The Classification of Diffeomorphism Classes of Real Bott Manifolds

A real Bott manifold (RBM) is obtained as the orbit space of the n-torus T^n by a free action of an elementary abelian 2-group ZZ_2^n. This paper deals with the classification of some particular types of RBMs of dimension n, so that we know the number of diffeomorphism classes in such RBMs.


I. INTRODUCTION
K AMISHIMA et al. [1], [2] defined a real Bott manifold of dimension n (RBM n ) as the total space B n of the sequence of RP 1 -bundles B n → B n−1 → · · · → B 2 → B 1 → {a point} (1) starting with a point, where each RP 1 -bundle B i → B i−1 is the projectivization of the Whitney sum of a real line bundle L i and the trivial line bundle over B i−1 . Then, from the viewpoint of group actions, it was explained that a RBM n is the quotient of the torus of dimension n, T n = S 1 × · · · × S 1 by the product (Z 2 ) n of cyclic group of order 2. Such RBM n can be expressed by an upper triangular matrix A of size n (called a Bott matrix of size n, BM n ) whose entries are either 1 or 0 except the diagonal entries which are 0. Each row of the BM n A express the free action of (Z 2 ) n on T n and the orbit space M n (A) = T n /(Z 2 ) n is the RBM n . In fact, M n (A) is a Riemannian flat manifold (compact Euclidean space form). To classify RBM n s, we can apply the Bieberbach Theorem [3] and by this theorem, it was obtained in [1], [4] the classification of RBM s up to dimension 4. Kamishima and Nazra proved in [2] that every RBM n M n (A) admits an injective Seifert fibred structure which has the form M n (A) = T k × (Z2) s M (B), that is there is a k-torus action on M n (A) whose quotient space is an (n − k)-dimensional real Bott orbifold M n−k (B)/(Z 2 ) s by some (Z 2 ) s -action (1 ≤ s ≤ k). Moreover, they have proved the smooth rigidity that two RBM n s M n (A 1 ) and M n (A 2 ) are diffeomorphic if and only if the corresponding actions ((Z 2 ) s1 , M n−k1 (B 1 )) and ((Z 2 ) s2 , M n−k2 (B 2 )) are equivariantly diffeomorphic. By the above rigidity we can determine the diffeomorphism classes of higher dimensional RBM s when the low dimensional ones with (Z 2 ) s -actions are classified. RBM s up to dimension 5 have been classified (see [5], [6]).
This paper aims to study the number of diffeomorphism classes in some particular types of RBM n s.

II. PRELIMINARIES
In this section, we shall review some concepts from [2] related to the RBM .

A. Seifert fiber space
In a BM n A, each i-th row defines a Z 2 -action on T n by wherez m is either z m orz m depending on whether (i, m)entry (i < m) is 0 or 1 respectively while (i, i)-(diagonal) entry 0 acts as z i → −z i . Note thatz is the conjugate of the complex number z ∈ S 1 . It is always trivial; z m → z m whenever m < i. Here (z 1 , . . . , z n ) are the standard coordinates of the n-dimensional torus T n = S 1 × · · · × S 1 whose universal covering is the n-dimensional Euclidean space R n . The projection p : R n → T n is denoted by p(x 1 , . . . , x n ) = (e 2πix1 , . . . , e 2πixn ) = (z 1 , . . . , z n ).
Those g 1 , . . . , g n constitute the generators of (Z 2 ) n . In fact, (Z 2 ) n acts freely on T n such that the orbit space M n (A) = T n /(Z 2 ) n is a smooth compact n-dimensional manifold. In this way, given a BM n A, we obtain a free action of (Z 2 ) n on T n .
Let π(A) = g 1 , . . . ,g n be the lift of (Z 2 ) n = g 1 , . . . , g n to R n . Then, we get wherex m is either x m or −x m . One can see that π(A) acts properly discontinuously and freely on R n as Euclidean motions. Note that π(A) is a Bieberbach group which is a discrete uniform subgroup of the Euclidean group E(n) = R n O(n) (cf. [3]). It follows that Now, we consider the following moves (I, II, III) to A under which the diffeomorphism class of RBM n M n (A) does not change. I If the j-th column has all 0-entries for some j > 1, then interchange the j-th column and the (j − 1)-th column. Next, interchange the j-th row and the (j − 1)-th row.
We perform move I iteratively to get a BM n A .
O k is a k × k zero matrix (1 ≤ k ≤ n) and we call it a block zero matrix of size k. Note the following. (1) O k is a maximal block of zero matrix.
(2) As B is an (n − k)-dimensional Bott matrix, we obtain a real Bott manifold (4) The matrix C corresponds to (Z 2 ) k -action on T n−k . II For an m-th row (1 ≤ m ≤ k) whose entries in C are all zero, divide T k × M n−k (B) by the corresponding Z 2 -action. III If there are two rows, p-th row and -th row (1 ≤ p < ≤ k), having the common entries in the C, then compose the Z 2 -action of p-th row and -th row and divide T k ×M n−k (B) by Z 2 -action.
By using II, III, the quotient is again diffeomorphic to Hereinafter, we write M n (A) in place of M n (A ).
wherez =z or z. So there induces an action of (Z 2 ) s on M n−k (B) by Moreover in [2], it was obtained the following theorem.
There exist a central extension of the fundamental group π(A) of M n (A): such that (i) Z k is the maximal central free abelian subgroup (ii) The induced group Q B is the semidirect product π(B) (Z 2 ) s for which R n−k /π(B) = M n−k (B). See [2] for the proof.
Using this theorem, a RBM n M n (A) which admits a maximal T k -action (k ≥ 1) can be created from an RBM n−k M n−k (B) by a (Z 2 ) s -action, and the corresponding BM n A has the form as in (2) above.

B. Affine maps between real Bott manifolds
Next, to check whether two RBM s are diffeomorphic, we can apply the following theorem.
Theorem 2 (Rigidity): Suppose that M n (A 1 ) and M n (A 2 ) are RBM n s and 1 → Z ki → π(A i ) → Q Bi → 1 is the associated group extensions (i = 1, 2). Then, the following are equivalent: ). See [2] for the proof. Here Bott matrices A 1 and A 2 are created from B 1 and B 2 respectively.
Note that two RBM n s M n (A 1 ) and M n (A 2 ) are diffeomorphic if and only if π(A 1 ) is isomorphic to π(A 2 ) by the Bieberbach theorem [3]. Moreover, by Theorem 1 and 2 we have, If two RBM s have the same maximal T k -action, then the quo- C. Type of fixed point set Note that from (4), the action of ( the action α lifts to a linear (affine) action on T n−k naturally: α(z 1 , . . . , z n−k ) = (z 1 , . . . ,z n−k ). Then, the fixed point set is characterized by the equation: (z 1 , . . . ,z n−k ) = g(z 1 , . . . , z n−k ) for some g ∈ (Z 2 ) n−k . It is also an affine subspace of T n−k . So the fixed point sets of (Z 2 ) s are affine subspaces in M n−k (B).
Let B be the Bott matrix as in above. By a repetition of move I, B has the form where rank Note that by the Bieberbach theorem (cf. [3]), if f is an isomorphism of π(A 1 ) onto π(A 2 ), then there exists an affine element g = (h, H) ∈A(n) = R n GL(n, R) such that f (r) = grg −1 (∀r ∈ π(A 1 )). Recall . This implies that B 1 and B 2 have the form as in (7). Using (8) and according to the form of B in (7) we obtain that which is equivariant with respect tof . The pair (f ,ḡ) induces an equivariant affine diffeomorphism . . , z b2 ; z b2+1 , . . . , z b2+b3 ; . . . . . . ; z b +1 , . . . , z b +b ]}. We say thatĝ preserves the type (b 2 , . . . , b ) of M n−k (B 1 ). Asĝ isf -equivariant, it also preserves the type corresponding to the fixed point sets between ((Z 2 ) s , M n−k (B 1 )) and ((Z 2 ) s , M n−k (B 2 )). In this part, we will review some results from [6] and prove some new results regarding the classification of certain ndimensional real Bott manifolds in order to obtain how many diffeomorphism classes of some particular types of RBM n s. Proposition 2: [6] There are 4 diffeomorphism classes of RBM n s (n ≥ 4) which admit the maximal T n−2 -actions (i.e. s = 1, 2 ): Proposition 5: For any k ≥ 1 and m ≥ 2 Proof: Similar with the proof of Proposition 4 (see [6]). ( . Proposition 6: [6] Let M n (A) = S 1 × Z2 M n−k (B) be a RBM n . Suppose that B is either one of the list in (11). Then M n−k (B) are diffeomorphic to each other and the number of diffeomorphism classes of such RBM n s M n (A) above is (k + 1)2 n−k−3 (k ≥ 2, n − k ≥ 3).
Lemma 1: Let M n (A) be an RBM n (n ≥ 5) corresponding to the Bott matrix Then the number of diffeomorphism classes of such M n (A) is . Proof: We associate with the pair (y, x) the Bott matrix (14) where y = n − x and x are the numbers of zero entries in the (k −1)-th row and k-th row respectively of the right-upper block matrix. Here 1 ≤ x ≤ − 1. Because of move I, we may assume that . For a fixed numbers and x, it is easy to check that the fixed point sets of ((Z 2 ) 2 , T ) corresponding to (14) are 2 x components T −x and 2 −x components T x .
For a fixed number , suppose that Bott matrices A 1 and A 2 correspond to the pairs (y 1 , x 1 ) and (y 2 , x 2 ) respectively. If Therefore for a fixed number , there are [ 2 ] diffeomorphism classes of such RBM n s. This implies the lemma.
Lemma 2: Let M n (A) be a RBM n (n ≥ 5) corresponding to the Bott matrix Then the number of diffeomorphism classes of such M n (A) is Proof: We associate with the pair (t, x) the Bott matrix (15) where x is the number of zero entries in the k-th row of the right-upper block matrix and t( = 0) is the number of columns having two non zero entries. Because of move I, we may assume that For fixed numbers , x and t, it is easy to check that the fixed point sets of ((Z 2 ) 2 , T ) corresponding to (15) are 2 t+x components T −x−t and 2 −x components T x .
For a fixed number , suppose that Bott matrices A 1 and A 2 correspond to the pairs (t 1 , x 1 ) and (t 2 , x 2 ) respectively. If t 1 = t 2 or(and) x 1 = x 2 , then by Proposition 1, A 1 is not equivalent to A 2 .
Therefore for fixed numbers and x, there are −x 2 −(x− 1) diffeomorphism classes of such RBM n s. Hence there are diffeomorphism classes of such M n (A) corresponding to Bott matrices as in (15). Since the fixed point sets of ((Z 2 ) 2 , T ) corresponding to Bott matrices (14) and (15) are different, the corresponding real Bott manifolds are not diffeomorphics.
Remark 10: It is hard task algebraically to determine the number of n-dimensional M n (A) = T k × (Z2) s T for 3 ≤ s ≤ min{n − , }. However we shall consider a special type in (12).
Next Choi considers an n × n Bott matrix A such that the rank of submatrix consisting of the first columns is −1 and the last n− columns are zero vectors (i.e, A = A 0 0 0 ).
By move I, the Bott matrix A is equivalent to By using the result of Masuda above, Choi [7] obtained that the number of diffeomorphism classes of RBM n s corresponding to Bott matrices (16) for = 2, . . . , n is n =2 2 ( −2)( −3)/2 . Masuda [8] found that 2 (n−2)(n−3)/2 ≤ N n , by considering the Bott matrices A above. Then, Choi [7] improved the Masuda's result where he considers Bott matrices (16). We assume that if u < u 0 in a summation u =u0 , the value of such summation is equal to zero.