easily check the following
proposition by deﬁnition of torus graphs (also see [14]):
Proposition 3.1. Let (Γ, A) be a torus graph induced from (Q, λO ). Then, Γ is an n-valent
regular graph, i.e., |Ep (Γ)| = n for all p ∈ V (Γ), and (Γ, A) satisﬁes the following conditions:6(1) A(e) = ±A(ē), where ē is the orientation reversed edge of e;(2) {A(e) | e ∈ Ep (Γ)} spans t∗Z for all vertices p ∈ V (Γ);(3) there is a bijection ∇pq : Ep (Γ) → Eq (Γ) for all edges whose initial vertex is p andterminal vertex is q such that the following conditions hold:(a) ∇ē = ∇−1e ;(b) ∇e (e) = ē;(c) A(∇pq (e)) − A(e) ≡ 0 mod A(pq) for all e ∈ Ep (Γ).
We call ∇ = {∇e | e ∈ E(Γ)} a connection on (Γ, A).
Remark 3.2. The original torus graph (induced from an omnioriented torus manifold) is
deﬁned by using the tangential representations, see [17], [14]. The deﬁnition of torus graph as
above is essentially same with this deﬁnition.
In [14], motivated by the GKM graph introduced by Guillemin-Zara in [4], an n-valent graphΓ with a label A : E(Γ) → t∗Z which satisﬁes three conditions in Proposition 3.1 is called an(abstract) torus graph (i.e., there might be no geometric objects which deﬁne (Γ, A)).
We next deﬁne the equivalence relation between two torus graphs. We call the map f : Γ1 =(V (Γ1 ), E(Γ1 )) → Γ2 = (V (Γ2 ), E(Γ2 )) a graph isomorphism, if the restricted map f |V : V (Γ1 ) →V (Γ2 ) and f |E : E(Γ1 ) → E(Γ2 ) are bijective and the following map commutes:
E(Γ1 )

f |E

πV1


V (Γ1 )/ E(Γ2 )
πV2

f |V


/ V (Γ2 )where πV : E(Γ) → V (Γ) is the map projecting onto the initial vertex, i.e., πV (pq) = p. In otherwords, the bijection f |V preserves the edges. Now we may deﬁne the equivalence relation.
Definition 3.3. Let (Γ1 , A1 ) and (Γ2 , A2 ) be torus graphs. We say (Γ1 , A1 ) and (Γ2 , A2 ) areequivalent if there is a graph isomorphism f : Γ1 → Γ2 such that the following diagram commutes:
E(Γ1 )

A1

f |E
E(Γ2 )

A2

/ t∗Z


Id

/ t∗Z

Assume (Γ, A) = Γ(Q, λO ). Let Pk (Γ, A) be the set of k-valent torus subgraphs in (Γ, A),i.e., k-valent subgraphs in Γ closed under the connection ∇, where −1 ≤ k ≤ n and we deﬁne
P−1 (Γ, A) = {∅}. Then, the setP(Γ, A) = ∪nk=−1 Pk (Γ, A)
admits the structure of a simplicial poset by inclusion (see [14]). We denote this structure by(P(Γ, A), ≼). Let P(Q) be the face poset of Q (see Section 2.2) and Pk (Q) be the set of all
k-dimensional faces, where −1 ≤ k ≤ n and P−1 (Q) = {∅}. Then, each element of Pk (Γ, A)
is nothing but the graph of an element in Pk (Q). This implies that the poset (P(Γ, A), ≼) isequivalent to the poset (P(Q), ≼). Therefore, we have the following lemma:
Lemma 3.4. The following two statements are equivalent:(1) two manifolds with faces with omnioriented characteristic functions (Q1 , λO1 ) and (Q2 , λO2 )are combinatorially equivalent;(2) their induced torus graphs Γ(Q1 , λO1 ) and Γ(Q2 , λO2 ) are equivalent.
By Lemma 2.6, Theorem 2.7 and Lemma 3.4, we have the following corollary:
Corollary 3.5. Let (M1 , T ) and (M2 , T ) be 6-dimensional simply connected torus manifolds
with vanishing odd degree cohomology. Then, the following statements are equivalent:(1) (M1 , T ) and (M2 , T ) are equivariantly diﬀeomorphic;
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(2) their orbit spaces, i.e., 3-dimensional disks with the structures of manifolds with faces,with characteristic functions (M1 /T, λ1 ) and (M2 /T, λ2 ) are combinatorially equivalent;(3) there are omnioriented characteristic functions λO1 and λO2 such that their induced 3valent torus graphs Γ(M1 /T, λO1 ) and Γ(M2 /T, λO2 ) are equivalent.
Therefore, in order to prove our main theorem (Theorem 7.1), it is enough to classify all
3-valent torus graphs (Γ, A), induced from (M, O), up to equivalent.
4. Basic six-dimensional torus manifolds
Let (M, T ) be a simply connected, 6-dimensional torus manifold with H odd (M ) = 0, and(ΓM , AM )(= (Γ, A)) be its torus graph induced by some omniorientation. As a preliminary toprove the main theorem (Theorem 7.1), in this section, we will introduce some of basic torus
graphs (Γ, A) and their corresponding 6-dimensional torus manifolds (M, T ).
4.1. 6-sphere. Because the induced torus graphs from (M, T ) are 3-valent, if there is a 3multiple edge, i.e., three edges that are incident to the same two vertices, then it follows from
Proposition 3.1 that such torus graph must be the torus graph in Figure 1. (We denote the torus
graph in Figure 1 as (Γsp , Aα,β,γ ).)
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Figure 1. The torus graph (Γsp , Aα,β,γ ), where α, β, γ ∈ t∗Z ≃ Z3 are Z-basis.
Put α = k11 e1 + k12 e2 + k13 e3 , β = k21 e1 + k22 e2 + k23 e3 and γ = k31 e1 + k32 e2 + k33 e3 , by
the standard basis e1 , e2 , e3 in t∗Z ≃ Z3 . Then, the following equation holds:
k11 k12 k13
det  k21 k22 k23  = ±1.
(4.1)
k31 k32 k33
Let S 6 ⊂ C3 ⊕ R be the unit sphere, i.e., the set (z1 , z2 , z3 , r) ∈ C3 ⊕ R such that |z1 |2 + |z2 |2 +|z3 |2 + r2 = 1. Deﬁne the T 3 -action on the ﬁrst three complex coordinates in S 6 by
(4.2)

(t1 , t2 , t3 )(z1 , z2 , z3 , r) 7→ (ρ1 (t)z1 , ρ2 (t)z2 , ρ3 (t)z3 , r)where t = (t1 , t2 , t3 ) ∈ T and ρi : T → S 1 , i = 1, 2, 3, is a one-dimensional complex representation
deﬁned by
ρi (t1 , t2 , t3 ) = tk1i1 tk2i2 tk3i3 .
Then, by choosing an appropriate omniorientation on S 6 , we have that its induced torus graph isequivalent to (Γsp , Aα,β,γ ). Therefore, by using Corollary 3.5, we have the following lemma:
Lemma 4.1. Let (M, O) be an omnioriented 6-dimensional simply connected torus manifoldwith H odd (M ) = 0. If its induced torus graph is (Γsp , Aα,β,γ ), then (M, T ) is equivariantly diﬀeomorphic to one of (S 6 , T ) deﬁned by (4.2).
4.2. S 4 -bundles over S 2 . Assume that a 3-valent torus graph (Γ, A) does not have 3multiple edges but have multiple edges, i.e., two edges that are incident to the same two vertices.
In this section, we classify the easiest case of such torus graphs.
Because Γ is a one-skeleton of 3-dimensional manifold with faces Q, the number of vertices|V (Γ)| ≥ 4. Assume that |V (Γ)| = 4. Then, we can easily check that such torus manifold is theone-skeleton of the 3-simplex (see Figure 3 in Section 4.3) or the graph drawn in Figure 2, say ΓS .
As is well known that the torus manifold whose torus graph is the one-skeleton of the 3-simplex isequivariantly diﬀeomorphic to the complex projective space with some T -action (see e.g. [2], andalso see Figure 3 in Section 4.3). So, we only study the torus manifold which induces the graph8ΓS . Because Q is homeomorphic to D3 , we may regard Q whose one-skeleton is ΓS as the productof D2 × I, where D2 is the 2-dimensional disk and I is the interval. By considering all functions
on facets of Q which satisﬁes (2.2), we can classify all omnioriented characteristic functions λO on
Q. Then, by the way to induce the axial function AS from (Q, λO ) demonstrated in Section 3, we
can obtain all possible axial functions on ΓS as shown in Figure 2.
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Figure 2. The torus graph (ΓS , AS ) = (ΓS , Aϵ,a,b
α,β,γ ), where ϵ = ±1 and a, b ∈ Z
and α, β, γ ∈ t∗Z are Z-basis of t∗Z .
The torus graph (ΓS , AS ) in Figure 2 can be induced from an S 4 -bundle over S 2 as follows.
First, by choosing ϵ = ±1, we may deﬁne two free T 1 -actions on S 3 ⊂ C2 as follows:(w, z) 7→ (t−1 w, tϵ z).
We denote S 3 with the above T 1 -action by Sϵ3 . Note that Sϵ3 /T 1 is diﬀeomorphic to the 2-sphereS 2 , and a complex line bundle over S 2 can be denoted by
Sϵ3 ×T 1 Ck ,where Ck is the complex 1-dimensional T 1 -representation space by k-times rotation for some k ∈ Z.
Let Sϵ3 ×T 1 R be the trivial real line bundle over S 2 . Take the unit sphere bundle of the following
Whitney sum of three vector bundles for a, b ∈ Z:
Sϵ3 ×T 1 (Ca ⊕ Cb ⊕ R).
Then, we obtain the S 4 -bundle over S 2 denoted by
M (ϵ, a, b) = Sϵ3 ×S 1 S(Ca ⊕ Cb ⊕ R),
for ϵ = ±1, a, b ∈ Z. Namely, we can identify elements in M (ϵ, a, b) by[(w, z), (x, y, r)] = [(t−1 w, tϵ z), (ta x, tb y, r)]for any t ∈ T 1 such that |w|2 + |z|2 = 1 and |x|2 + |y|2 + r2 = 1. Deﬁne a T 3 -action on M (ϵ, a, b)
by[(w, z), (x, y, r)] 7→ [(t1 w, z), (t2 x, t3 y, r)],3
where (t1 , t2 , t3 ) ∈ T . Fix an omniorientation on M (ϵ, a, b) by the induced orientations from
Sϵ3 × S 4 ⊂ C2 × (C ⊕ C ⊕ R). Then, considering the tangential representations around each ﬁxed
point, it is easy to check that the induced torus graph is (ΓS , Aϵ,a,be1 ,e2 ,e3 ), where e1 , e2 , e3 are thestandard basis of tZ ≃ Z3 . Therefore, by taking the appropriate automorphism of T 3 , we canconstruct each torus graph (ΓS , AS ) in Figure 2 from M (ϵ, a, b). Note that if ϵ = −1 and a = b,then this is nothing but one of the torus manifolds which appeared in the classiﬁcations of torus
manifolds with codimension one extended actions in [12].
By the argument as above and Corollary 3.5, we establish the following lemma:
Lemma 4.2. Let (M, O) be an omnioriented 6-dimensional simply connected torus manifoldwith H odd (M ) = 0. If its induced torus graph has 4-vertices, then (M, T ) is equivariantly diﬀeomorphic to one of the followings:(1) CP 3 with the standard T 3 -action up to automorphism of T 3 ;9(2) M (ϵ, a, b) for some ϵ = ±1 and a, b ∈ Z.
4.3. 6-dimensional quasitoric manifolds. Assume that there are no multiple edges in a
3-valent torus graph (Γ, A), i.e., there are no two edges that are incident to the same two vertices.
A graph Γ is called a simple if Γ does not have both of multiple edges and loops. In this andSection 5, we study simple torus graphs which can be realized as the one-skeleton of manifold withfaces homeomorphic to D3 .
The typical example of such torus manifolds whose torus graphs are simple is a quasitoric
manifold (introduced by Davis-Januszkiewicz in [2] (also see [1])). A quasitoric manifold is deﬁned
by a torus manifold whose orbit space is a simple convex polytope, i.e., a convex polytope admitting
the structure of a manifold with faces. For example, the complex projective space CP n with thestandard T n -action is the quasitoric manifold whose orbit space is the n-dimensional simplex.
The Figure 3 shows the torus graph induced from (CP 3 , OC ), i.e., the omniorientation OC induced
from the standard complex structure on CP 3 and the standard T -action on CP 3 .
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Figure 3. Torus graph induced from (CP 3 , OC ).
We next characterize when torus graphs are induced from simple convex polytopes, i.e., induced from quasitoric manifolds. The Steinitz theorem (see [22, Chapter 4]) tells us that a graphΓ is the one skeleton of a 3-dimensional convex polytope if and only if Γ is a simple, planner and3-connected graph, where Γ is called a 3-connected graph if it remains connected whenever fewer
than 3 vertices are removed. It easily follows from the Steinitz theorem that we have the following
lemma:
Lemma 4.3. Let Q be a manifold with faces and Γ be its graph. Assume that Q is homeomorphic to the 3-disk D3 and there are no multiple edges. Then, the following two statements areequivalent:(1) Q is combinatorially equivalent to a 3-dimensional simple convex polytope P ;(2) Γ is a 3-connected graph.
Therefore, together with Corollary 3.5, we have the following fact:
Lemma 4.4. Let (M, O) be an omnioriented 6-dimensional simply connected torus manifoldwith H odd (M ) = 0. Then, the following two statements are equivalent:(1) (M, T ) is equivariantly diﬀeomorphic to a quasitoric manifold;(2) its induced torus graph Γ is a 3-connected graph with no multiple edges.
5. Connected sum of torus graphs and other 6-dimensional torus manifolds
By the arguments in Section 4, only the following case remains: the simply connected 6dimensional torus manifolds with H odd (M ) = 0 whose induced torus graphs are simple but not
3-connected. Such torus manifolds can be constructed by using the connected sum of “oriented”
torus graphs. The purpose of this section is to introduce oriented torus graphs and their connectedsum.
We ﬁrst recall the equivariant connected sum of torus manifolds. Let M1 , M2 be 2n-dimensional
torus manifolds and p ∈ M1T , q ∈ M2T be ﬁxed points. By using the slice theorem, we may takeT -invariant open neighborhoods U1 ⊂ M1 of p and U2 ⊂ M2 of q. Assume that U1 and U2
10

are equivariantly diﬀeomorphic. Then, U1 \ {p} and U2 \ {q} are equivariantly diﬀeomorphic to
S 2n−1 × I, where S 2n−1 ⊂ Cn with some eﬀective T n -action and I = (−ϵ, ϵ) with the trivial
T n -action for some ϵ > 0. Then, we glues these two neighborhood by φ deﬁned by the identity on
S 2n−1 and the map r 7→ −r on I for r ∈ I. Namely, we can glue M1 \ {p} and M2 \ {q} by thefollowing identiﬁcation:
(5.1)

≃

≃

φ

M1 \ {p} ⊃ U1 \ {p} −→ S 2n−1 × I −→ S 2n−1 × I −→ U2 \ {q} ⊂ M2 \ {q}.

The T n -manifold obtained by this way is denoted by M1 #M2 or M1 #(p,q) M2 (if we emphasize
ﬁxed points p ∈ M1T and q ∈ M2T ). Because each torus manifold has more than two ﬁxed points,
M1 #M2 is again a torus manifold. We call this operation an equivariant connected sum. The
following lemma holds:
Lemma 5.1. If two torus manifolds M1 and M2 are simply connected and H odd (M1 ) =H (M2 ) = 0, then M1 #M2 is also simply connected and H odd (M1 #M2 ) = 0.
odd

Proof. It is easy to check the statement by using van-Kampen’s theorem and the MayerVietoris exact sequence.
Assume that (M1 , O1 ), (M2 , O2 ) are 6-dimensional omnioriented simply connected torus manifolds with H odd (M1 ) = H odd (M2 ) = 0. Let (Γ1 , A1 ) and (Γ2 , A2 ) be their induced 3-valent torus
graphs. Assume that we can glue p ∈ M1T and q ∈ M2T by the connected sum. Then, by considering the restriction of φ in (5.1) onto S 2n−1 ⊂ Cn , i.e., the identity map, the axial functions
around p ∈ V (Γ1 ) and q ∈ V (Γ2 ) must satisfy(5.2){A1 (e) | e ∈ Ep (Γ1 )} = {A2 (e) | e ∈ Eq (Γ2 )}.

However, at this stage, torus graphs (Γ1 , A1 ) and (Γ2 , A2 ) do not have the information of orientations of M1 and M2 . Note that, to do connected sum, we need the orientations around p ∈ M1T
and q ∈ M2T . To encode the orientations around ﬁxed points, we need the following notion:
Definition 5.2. Let (Γ, A) be a torus graph. We call a triple (Γ, A, σ) with a map σ : V (Γ) →{+1, −1} an oriented torus graph, if σ : V (Γ) → {+1, −1} satisﬁes the following condition for alle ∈ E(Γ):σ(πV (e))A(e) = −σ(πV (ē))A(ē),
where πV (e) ∈ V (Γ) is the initial vertex of e ∈ E(Γ), i.e., for e = pq, πV (e) = p and πV (ē) = q.
We call a map σ an orientation of (Γ, A).
Remark 5.3. Let (M, O) be an omnioriented torus manifold. The oriented torus graph(Γ, A, σ) of (M, O) is deﬁned as follows. Let p ∈ M T . Then, there exist exactly n characteristic
submanifolds M1 , . . . , Mn such that p is a connected component of ∩ni=1 Mi . Now the ﬁxed
orientations of M1 , . . . , Mn determine the decomposition of the tangential representation, i.e.,≃ψp : Tp M → V (α1 ) ⊕ · · · ⊕ V (αn ) is determined by ﬁxing the orientations of M1 , . . . , Mn . On
the other hand, the orientation of M determines the orientation of Tp M . So, we deﬁne the map
σ : V (Γ) = M T → {+1, −1} by{+1 if ψp preserves the orientations
σ(p) =−1 if ψp reverses the orientations
Let (Γ1 , A1 , σ1 ) and (Γ2 , A2 , σ2 ) be the induced oriented torus graphs from (M1 , O1 ) and(M2 , O2 ). If we can glue p ∈ M1T and q ∈ M2T by the connected sum, then both of the relation(5.2) and the relation(5.3)

σ1 (p) ̸= σ2 (q)hold ((5.3) corresponds to that the orientations on Tp M1 and Tq M2 are diﬀerent). Note the induced (oriented) torus graph by M1 #(p,q) M2 is nothing but the one-skeleton of the connected sum
Q1 #(p,q) Q2 of manifolds with faces, where Qi is the orbit space of Mi , i = 1, 2 (also see [7, Deﬁnition 3], [11, Section 3.1] for details about the connected sum of polytopes). Therefore, conversely,
if p ∈ V (Γ1 ) and q ∈ V (Γ2 ) satisfy (5.2) and (5.3), then we can do the connected sum of (oriented)
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torus graphs between (Γ1 , A1 , σ1 ) and (Γ2 , A2 , σ2 ), say (Γ, A, σ) = (Γ1 , A1 , σ1 )#(Γ2 , A2 , σ2 ) or(Γ1 , A1 , σ1 )#(p,q) (Γ2 , A2 , σ2 ) (if we emphasize the vertices p ∈ V (Γ1 ) and q ∈ V (Γ2 )). Moreprecisely, (Γ, A, σ) = (Γ1 , A1 , σ1 )#(Γ2 , A2 , σ2 ) is deﬁned as follows (also see Figure 4).
(1) V (Γ) = V (Γ1 ) \ {p} ⊔ V (Γ2 ) \ {q};(2) E(Γ) = (E(Γ1 ) \ {pp1 , pp2 , pp3 }) ⊔ (E(Γ2 ) \ {qq1 , qq2 , qq3 }) ⊔ {p1 q1 , p2 q2 , p3 q3 }, whereA1 (ppi ) = A2 (qqi ) for i = 1, 2, 3;(3) A : E(Γ) → (t3Z )∗ such that A(e) = A1 (e) and A(f ) = A2 (f ) for e ∈ E(Γ1 ) \{pp1 , pp2 , pp3 } and f ∈ E(Γ2 ) \ {qq1 , qq2 , qq3 }, and A(pi qi ) = A1 (pi p) and A(qi pi ) =A2 (qi q);(4) σ : V (Γ) → {+1, −1} is deﬁned by σ(r) = σ1 (r) for r ∈ V (Γ1 ) \ {p} and σ(r′ ) = σ2 (r′ )for r′ ∈ V (Γ2 ) \ {q}.
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Figure 4. This is the local ﬁgure of the connected sum #(p,q) of torus manifold(from left to right) and its inverse #−1(p,q) (from right to left), where σ1 (p) ̸= σ2 (q).
3 ∗Here, α, β, γ are Z-basis of (tZ ) and a, a′ , b, b′ , c, c′ = ±1.
Then, we can easily check that (Γ, A, σ) is an oriented torus graph. Using Corollary 3.5 andthe arguments as above, we have the following lemma:
Lemma 5.4. Let M1 , M2 be 6-dimensional simply connected torus manifolds with H odd (M1 ) =H (M2 ) = 0, and (Γ1 , A1 , σ1 ), (Γ2 , A2 , σ2 ) be their induced oriented torus graphs from someomniorientations, respectively. If (Γ, A, σ) = (Γ1 , A1 , σ1 )#(p,q) (Γ2 , A2 , σ2 ), then (Γ, A, σ) is theoriented torus graph induced from M = M1 #(p,q) M2 with some omniorientation.
odd

By using the connected sum, we can construct the torus manifolds which are not appeared in
Section 4. For example, the following torus manifold is one of such examples:
CP 3 #(S 2 × S 4 )#CP 3 ,where CP 3 is the reversed orientation of CP 3 . The Figure 5 shows the torus graph induced fromCP 3 #(S 2 × S 4 )#CP 3 (see the axial functions on Figure 2, 3 for details). We can easily checkthat this graph is 3-valent, simple and planner but not 3-connected; therefore, due to Lemma 4.4,
this manifold is not a quasitoric manifold.
6. Some combinatorial lemmas
To prove the main theorem (Theorem 7.1), in this section, we show the following two lemmas.
Lemma 6.1. Let Q be a 3-dimensional manifold with faces which is homeomorphic to D3 andΓ be its graph. Then, Γ \ {p} is connected, for all vertices p ∈ V (Γ).
Proof. Because Q is homeomorphic to the 3-disk D3 , Γ may be regarded as a planner graphby the stereographic projection of ∂Q = S 2 . Assume Γ \ {p} is not connected. Because Q is a3-dimensional manifold with faces, there are exactly three out-going edges from p, say pp1 , pp2 andpp3 . Therefore, we may assume that there exists a connected component Γ1 in Γ \ {p} such thatp1 ∈ V (Γ1 ) but p2 , p3 ̸∈ V (Γ1 ) (see Figure 6). Since Γ1 is also a planner 3-valent graph exceptson the vertex p1 (because p ̸∈ V (Γ1 )), there is a 2-valent subgraph in Γ1 , say ∂Γ1 , such that ∂Γ1
splits ∂Q = S 2 into two connected components H+ and H− , where Γ1 \ ∂Γ1 ⊂ H+ \ ∂Γ1 butΓ1 ̸⊂ H− . This implies that there is a facet F in Q such that ∂F contains ∂Γ1 and p1 p. However,in this case, p1 p must be a self-intersection edge of F (see Figure 6). This gives a contradiction
to that Q is a manifold with faces. This establishes the statement.
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Figure 5. Torus graph (with appropriate orientations, e.g, σ(p) = +1, σ(q) =−1, σ(r) = +1, σ(s) = −1) induced from CP 3 #(S 2 × S 4 )#CP 3 .

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Figure 6. The ﬁgure explained the proof of Lemma 6.1. The facet F has theself-intersection on the edge p1 p.
By Lemma 6.1, if Γ is not 3-connected, then there are two vertices p, q ∈ V (Γ) such thatΓ \ {p, q} is not connected but both of Γ \ {p} and Γ \ {q} are connected. More precisely, we havethe following lemma:
Lemma 6.2. Let Q be a 3-dimensional manifold with faces which is homeomorphic to D3 andΓ be its graph. Assume that there are two vertices p, q ∈ V (Γ) such that {p, q} ̸⊂ V (F ) for any
facets F , i.e, p and q are not on the same facet F . Then, Γ \ {p, q} is connected.
Proof. Assume that p, q are not on the same facet of Q. Because Q is a manifold with faces,
there are mutually distinct facets F1 , . . . , F6 such that {p} is a component of F1 ∩ F2 ∩ F3 and {q}is that of F4 ∩ F5 ∩ F6 and we can take vertices p1 , p2 , p3 and q1 , q2 , q3 such that ppi and qqi are all
outgoing edges from p and q, for i = 1, 2, 3. Take two vertices r, s from Γ \ {p, q}. By Lemma 6.1,
Γ \ {q} is connected. So there is a path γ from r to s in Γ \ {q}. If γ does not go through p, then rand s are connected in Γ \ {p, q}. Assume that this path γ goes through p. Then γ goes throughexactly two vertices pi , pj (we may assume p1 and p2 ). Moreover, one of the facets F1 , F2 , F3 , say
F1 , contains both of p1 , p2 . Note that F1 corresponds to the 2-valent subgraph in Γ. Therefore,
we can take the path γp connecting p1 , p2 on F1 which is not the path p1 pp2 . Because p, q are noton the same facet, in particular q ̸∈ V (F1 ), the path γp does not contain q. Hence, the connected13

subgraph γ ∪ γp contains both of r, s but does not contain both of p, q. Thus, we can take thepath γ ′ from r to s in γ ∪ γp ⊂ Γ \ {p, q}. This establishes that Γ \ {p, q} is connected.
In summary, by Lemma 6.1 and 6.2, we have the following fact.
Corollary 6.3. Let Γ be a one-skeleton of 3-dimensional manifold with faces Q. Then, forall p ∈ V (Γ), Γ \ {p} is connected. Furthermore, if Γ \ {p, q} is not connected, then p, q is on thesame facet.
7. Proof of main theorem
The main theorem of this paper can be stated as follows:
Theorem 7.1. Let M be a simply connected, 6-dimensional torus manifold with H odd (M ) = 0.
Then, M is equivariantly diﬀeomorphic to one of the following manifolds:(1) S 6 ⊂ C3 ⊕ R with a torus action induced from a (faithful) representation of T 3 on C3 ;(2) 6-dimensional quasitoric manifold X;(3) S 4 -bundle over S 2 which is equivariantly diﬀeomorphic to M (ϵ, a, b) for some ϵ = ±1, a, b ∈Z,
or otherwise, there are some 6-dimensional quasitoric manifolds Xh , for some h = 1, . . . , k, andsome S 4 -bundles over S 2 , say Si = M (ϵi , ai , bi ) (for some ϵi = ±1, ai , bi ∈ Z and i = 1, . . . , ℓ),such that M is equivariantly diﬀeomorphic to(#kh=1 Xh )#(#ℓi=1 Si )where # represents the equivariant connected sum around ﬁxed points, k + ℓ ≥ 2 for k ≥ 0, ℓ ≥ 1,
and the case k = 0 means that there is no Xh factor.
In this ﬁnal section, we prove Theorem 7.1.
Let M be a simply connected, 6-dimensional torus manifold with H odd (M ) = 0, Q be its orbit
space which is homeomorphic to D3 and (ΓM , AM ) be its induced oriented torus graph (we omit
the orientation).
Because ΓM is a one-skeleton of a manifold with faces which is homeomorphic to D3 , it is easy
to check that |V (ΓM )| ̸= 1, 3. If |V (ΓM )| = 2, by Lemma 4.1, we have that M is equivariantly
diﬀeomorphic to S 6 , i.e., the statement (1). If |V (ΓM )| = 4, it follows from Lemma 4.2 that M isequivariantly diﬀeomorphic to a quasitoric manifold CP 3 or M (ϵ, a, b) for some ϵ = ±1, a, b ∈ Z,
i.e., the statement (2) or (3) occurs. So we may only prove the case when |V (ΓM )| ≥ 5.
We ﬁrst claim the following lemma:
Lemma 7.2. Assume that |V (ΓM )| ≥ 5 and there is a multiple edge in ΓM . Then, (ΓM , AM )can be decomposed into the connected sum of the following torus graphs:(ΓM , AM ) = (ΓX , AX )#(ΓS1 , AS1 )# · · · #(ΓSℓ′ , ASℓ′ )or(ΓM , AM ) = (ΓS1 , AS1 )# · · · #(ΓSℓ′ , ASℓ′ ),where (ΓX , AX ) is a torus graph without multiple edges and Si = M (ϵi , ai , bi ) for i = 1, . . . , ℓ′ .
Proof. Assume two vertices p, q are connected by a multiple edge, i.e., two edges (see thebottom graph in Figure 7).
Then, by the connection of the torus graph (see Proposition 3.1), it is easy to check that theaxial functions around the vertex r of the bottom graph in Figure 7 satisfy the axial functions
expressed in Figure 7, where we can take α, β, γ as Z-basis (t3Z )∗ . In this case, we can do (inverse)
connected sum such as expressed in Figure 7 (from the bottom to the top in Figure 7). Then,the induced torus graph (ΓM , AM ) is decomposed into two induced torus graphs (ΓS1 , AS1 ) and(ΓM ′ , AM ′ ), where M ′ is some simply connected 6-dimensional torus manifold with H odd (M ′ ) = 0
by Lemma 5.1. Namely, we have(ΓM , AM ) = (ΓM ′ , AM ′ )#(ΓS1 , AS1 ).
14

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Figure 7. We may regard α, β, γ as any generators in (t3Z )∗ and a, a′ , b, b′ ∈ Zand ϵ, ϵ′ = ±1. Here, the bottom graph is (ΓM , AM ) and the left upper graphis (ΓS1 , AS1 ) and the right upper graph is (ΓM ′ , AM ′ ). Note that if we ﬁx theorientation of (ΓM , AM ) then the orientations of (ΓS1 , AS1 ) and (ΓM ′ , AM ′ ) areautomatically determined.

If there is no multiple edges in ΓM ′ , then we may put ΓM ′ = ΓX . Assume that there is a multipleedge in ΓM ′ . If there are only 4 vertices in ΓM ′ , then we may put M ′ as S2 = M (ϵ2 , a2 , b2 ) byLemma 4.2. When there are more than 4 vertices in ΓM ′ , with iterating the arguments as above,
ﬁnally we establish the statement of this lemma.

Therefore, to prove Theorem 7.1, it is enough to show the following lemma:
Lemma 7.3. Assume that |V (ΓM )| ≥ 5 and there are no multiple edges in ΓM . Then,(ΓM , AM ) can be decomposed into the connected sum of the following torus graphs:(ΓM , AM ) = (ΓX1 , AX1 )# · · · #(ΓXk , AXk )#(ΓS1 , AS1 )# · · · #(ΓSℓ′′ , ASℓ′′ )where (ΓXh , AXh ) for h = 1, . . . , k is the torus graph induced from a quasitoric manifold Xh , and
Si = M (ϵi , ai , bi ) for i = 1, . . . , ℓ′′ .
Proof. If ΓM (= Γ) is 3-connected, then it follows from Lemma 4.4 that the statement holds,
i.e., k = 1, ℓ′′ = 0. Therefore, we may assume Γ is not 3-connected. In this case, by Corollary 6.3,
there is a 2-valent torus subgraph F ⊂ Γ such that some p, q ∈ V (F ) satisfy that Γ \ {p, q} is notconnected.
If F is a triangle (i.e., |V (F )| = 3), with the method similar to that demonstrated in the proof
of Lemma 6.1, we have that there is a face in Q which has the self-intersection edge. This gives acontradiction to that Q is a manifold with faces. Therefore, we may assume |V (F )| ≥ 4. We ﬁrst
assume that pq is an edge of F . Then, there are two graphs Γ1 and Γ2 which are the connectedcomponents of Γ \ {p, q} expressed in Figure 8. If we remove the two vertices r and q from Γ
15

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Figure 8. If we remove r and q from Γ instead of p and q, the graph is also disconnected.
instead of p, q, where r ∈ V (Γ2 ) such that pr is an edge, then Γ \ {r, q} is also not connected (seeFigure 8). Therefore, from now, we may assume the followings:(1) p, q ∈ V (Γ) satisfy Γ \ {p, q} is not connected;(2) pq ̸∈ E(Γ);(3) there is a 2-valent torus subgraph (facet) F with |V (F )| ≥ 4 in Γ such that p, q ∈ V (F ).
We call such facet F a singular facet.
Let F be a singular facet. Assume |V (F )| ≥ 6. In this case, by the similar argument just

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Figure 9. The axial functions around F when |V (F )| ≥ 6, where ϵ, ϵ′ , ϵ′′ = ±1
and a, b, c, d, e, f ∈ Z. Here, F ′ is a facet which intersects with F on pr and qs.
before, we may take p, q as in the position of Figure 9, i.e., p and q are on two separated edges
rp and sq which are common edges of two facets F and F ′ in Figure 9 (Note that r and s might
be connected by an edge). Moreover, by considering the omnioriented characteristic functions of
the facets F and F ′ , we may take the axial functions around the facet F as in Figure 9.
By taking an appropriate orientation, we can do the connected sum as in Figure 10; here wee 1 , Ae1 ) and that containing Γ2 as (Γe ′ , Ae′ ).
denote the (oriented) torus graph containing Γ1 as (Γ22
The torus graph obtained by this connected sum is nothing but the torus graph (Γ, A) ine 1 is simple and planner, and Γe ′ is just planner. With the method similar to
Figure 9. Note that Γ2′ e′ethat demonstrated in Figure 7, (Γ2 , A2 ) can be obtained from the connected sum of (ΓS , AS ) ande 2 , Ae2 ) (containing Γ2 ), where (ΓS , AS ) is one of the torus graphs (bythe simple, planner graph (Γtaking the appropriate axial functions) in Figure 2. Namely, the torus graph in Figure 9 can beobtained from the following connected sum:e 1 , Ae1 )#(ΓS , AS )#(Γe 2 , Ae2 ).
(Γ, A) = (Γe i consists of Γi and the other two facets, say Fe (i) and Fe′ (i) (induced
Here, it is easy to check that Γ′from F and F in Γ). Because of Figure 10, the number of vertices of other two facets Fe(i) and
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Figure 10. The torus graph (Γ, A) in Figure 9 splits into two torus graphs
e 1 , Ae1 ) (upper) and (Γe ′ , Ae′ ) (lower). Here, we omit the axial functions around(Γ22
the vertices p, q, r, s because they are exactly same with that of Figure 9.
Fe′ (i) are reduced; in particular, the number of vertices of the facet Fe(i) induced from the singulare 1 , Ae1 ) and (Γe 2 , Ae2 ) are 3-connected, then these torus
facet F is strictly less than 6. If both of (Γgraphs are induced from quasitoric manifolds, i.e, the statements of Lemma 7.3 hold. Assume
e 1 , Ae1 ) is not 3-connected. Then, by the arguments before, there is a singular facet F inthat (Γeee 1 , Ae1 ) also decomposes into
(Γ1 , A1 ). If |V (F )| ≥ 6, then (Γe 1 , Ae1 ) = (Γe 3 , Ae3 )#(ΓS ′ , AS ′ )#(Γe 4 , Ae4 ),(Γ
by using the similar arguments explained in Figure 10. Iterating this arguments, we may reduce
all singular facets with |V (F )| ≥ 6. More precisely, we may decompose (Γ, A) in Figure 9 into
(Γ, A) = #ℓi=1 {(Γi , Ai )#(ΓSi , ASi )#(Γi+ℓ , Ai+ℓ )} ,where (ΓSi , ASi ), for i = 1, . . . , ℓ, is a torus graph in Figure 2 and (Γh , Ah ), for h = 1, . . . , 2ℓ, is a
3-valent simple and planner torus graph which satisﬁes one of the followings:• 3-connected (in this case, (Γh , Ah ) is induced from a quasitoric manifold);• otherwise, all singular facets F satisfy |V (F )| = 4 or 5.
Assume that the number of vertices in every singular facet of the torus graph (Γ, A) is lessthan or equal to 5. Then, such torus graph is one of torus graphs expressed in Figure 11. However,
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Figure 11. The singular facets F with |V (F )| = 5 or 4. Here, F ′ is a facet whichintersects with F on pr and qs.
17because Γ is the one-skeleton of a manifold with faces and not 3-connected, it is easy to checkthat there exists the singular facet F ′ such that F ′ ∩ F = {pr, qs} and |V (F ′ )| ≥ 6. This gives acontradiction. Hence, this case does not occur. This establishes Lemma 7.3.
Consequently, by Lemma 5.4, 7.2 and 7.3, we have the statement of Theorem 7.1
Finally, by Theorem 7.1 and the Mayer-Vietoris exact sequence, we also have the following
well-known result.
Corollary 7.4. Let M be a simply connected 6-dimensional torus manifold whose cohomology
ring is generated by the 2nd degree cohomology. Then, M is a quasitoric manifold.
Acknowledgments
The author would like to thank Professors Yael Karshon and Takashi Tsuboi for providing
him excellent circumstances to do research.
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