A TOPOLOGICAL DEFINITION OF HYPERTORIC MANIFOLDS ANDITS EQUIVARIANT COHOMOLOGYSHINTARÔ KUROKI

1. Introduction
In [2], Bielawski and Dancer deﬁned the hyperKähler analogue of symplectic toric manifolds, called a toric hyperKähler manifold or hypertoric manifold (also see [1, 6, 7, 8, 9, 10]).
On the other hand, there is the topological deﬁnition of toric manifolds introduced by
Davis-Januszkiewicz [4], called a quasitoric manifold (also see [3, 5]).
In this article, we give a constructive deﬁnition of a topological hypertoric manifold (seeSection 2), and compute its equivariant cohomology (see Theorem 3.2).
The details of this article will be appeared in the forthcoming paper [13].
2. Definition
Let T n be the n-dimensional compact abelian group, i.e., n-dimensional torus, and Rn ×
Cn be the product of n-dimensional real and complex Euclidean spaces. We let S ={S1 , . . . , Sm } denote a hypersurface arrangement in Rn , i.e., each Si ∈ S, i = 1, . . . , m,is a smooth surface in Rn which is diﬀeomorphic to Rn−1 . Note that Si ∈ S might not be
the aﬃne space in Rn . Assume that if ∩i∈I Si ̸= ∅ then ∩i∈I Si ∼= Rn−|I| , where |I| is thecardinality of I ⊂ [m].
We attach a label to each of Si ’s by using the characteristic function
λ : S → Zn
such thatdet(λ(Si1 ) · · · λ(Sin )) = ±1 when ∩nj=1 Sij ̸= ∅.
Now we may deﬁne the space M as follows:
M = T n × Rn × Cn / ∼λ .
Here, the equivalence relation ∼λ is deﬁned by the following relations:⊥(t1 , r, z) ∼λ (t2 , r, z) ⇔ t−1
1 t2 ∈ Tλ(S) if (r, z) ∈ S × λ(S) ;(t1 , r, z) ∼λ (t2 , r, z) ⇔ t1 = t2otherwise,Date: June 29, 2010.
2010 Mathematics Subject Classification. Principal:55N91; 57S25, Secondly:05C25; 05C65; 14M25.
Key words and phrases. Arrangement of hyper-surfaces, Equivariant cohomology, GKM manifold, GKM
graph, Toric hyperKähler manifold (hypertoric manifold), (Quasi)toric manifold.
The author was supported in part by Basic Science Research Program through the NRF of Korea fundedby the Ministry of Education, Science and Technology (2010-0001651) and the Fujyukai foundation.
12S. KUROKIwhere Tλ(S) is the connected 1-dimensional subgroup in T n whose Lie algebra is generatedby λ(S) ∈ Zn = tnZ for S ∈ S, and λ(S)⊥ is the normal space of the vector λ(S) inCn with ∑respect to the standard Hermitian inner product on Cn , i.e., the inner product⟨z, w⟩ = ni=1 zi wi for z = (z1 , . . . , zn ) and w = (w1 , . . . , wn ). Note that λ(S)⊥ is a complex(n − 1)-dimensional linear subspace in Cn .
Then we have the following theorem.
Theorem 2.1. For the space M defined above, the following statements hold.
(1) M is a 4n-dimensional non-compact manifold.
(2) M has the natural T n -action on the T n -factor in T n × Rn × Cn .
(3) M has the extra S 1 -action on the Cn -factor in T n × Rn × Cn by the scaler product.
We call M deﬁned as above a topological hypertoric manifold.
Remark 2.2. Let X be a hypertoric manifold. Then, X satisﬁes the above conditions, i.e.,
X is a topological hypertoric manifold. Moreover, in this case, all Si ’s are codimension-one
aﬃne subspaces in Rn .
Remark 2.3. For all 2n-dimensional quasitoric manifolds V , there is a 4n-dimensional
topological hypertoric manifold M such that M is T n -homotopy equivalent to V .
It follows from Remark 2.2 and 2.3 that the set of all topological hypertoric manifolds
is strictly wider than that of all hypertoric manifolds.
3. Equivariant cohomology
In order to compute T n ×S 1 -equivariant cohomologies of topological hypertoric manifolds
M , we ﬁrst prepare the following lemma.
Lemma 3.1. A topological hypertoric manifold with the extra S 1 -action (M 4n , T n × S 1 ) isa GKM manifold whose GKM graph is a 2n-valent hypertorus graph.
Furthermore, the hypertorus graph defined by (M 4n , T n × S 1 ) satisfies the following conditions:(1) For each L ∈ L, there is the unique pair of a hyperfacet H and its opposite side Hsuch that H ∩ H = L;
(2) For each subset L′ ⊂ L, the intersection of all elements in L′ is empty or connected.
where L = {L1 , · · · , Lm } is the set of all (2n − 2)-valent hypertorus subgraphs.
Here, a hypertorus graph is the special type of GKM graphs deﬁned in [12] (also see [11]),and all notations appeared in Lemma 3.1 are deﬁned in [12].
Using Lemma 3.1 and the main theorem in [12], we have the following theorem.
Theorem 3.2. Let (M 4n , T n × S 1 ) be the topological hypertoric manifold defined by the labeled, hypersurface arrangement S = {S1 , . . . , Sm } in Rn . Then its equivariant cohomology
HT n ×S 1 (M ; Z) is isomorphic to the following ring:Z[τ1 , . . . , τm , τm+1 , . . . , τ2m , χ]/I,

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where deg τi = deg χ = 2, i = 1, . . . , 2m, and the ideal I is generated by the following
homogeneous polynomials:
τi + τm+i − χ for all i = 1, . . . , m;
∏τi for ∩i∈I Hi = ∅ (I ⊂ [2m]).
i∈IHere, Hj in Theorem 3.2 is deﬁned as follows. Now Rn can be divided into two half
(topological) spaces by Sj , j = 1, . . . , m; we denote one of them and another by Hj andHj+m , respectively. That is, Hj and Hj+m are diﬀeomorphic to the half spaces Rn+ and Rn−(respectively) in Rn ; moreover, they satisfy that Hj ∪ Hj+m = Rn and Hj ∩ Hj+m = Sj .
See [13] for further studies.
Acknowledgements. The author would like to thank all speakers and participants inKAIST Toric Topology Workshop 2010 from February 22nd to 26th 2010. I also would
like to express my gratitude to Professor DongYoup Suh for providing me excellent circumstances to do research.
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(Japanese)[12] S. Kuroki, Hypertorus graphs and graph equivariant cohomologies, preprint (2007).
[13] S. Kuroki, Topological generalization of hypertoric manifolds, in preparation.
Department of Mathematical Sciences, KAIST, Daejeon 305-701, R. Korea
E-mail address: kuroki@kaist.ac.kr