QUASITORIC MANIFOLDS, ROOT SYSTEMS AND
J-CONSTRUCTIONS OF POLYTOPES
岡山理科大学理学部応用数学科 黒木 慎太郎
Shintarô Kuroki
Dep. of Applied Math. Faculty of Science, Okayama U of Science

1. Introduction
This article is a research announcement of the progress paper [Ku3].
Let (M, T ) be a quasitoric manifold over P . In this article, we assume
that there is an extended G-action of (M, T ), say (M, G). See [Ku1]
about the extended action and the definition of quasitoric manifolds.
The purpose of this article is to introduce the following relation among
geometry, algebra and combinatorics.
t 5=tttttttttttu} tttt
P (J) ks(M, G)

dl PP
PPPP
PPPP
P $,+3R(M ) ⊂ H 2 (BT )Here, P (J) is the J-construction of a polytope P which is introduced by
Bahri-Bendersky-Cohen-Gitler in [BBCG1] (see Section 3), and R(M )is the root systems of a quasitoric manifold introduced by the authorand Masuda in [KM] (see Section 2). This relation can be summarized
as the following theorem (also see Theorem 4.1):
Theorem 1.1. If R(M ) has a simple factor Aj (= ⟨α1 , . . . , αj+1 |p∗ (αi ) =τi −τi+1 ⟩) (i.e., R(M ) ̸= ∅), then there is an (n−j)-dim simple polytope
P ′ such that one of the following holds:(1) P ≃ P ′ × ∆j ;(2) P ≃ P ′ (j + 1, 1, . . . , 1)(=: P ′ (J)),
where ∆j is the j-dimensional simplex and P ′ (J) is the J-constructionof polytope.
In [BBCG1], the J-construction of P is defined as purely combinatorial way. So this theorem may be regarded as the geometric meaningof the J-construction of P in the context of toric topology.
As a corollary of this result, we also have1

Corollary 1.2. There is an extended (M, G) of (M, T ) if and only if
R(M ) ̸= ∅ .
2. Root systems of quasitoric manifolds
We first introduce the definition of root systems of quasitoric manifolds (see [KM] for details).
To do that we need to recall the generators in the 2nd degree equivariant cohomology of quasitoric manifolds. Let π : M → P be theorbit projection of quasitoric manifold and Mi := π −1 (Fi ) be the characteristic submanifold corresponding to the facet Fi , i = 1, . . . , m. Set
the equivariant Thom class of Mi as τi (for the fixed omniorientationof M ). Then, the following isomorphism holds:
HT2 (M ) = Zτ1 ⊕ · · · ⊕ Zτm ,that is the 2nd degree equivariant cohomology is generated by the equivariant Thom classes of characteristic submanifolds. Let p : ET ×T
M → BT be the projection of the Borel construction of M . Then, theinduced injective map p∗ : H 2 (BT ) → HT2 (M ) is defined asm∑∗p (α) =⟨α, λ(Fi )⟩τi ,i=1where λ : {F1 , . . . , Fm } → tZ is the characteristic function on P and ⟨, ⟩is the evaluation of H 2 (BT ) ≃ t∗Z and H2 (BT ) ≃ tZ . The root systemsof a quasitoric manifold can be defined as follows:
Definition 2.1. R(M ) := {α ∈ H 2 (BT ) | p∗ (α) = τi − τj } is called aroot systems of a quasitoric manifold.
We proved the following result in the previous paper [KM]:
Theorem 2.2. If there is an extended (M, G) of a quasitoric (M, T ),thenR(G) ⊂ R(M ),where R(G) is the root systems of G.
Note that the following corollary holds:
Corollary 2.3. If R(M ) = ∅, then there is no extended action (M, G)of a quasitoric (M, T ).
This shows that the root systems of a quasitoric manifold is an invariant of the existence of an extended actions. Therefore, the following
question is the natural question: Is R(M ) the complete invariant ofthe existence of an extended actions? Corollary 1.2 answers to thisquestion.
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3. J-construction of polytopes
We next introduce the J-construction of a simple polytope. Notethat in [BBCG1] the J-construction is defined for the simplicial complex. The following definition is the dual of the definition in [BBCG1].
Let P n be a simple convex polytope with m-facets. Let J = (j, 1, . . . , 1) ∈Nm .
Definition 3.1. The J-construction of polytope, denoted by P (J), isdefined by the following way:
(1) Fix a facet F ⊂ P and take j copies of P say P1 , . . . , Pj ;(2) Embed Pi ⊂ Rn−1 × [0, ∞) such that F = Pi ∩ Rn−1 , then wehave P1 ∪· · ·∪Pj ⊂ Rn−1 ×[0, ∞)j and P1 ∩· · ·∩Pj = F ⊂ Rn−1 ;(3) P (J) := Conv(P1 ∪ · · · ∪ Pj ).
The following figures show two examples of J-constructions of polytopes.
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

,
,

,)

, 

Ǽ

Remark 3.2. When j = 2, this is also called a (simplicial) wedge operation (Ewald). For J = (j1 , . . . , jm ), we can also define J-construction
by the iteration of this construction (Bahri-Bendersky-Cohen-Gitler).
4. Main theorem and some conclusions
In [Ku3], we prove the following theorem:
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Theorem 4.1. If R(M ) has a simple factor Aj (= ⟨α1 , . . . , αj+1 |p∗ (αi ) =τi −τi+1 ⟩) (i.e., R(M ) ̸= ∅), then there is an (n−j)-dim simple polytope
P ′ such that one of the following holds:(1) P ≃ P ′ × ∆j ;(2) P ≃ P ′ (j + 1, 1, . . . , 1)(=: P ′ (J)),
where ∆j is the j-dimensional simplex and P ′ (J) is the J-constructionof polytope.
For example, if R(M ) = A1 (j = 1), then M/T is one of the following
type of polytopes: Each case satisfies p∗ (α) = τ1 − τ2 , where τi is aThom class corresponds to up and down facets P .

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3 

Remark 4.2. The left polytope may be regarded as a blow-up of theright polytope (also see [Ku2]). This also gives another combinatorial
meaning of the J-construction of P , i.e., if we blow-up P (J) alongsome face, then it becomes the product with simplicies. This might be
not known fact in combinatorics but it is the known fact in geometry(see [Wi1]).
Finally, by using theorem of [BBCG2], we also have the following
proposition.
Proposition 4.3. If R(M ) has an Aj simple factor and P = P ′ × ∆jthenM ≃ S 2j+1 ×S 1 M (P ′ , λ′ ) ≃ (S 2j+1 × ZP ′ )/H(= Z/H),
where M (P ′ , λ′ ) is the quasitoric of (P ′ , λ′ ).
If R(M ) has an Aj simple factor and P = P ′ (j + 1, 1, . . . , 1) thenM ≃ Z(KP ′ , (D2J , S 2J−1 ))/ ker λ(= Z/H),
where (D2J , S 2J−1 ) = {(D2ji , S 2ji −1 )}m
i=1 (j1 = j + 1, ji = 1(i ̸= 1)).
In summary, we have that
Corollary 4.4. Every extended action (M, G) of quasitoric (M, T ) ise
e is theinduced from the G-action
on the polyhedral product, where G
finite covering of G.
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This is the generalization of [Ku1, Theorem11.2].
The details of the facts and notations in this article will be appearedin [Ku3].
Acknowledgment
This work was supported by JSPS KAKENHI Grant Number 17K14196.
References[BBCG1] A. Bahri, M. Bendersky, F.R. Cohen, S. Gitler, Operations on polyhedral products and a new topological construction of infinite families of toricmanifolds, Homology Homotopy Appl. 17 (2015), no. 2, 137–160.
[BBCG2] A. Bahri, M. Bendersky, F.R. Cohen, S. Gitler, A generalization of theDavis-Januszkiewicz construction and applications to toric manifold and iterated polyhedral products, arXiv:1311.4256.
[Ku1] S. Kuroki, Classification of torus manifolds with codimension one extended
actions, Transformation Groups: 16 (2011) 481–536.
[Ku2] S. Kuroki, A remark on torus graph with root systems of type A, RIMS
Kokyuroku 1968 (2015) 55–59.
[Ku3] S. Kuroki Extended actions of quasitoric manifolds and J-constructions ofpolytopes, in preparation.
[KM] S. Kuroki, M. Masuda Root systems and symmetries of a torus manifold,Transform. Groups 22 (2017), Issue 2, 453–474.
[Wi1] M. Wiemeler, Torus manifolds with non-abelian symmetries, Trans. Amer.
Math. Soc. 364 (2012), no. 3, 1427–1487.
[Wi2] M. Wiemeler, Non-abelian symmetries of quasitoric manifolds, M´’unster J.
of Math. 7 (2014), 753-769.
Department of Applied Mathematics Faculty of Science, Okayama
Uniervsity of Science, 1-1 Ridai-cho Kita-ku Okayama-shi Okayama 7000005, Okayama, Japan
E-mail address: kuroki@xmath.ous.ac.jp

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