# Small Covers Over the Product of 3-Sided Prism with n -Simplex

Small Covers Over the Product of 3-Sided Prism with n -Simplex In this paper, the author calculates the number of equivariant homeomorphism classes of (orientable) over the product of 3-sided prism with n-simplex. Mathematics Subject Classification 2010: 57S10, 57S25, 52B11. Key words: small cover, equivariant homeomorphism, polytope. 1. Introduction A small cover, defined by Davis and Januszkiewicz in [9], is a smooth closed manifold M n with a locally standard (Z2 )n -action such that its orbit space is a simple convex polytope. For instance, the real projective space RP n with a natural (Z2 )n -action is a small cover over an n-simplex. This gives a direct connection between equivariant topology and combinatorics and makes it possible to study the topology of through the combinatorial structure of quotient spaces. ¨ In [13], Lu and Masuda showed that the equivariant homeomorphism class of a small cover over a simple convex polytope P n agrees with the equivalence class of its corresponding (Z2 )n -coloring under the action of automorphism group of face poset of P n . This holds for orientable by the orientability condition in [15] (see Theoerem 4.4). But there aren't general formulas to calculate the number of equivariant homeomorphism classes of (orientable) over an arbitrary simple convex polytope. In recent years, several studies have attempted to enumerate the number of equivalence classes of all over a specific polytope. Garrison and Scott [10] used a computer program to calculate the number of homeomorphism classes of all over a dodecahedron. In 2008, Choi [6] determined the number of equivariant homeomorphism classes of over cubes. Prisms are an interesting class of 3-dimensional ¨ polytopes. Cao and Lu [4] classified over prisms up to homeomorphism and calculated the number of homeomorphism classes of small ¨ covers over prisms. Cai, Chen and Lu [3] calculated the number of equivariant homeomorphism classes of over prisms. As a generalization, Wang and Chen [17] determined the number of equivariant homeomorphism classes of over products of a polygon with a simplex. Furthermore, Chen and Wang [5] calculated the number of D-J equivalence classes of all orientable over products of a prism with a simplex. ¨ The smallest interesting prism is the 3-sided prism. Lu and Yu [14] showed that one can obtain each 3-dimensional small cover from the small cover S 1 × RP 2 over the 3-sided prism and the small cover RP 3 over the 3-simplex by using cut and paste strategies in the sense of six equivariant operations. Let M be the set of equivariant unoriented cobordism classes of all 3-dimensional . Then M is generated by the classes of S 1 ×RP 2 over the 3-sided prism and RP 3 over the 3-simplex ([12, 14]). Much further research on over the 3-sided prism has been carried on ([11, 16]). An example of a cohomologically non-rigid polytope was obtained from a 3-sided prism by iterating the operation of vertex cut twice ([8]). Products of polytopes are another interesting class of polytopes and more complicated than one might think ([18]). And over products of simple convex polytopes have become an important search object ([5, 6, 7, 17]). Motivated by these, the author calculates the number of equivariant homeomorphism classes of over the product of 3-sided prism with n-simplex in this paper. From [15], there exist orientable over every simple convex 3-polytope and there exist non-orientable over every simple convex 3-polytope, except the 3-simplex. An orientable 3-dimensional small cover corresponds to a 4-colored simple convex 3-polytope and the existence of an orientable small cover over every simple convex 3-polytope is closely related to the four color theorem (see [2] for the four color theorem). In [7], Choi calculated the number of D-J equivalence classes of orientable over cubes. The number of equivariant homeomorphism classes of orientable over the product of 3-sided prism with n-simplex is also calculated in this paper. Let P3 , n be a 3-sided prism (i.e., the product of a 3-gon and the interval I) and an n-simplex respectively. The main results of this paper are stated as follows: Theorem 1.1. Let E(P3 × n ) be the number of equivariant homeomorphism classes of over P3 × n . When n 3, E(P3 × n ) = (5 · 4n + 9 · 2n+1 + 33) n+3 (2n+3 - 2k-1 ) . 12(n + 1)! Up to equivariant homeomorphism, there are 12846400 over P3 × 2 , 47460 over P3 × I and 98 over P3 . Theorem 1.2. Let Eo (P3 × n ) be the number of equivariant homeomorphism classes of orientable over P3 × n . When n 3 and n is odd, (5 · 4n-1 + 2n + 6) n+3 (2n+3 - 2k-1 ) . Eo (P3 × ) = 12(n + 1)! When n 4 and n is even, Eo (P3 × n ) = (2n+1 + 6) n+3 (2n+3 - 2k-1 ) . 12(n + 1)! Up to equivariant homeomorphism, there are 972160 orientable over P3 × 2 , 8400 orientable over P3 × I and 28 orientable over P3 . The paper is organized as follows. In Section 2, we review the basic theory about and calculate the automorphism group of face poset of P3 × n . In Section 3, we determine the number of all (Z2 )n+3 colorings on P3 × n and prove Theorem 1.1. In Section 4, we calculate the number of all orientable colorings on P3 × n and prove Theorem 1.2. 2. Preliminaries A convex polytope P n of dimension n is said to be simple if every vertex of P n is the intersection of exactly n facets (i.e. faces of dimension (n - 1)) (see [18]). An n-dimensional smooth closed manifold M n is said to be a small cover if it admits a smooth (Z2 )n -action such that the action is locally isomorphic to a standard action of (Z2 )n on Rn and the orbit space M n /(Z2 )n is a simple convex polytope of dimension n. Let P n be a simple convex polytope of dimension n and F(P n ) = {F1 , . . ., F } be the set of facets of P n . Suppose that : M n P n is a small cover over P n . Then there are connected submanifolds -1 (F1 ), . . ., -1 (F ). Each submanifold -1 (Fi ) is fixed pointwise by a Z2 -subgroup Z2 (Fi ) of (Z2 )n , so that each facet Fi corresponds to the Z2 -subgroup Z2 (Fi ). Obviously, the Z2 -subgroup Z2 (Fi ) actually agrees with an element i in (Z2 )n as a vector space. For each face F of codimension u, since P n is simple, there are u facets Fi1 , . . . , Fiu such that F = Fi1 . . .Fiu . Then, the corresponding submanifolds -1 (Fi1 ), . . . , -1 (Fiu ) intersect transversally in the (n - u)-dimensional submanifold -1 (F ), and the isotropy subgroup Z2 (F ) of -1 (F ) is a subtorus of rank u and is generated by Z2 (Fi1 ), . . . , Z2 (Fiu ) (or is determined by i1 , . . . , iu in (Z2 )n ) (see [9]). Consider a map : F(P n ) - (Z2 )n which satisfies the nonsingularity condition: {(Fi1 ), . . . , (Fin )} is a basis of (Z2 )n whenever the intersection Fi1 . . . Fin is non-empty. We call a characteristic function. If we regard each nonzero vector of (Z2 )n as being a color, then the characteristic function means that each facet is colored by a color. Here we also call a (Z2 )n -coloring on P n . In fact, Davis and Januszkiewicz [9] gave a reconstruction process of a small cover by using a (Z2 )n -coloring : F(P n ) - (Z2 )n . Let Z2 (Fi ) be the subgroup of (Z2 )n generated by (Fi ). Given a point p P n , by F (p) we denote the minimal face containing p in its relative interior. Assume F (p) = Fi1 . . . Fiu and Z2 (F (p)) = u Z2 (Fij ). Note that Z2 (F (p)) is j=1 a u-dimensional subgroup of (Z2 )n . Let M () denote P n × (Z2 )n / , where (p, g) (q, h) if p = q and g-1 h Z2 (F (p)). The free action of (Z2 )n on P n × (Z2 )n descends to an action on M () with quotient P n . Thus M () is a small cover over P n . Two M1 and M2 over P n are said to be weakly equivariantly homeomorphic if there is an automorphism : (Z2 )n (Z2 )n and a homeomorphism f : M1 M2 such that f (t · x) = (t) · f (x) for every t (Z2 )n and x M1 . If is an identity, then M1 and M2 are equivariantly homeomorphic. Following [9], two M1 and M2 over P n are said to be Davis-Januszkiewicz equivalent (or simply, D-J equivalent) if there is a weakly equivariant homeomorphism f : M1 M2 covering the identity on P n . By (P n ) we denote the set of all (Z2 )n -colorings on P n . Then we have: Theorem 2.1. All over P n are given by {M ()| (P n )}, i.e. for each small cover M n over P n , there is a (Z2 )n -coloring with an equivariant homeomorphism M () - M n covering the identity on P n . Remark 2.2. Generally speaking, we can't make sure that there always exist over a simple convex polytope P n when n 4. For example, see Nonexample 1.22 of [9]. However, the Four Color Theorem makes sure that there always exist over every 3-dimensional simple convex polytope, so there exist over P3 . Since RP n is a small cover over n , there exist over P3 × n . There is a natural action of GL(n, Z2 ) on (P n ) defined by the correspondence - , and the action on (P n ) is free. Without loss of generality, we assume that F1 , . . . , Fn of F(P n ) meet at one vertex p of P n . Let e1 , . . . , en be the standard basis of (Z2 )n . Write A(P n ) = { (P n )|(Fi ) = ei for i = 1, . . . , n}. In fact, A(P n ) is the orbit space of (P n ) under the action of GL(n, Z2 ). Then we have Lemma 2.3. |(P n )| = |A(P n )| × |GL(n, Z2 )|. n n k-1 ([1]). Two Note that |GL(n, Z2 )| = 2 - 2 M (1 ) and M (2 ) over P n are D-J equivalent if and only if there is GL(n, Z2 ) such that 1 = 2 . So the number of D-J equivalence classes of over P n is |A(P n )|. Let P n be a simple convex polytope of dimension n. All faces of P n form a poset (i.e. a partially ordered set by inclusion). An automorphism of F(P n ) is a bijection from F(P n ) to itself which preserves the poset structure of all faces of P n , and by Aut(F(P n )) we denote the group of automorphisms of F(P n ). One can define the right action of Aut(F(P n )) on (P n ) by × h - h, where (P n ) and h Aut(F(P n )). The following theorem is well known (see [13]). Theorem 2.4. Two over an n-dimensional simple convex polytope P n are equivariantly homeomorphic if and only if there is h Aut(F(P n )) such that 1 = 2 h, where 1 and 2 are their corresponding (Z2 )n -colorings on P n . So the number of orbits of (P n ) under the action of Aut(F(P n )) is just the number of equivariant homeomorphism classes of over P n . Thus, we are going to count the orbits. Burnside Lemma is very useful in the enumeration of the number of orbits. Burnside Lemma. Let G be a finite group acting on a set X. Then 1 the number of orbits of X under the action of G equals |G| gG |Xg |, where Xg = {x X|gx = x}. Burnside Lemma suggests that we need to understand the structure of Aut(F(P n )) in order to determine the number of the orbits of (P n ) under the action of Aut(F(P n )). We shall particularly be concerned with the case in which the simple convex polytope is P3 × n . To be convenient, we introduce the following marks. By s and s we 2 1 denote the top and bottom facets of P3 respectively, and by a , a , a we 1 2 3 denote three sided facets of P3 in their general order. For n-simplex n , n n by b , b , . . . , b 1 2 n+1 we denote all facets of . Set F = {s1 = s1 × , s2 = s × n , ai = a × n |1 i 3}, F = {bj = P3 × b |1 j n + 1}. Then 2 i j F(P3 × n ) = F F . Lemma 2.5. Let P3 , n be a 3-sided prism and an n-simplex respectively. The automorphism group Aut(F(P3 × n )) is isomorphic to S3 × (Z2 )3 , n = 1, 2 × (Z )2 , (S ) n = 2, 2 3 S3 × Z2 × Sn+1 , n 3, where Sn+1 is the symmetric group of rank n + 1. Proof. First, we show that the automorphism group Aut(F(P3 )) is isomorphic to S3 × Z2 . Let three sided facets of P3 interchange and the top and bottom facets stay unchanged. Then these automorphisms form a group S3 . Let the top and bottom facets interchange and three sided facets of P3 stay unchanged. These automorphisms form a group Z2 . Since any one of all sided facets cannot be mapped to the top facet or bottom facet under the automorphisms of F(P3 ), Aut(F(P3 )) is isomorphic to S3 × Z2 . When n 3, the facets of F and F are mapped to F and F respectively under the automorphisms of Aut(F(P3 × n )). Since Aut(F(P3 )) is isomorphic to S3 × Z2 and Aut(F(n )) is isomorphic to Sn+1 , Aut( F(P3 × n )) is isomorphic to S3 × Z2 × Sn+1 . Next, we consider the case in which n=2. Below we show that the automorphism group Aut(F(2 × 2 )) is isomorphic to S3 × S3 × Z2 . 2 is a 3-gon. By c , c , c we denote three edges of 2 in their general order. 1 2 3 Let F1 = {ci = c × 2 |1 i 3}, F1 = {dj = 2 × c |1 j 3}. Then i j F(2 × 2 ) = F1 F1 . There are automorphisms of F(2 × 2 ) under which the facets of F1 and F1 are mapped to F1 and F1 respectively. These automorphisms form a group S3 × S3 . We choose an automorphism f such that f (ci ) = di for 1 i 3 and f (dj ) = cj for 1 j 3. Let Z2 = {f, 1}. Then we get a new group S3 × S3 × Z2 , each of which is an automorphism under which the facets in F1 and F1 are mapped to F1 and F1 or to F1 and F1 respectively. Since other bijections from F(2 × 2 ) to itself don't preserve the poset structure of all faces of 2 × 2 , Aut(F(2 × 2 )) is isomorphic to S3 × S3 × Z2 . When n=2, P3 ×n is 2 ×I ×2 . Since Aut(F(2 ×2 )) is isomorphic to S3 × S3 × Z2 and Aut(F(I)) is isomorphic to Z2 , Aut(F(P3 × 2 )) is isomorphic to S3 × S3 × Z2 × Z2 . Finally, we consider the case in which n=1. Below we show that the automorphism group Aut(F(I 2 )) is isomorphic to (Z2 )3 . Aut(F(I 2 )) contains a Z2 -subgroup since there is one automorphism for the permutation of the two pairs of opposite facets of I 2 = I × I. All elements of Aut(F(I 2 )) can be written in a simple form as follows: ae1 be2 · u, where e1 , e2 Z2 , with reflections a, b and u the former Z2 -subgroup. Thus, Aut(F(I 2 )) is isomorphic to (Z2 )3 . When n = 1, P3 × n is 2 × I 2 . Since the automorphism group Aut(F(2 )) is isomorphic to S3 and Aut(F(I 2 )) is isomorphic to (Z2 )3 , Aut(F(P3 × I)) is isomorphic to S3 × (Z2 )3 . 3. The number of over P3 × n First, we give a criterion for a map : F(P3 × n ) - (Z2 )n+3 to be a characteristic function. The nonsingularity condition of the characteristic function means the following: (1) {(s1 ), (a1 ), (a2 ), (b1 ), . . . , (bn )} is a basis of (Z2 )n+3 . (2) (bn+1 ) satisfies that {(bn+1 ), (bk1 ), . . . , (bkn-1 ), (s1 ), (a1 ), (a2 )} is a basis of (Z2 )n+3 , where k1 < k2 < . . . < kn-1 and k1 , . . . , kn-1 {1, 2, . . . , n}. (3) (a3 ) satisfies that {(a3 ), (s1 ), (al1 ), (bh1 ), . . . , (bhn )} is a basis of (Z2 )n+3 , where l1 {1, 2}, h1 < h2 < . . . < hn and h1 , . . . , hn {1, 2, . . . , n + 1}. (4) (s2 ) satisfies that {(s2 ), (aj1 ), (aj2 ), (bh1 ), . . . , (bhn )} is a basis of (Z2 )n+3 , where j1 < j2 , j1 , j2 {1, 2, 3}, h1 < h2 < . . . < hn and h1 , . . . , hn {1, 2, . . . , n + 1}. Next, we calculate the number of (Z2 )n+3 -colorings on P3 × n . n Lemma 3.1. When n 1, the number of (Z2 )n+3 -colorings over P3 × is |(P3 × n )| = (5 · 4n + 7 · 2n+1 + 21) n+3 (2n+3 - 2k-1 ). Proof. Let e1 , e2 , . . . , en+3 be the standard basis of (Z2 )n+3 , then (Z2 )n+3 contains 2n+3 - 1 nonzero elements (or 2n+3 - 1 colors). We choose s1 , a1 , a2 from F and b1 , . . . , bn from F , then s1 , a1 , a2 , b1 , . . . , bn meet at one vertex of P3 × n . Then A(P3 × n ) = { (P3 × n )|(s1 ) = e1 , (a1 ) = e2 , (a2 ) = e3 , (bi ) = ei+3 , 1 i n}. Then, by Lemma 2.3, we have that n+3 |(P3 × )| = |A(P3 × )|×|GL(n+3, Z2 )| = (2n+3 -2k-1 )|A(P3 ×n )|. Write A0 (P3 × n ) = { A(P3 × n )|(bn+1 ) = e4 + . . . + en+3 }, A1 (P3 × n ) = { A(P3 × n )|(bn+1 ) = e4 + . . . + en+3 + eh1 , 1 h1 3}, A2 (P3 × ) = { A(P3 × n )|(bn+1 ) = e4 + . . . + en+3 + ek1 + ek2 , 1 k1 < k2 3}, A3 (P3 × ) = { A(P3 × n )|(bn+1 ) = e4 + . . . + en+3 + e1 + e2 + e3 }. By the definition of (Z2 )n+3 -colorings, we have that 3 n n |A(P3 × n )| = i=0 |Ai (P3 × n )|. Then, our argument proceeds as follows: (I) Calculation of |A0 (P3 × n )|. In this case, (a3 ) = e2 + e3 + em1 . . . + eml , where 1 m1 < . . . < ml n + 3, m1 = 2, m1 = 3, . . . , ml = 2, ml = 3 and 0 l n + 1. When (a3 ) = e2 + e3 + em1 + . . . + eml , 4 m1 < . . . < ml n + 3 and 0 l n, (s2 ) = e1 + en1 + . . . + enj , where 2 n1 < . . . < nj n + 3 and 0 j n+2. When (a3 ) = e2 +e3 +e1 +em1 +. . .+eml , 4 m1 < . . . < ml n+3 and 0 l n, (s2 ) = e1 + eg1 + . . . + egk , where 4 g1 < . . . < gk n + 3 and 0 k n. Thus, we have |A0 (P3 × n )| = 5 · 4n . (II) Calculation of |A1 (P3 × n )|. The argument is divided into two cases. Case 1. (bn+1 ) = e4 + . . . + en+3 + e1 In this case, (a3 ) = e2 + e3 + em1 . . . + eml , where 1 m1 < . . . < ml n + 3, m1 = 2, m1 = 3, . . . , ml = 2, ml = 3 and 0 l n + 1. When (a3 ) = e2 + e3 , (s2 ) = e1 , e1 + e2 , e1 + e3 , e1 + e2 + e3 . When (a3 ) = e2 + e3 + em1 . . . + eml , 1 m1 < . . . < ml n + 3, m1 = 2, m1 = 3, . . . , ml = 2, ml = 3 and 1 l n + 1, (s2 ) = e1 . Case 2. (bn+1 ) = e4 + . . . + en+3 + e2 or e4 + . . . + en+3 + e3 In this case, no matter which value of (bn+1 ) is chosen, we have (a3 ) = e2 + e3 , e2 + e3 + e1 . When (a3 ) = e2 + e3 , (s2 ) = e1 + en1 + . . . + enj , 2 n1 < . . . < nj n + 3 and 0 j n + 2. When (a3 ) = e2 + e3 + e1 , (s2 ) = e1 . Thus, we have |A1 (P3 × n )| = 5 · 2n+1 + 5. (III) Calculation of |A2 (P3 × n )|. The argument is also divided into two cases. Case 1. (bn+1 ) = e4 + . . . + en+3 + e1 + e2 or e4 + . . . + en+3 + e1 + e3 In this case, no matter which value of (bn+1 ) is chosen, we have (a3 ) = e2 + e3 , e2 + e3 + e1 . When (a3 ) = e2 + e3 , (s2 ) = e1 , e1 + e2 , e1 + e3 , e1 + e2 + e3 . When (a3 ) = e2 + e3 + e1 , (s2 ) = e1 . Case 2. (bn+1 ) = e4 + . . . + en+3 + e2 + e3 In this case, (a3 ) = e2 + e3 , e2 + e3 + e1 . When (a3 ) = e2 + e3 , (s2 ) = e1 + en1 + . . . + enj , 2 n1 < . . . < nj n + 3 and 0 j n + 2. When (a3 ) = e2 + e3 + e1 , (s2 ) = e1 . Thus, we have |A2 (P3 × n )| = 2n+2 + 11. (IV) Calculation of |A3 (P3 × n )|. In this case, (a3 ) = e2 + e3 , e2 + e3 + e1 . When (a3 ) = e2 + e3 , (s2 ) = e1 , e1 + e2 , e1 + e3 , e1 + e2 + e3 . When (a3 ) = e2 + e3 + e1 , (s2 ) = e1 . Thus, we have |A3 (P3 × n )| = 5. Remark 3.2. From Lemma 3.1, the number of D-J equivalence classes of over P3 × n is 5 · 4n + 7 · 2n+1 + 21. The proof of Theorem 1.1. When n 3, from Theorem 2.4, Burnside Lemma and Lemma 2.5, we have that E(P3 × n ) = 1 12(n + 1)! |g |, gAut(F (P3 ×n )) where g = { (P3 × n )| = g}. When n 3, the automorphism group Aut(F(P3 ×n )) is isomorphic to S3 ×Z2 ×Sn+1 . If g is the generator of the Z2 -subgroup of Aut(F(P3 × n )) and g , then (s1 ) = (s2 ). By the argument of Lemma 3.1, we have |g | = (2n+2 + 12) n+3 (2n+3 - 2k-1 ). If g is other automorphism and isn't unit element of Aut(F(P3 × n )), by the nonsingularity condition of characteristic functions, we have that g is empty. Thus, from Lemma 3.1, when n 3, E(P3 × n ) = 1 (2n+2 + 12 + 5 · 4n + 7 · 2n+1 + 21)· 12(n + 1)! n+3 (2n+3 - 2k-1 ) (5 · 4n + 9 · 2n+1 + 33) n+3 (2n+3 - 2k-1 ) . 12(n + 1)! When n=2, similarly we have E(P3 × 2 ) = 12846400. When n=1, n is the interval I. Aut(F(P3 × I)) is isomorphic to S3 × Z2 ×Z2 ×Z2 . If g is the generator a of the first Z2 -subgroup of Aut(F(P3 ×I)) and g , then (s1 ) = (s2 ). By the argument of Lemma 3.1, we have |g | = 20 4 (24 - 2k-1 ). If g is the generator b of the second Z2 subgroup of Aut(F(P3 × I)) and g , then (b1 ) = (b2 ). By the argument of (I) of Lemma 3.1, we also have |g | = 20 4 (24 - 2k-1 ). If g is the automorphism ab of Aut(F(P3 × I)), similarly we have |g | = 4 4 (24 - 2k-1 ). If g is other automorphism and isn't unit element of Aut(F(P3 × I)), by the nonsingularity condition of characteristic functions, we have that g is empty. Thus, from Lemma 3.1, we have E(P3 × I) = 1 (20 + 20 + 4 + 69) 48 (24 - 2k-1 ) = 47460. In the similar way, we have that there are 98 equivariant homeomorphism classes of over P3 . 4. The number of orientable over P3 × n Nakayama and Nishimura [15] found an orientability condition for a small cover. Theorem 4.1. For a basis {e1 , . . . , en } of (Z2 )n , a homomorphism : (Z2 )n - Z2 = {0, 1} is defined by (ei ) = 1(i = 1, . . . , n). A small cover M () over a simple convex polytope P n is orientable if and only if there exists a basis {e1 , . . . , en } of (Z2 )n such that the image of is {1}. We call a (Z2 )n -coloring which satisfies the orientability condition in Theorem 4.1 an orientable coloring of P n . We can know the existence of orientable over P3 × n by the existence of orientable colorings and determine the number of equivariant homeomorphism classes. By O(P n ) we denote the set of all orientable colorings on P n . There is a natural action of GL(n, Z2 ) on O(P n ) defined by the correspondence - , and the action on O(P n ) is free. Assume that F1 , . . . , Fn of F(P n ) meet at one vertex p of P n . Let e1 , . . . , en be the standard basis of (Z2 )n . Write B(P n ) = { O(P n )|(Fi ) = ei for i = 1, . . . , n}. It is easy to check that B(P n ) is the orbit space of O(P n ) under the action of GL(n, Z2 ). Remark 4.2. In fact, we have B(P n ) = { O(P n )|(Fi ) = ei for i = 1, . . . , n, and for n + 1 j , (Fj ) = ej1 + ej2 + . . . + ej2hj +1 , where 1 j1 < j2 < . . . < j2hj +1 n}. Below we show that (Fj ) = ej1 + ej2 + . . . + ej2hj +1 for n + 1 j . If O(P n ), there exists a basis {e , . . . , e } of (Z2 )n such that for 1 i , (Fi ) = e 1 + . . . + e 2f +1 , n 1 i i i where 1 i1 < . . . < i2fi +1 n. Since (Fi ) = ei for i = 1, . . . , n, we have ei = e 1 + . . . + e 2f +1 . So we obtain that for n + 1 j , there aren't i i i j1 , . . . , j2k such that (Fj ) = ej1 + . . . + ej2k , where 1 j1 < . . . < j2k n. Since B(P n ) is the orbit space of O(P n ), we have Lemma 4.3. |O(P n )| = |B(P n )| × |GL(n, Z2 )|. One can define the right action of Aut(F(P n )) on O(P n ) by × h - h, where O(P n ) and h Aut(F(P n )). By improving the classifying result on in [13], we have Theorem 4.4. Two orientable over an n-dimensional simple convex polytope P n are equivariantly homeomorphic if and only if there is h Aut(F(P n )) such that 1 = 2 h, where 1 and 2 are their corresponding orientable colorings on P n . Proof. We know Theorem 4.4 is true by combining Lemma 5.4 in [13] with Theorem 4.1. By Theorem 4.4, the number of orbits of O(P n ) under the action of Aut(F(P n )) is the number of equivariant homeomorphism classes of orientable over P n . So we also are going to count the orbits. Lemma 4.5. When n 1, the number of orientable colorings on P3 × n is n+3 (5 · 4n-1 + 3) (2n+3 - 2k-1 ), n odd, |O(P3 × n )| = n+3 n (2 + 3) (2n+3 - 2k-1 ), n even. Proof. Let e1 , e2 , . . . , en+3 be the standard basis of (Z2 )n+3 . Then B(P3 × n ) = { O(P3 × n )|(s1 ) = e1 , (a1 ) = e2 , (a2 ) = e3 , (bi ) = ei+3 , 1 i n}. Then, by Lemma 4.3, we have that |O(P3 × n )| = |B(P3 × n )| × |GL(n + 3, Z2 )| = n+3 (2n+3 - 2k-1 )|B(P3 × n )|. The calculation of |B(P3 × n )| is divided into two cases: (I) n odd, (II) n even. (I) n odd Write: B0 (P3 × n ) = { B(P3 × n )|(bn+1 ) = e4 + . . . + en+3 }, B1 (P3 × n ) = { B(P3 × n )|(bn+1 ) = e4 + . . . + en+3 + ek1 + ek2 , 1 k1 < k2 3}. By the definition of B(P n ), we have that |B(P3 × n )| = |B0 (P3 × n )| + |B1 (P3 × n )|. Then, our argument proceeds as follows. Case 1. Calculation of |B0 (P3 × n )|. In this case, (a3 ) = e2 + e3 + em1 . . . + eml , where 1 m1 < . . . < ml n + 3, m1 = 2, m1 = 3, . . . , ml = 2, ml = 3, l odd and 1 l n + 1. When (a3 ) = e2 + e3 + em1 + . . . + eml , 4 m1 < . . . < ml n + 3, l odd and 1 l n, (s2 ) = e1 + en1 + . . . + enj , 2 n1 < . . . < nj n + 3, j even and 0 j n + 2. When (a3 ) = e2 + e3 + e1 + em1 + . . . + eml , 4 m1 < . . . < ml n + 3, l even and 0 l n, (s2 ) = e1 + eg1 + . . . + egk , 4 g1 < . . . < gk n + 3, k even and 0 k n. Thus, we have |B0 (P3 × n )| = 5 · 4n-1 . Case 2. Calculation of |B1 (P3 × n )|. In this case, no matter which value of (bn+1 ) is chosen, we have (a3 ) = e2 + e3 + e1 , (s2 ) = e1 . Thus, we have |B1 (P3 × n )| = 3. (II) n even Write B0 (P3 × n ) = { B(P3 × n )|(bn+1 ) = e4 + . . . + en+3 + eh1 , 1 h1 3}, B1 (P3 ×n ) = { B(P3 × n )|(bn+1 )=e4 + . . . +en+3 +e1 + e2 + e3 }. n )| By the definition of B(P n ), we have that |B(P3 × n )| = |B0 (P3 × (P × n )|. Then, our argument proceeds as follows. + |B1 3 Case 1. Calculation of |B0 (P3 × n )|. The argument is divided into two cases: (1) (bn+1 ) = e4 + . . . + en+3 + e1 . In this case, (a3 ) = e2 + e3 + em1 . . . + eml , where 1 m1 < . . . < ml n + 3, m1 = 2, m1 = 3, . . . , ml = 2, ml = 3, l odd and 1 l n + 1, (s2 ) = e1 . (2) (bn+1 ) = e4 + . . . + en+3 + e2 or e4 + . . . + en+3 + e3 . In this case, no matter which value of (bn+1 ) is chosen, we have (a3 ) = e2 + e3 + e1 , (s2 ) = e1 . Thus, we have |B0 (P3 × n )| = 2n + 2. Case 2. Calculation of |B1 (P3 × n )|. In this case, (a3 ) = e2 + e3 + e1 , (s2 ) = e1 . Thus, we have |B1 (P3 × n )| = 1. The proof of Theorem 1.2. When n 3, from Theorem 4.4, Burnside Lemma and Lemma 2.5, we have that Eo (P3 × n ) = 1 12(n + 1)! |g |, gAut(F (P3 ×n )) where g = { O(P3 × n )| = g}. When n 3 and n is odd, the automorphism group Aut(F(P3 × n )) is isomorphic to S3 × Z2 × Sn+1 . If g is the generator of the Z2 -subgroup of Aut(F(P3 × n )) and g , then (s1 ) = (s2 ). By the argument of (I) of Lemma 4.5, we have |g | = (2n + 3) n+3 (2n+3 - 2k-1 ). If g is other automorphism and isn't unit element of Aut(F(P3 × n )), then g is empty. Thus, from Lemma 4.5, when n 3 and n is odd, Eo (P3 × n ) = = 1 (2n + 3 + 5 · 4n-1 + 3) 12(n + 1)! (5 · 4n-1 + 2n + 6) 12(n + 1)! n+3 n+3 (2 n+3 (2n+3 - 2k-1 ) 2k-1 ) When n 4 and n is even, using the above method we have Eo (P3 × n ) = (2n+1 + 6) n+3 (2n+3 - 2k-1 ) . 12(n + 1)! When n=2, similarly we have Eo (P3 × 2 ) = 972160. When n=1, Aut(F(P3 × n )) is isomorphic to S3 × Z2 × Z2 × Z2 . If g is the generator a of the first Z2 -subgroup of Aut(F(P3 × I)) and g , then (s1 ) = (s2 ). By the argument of (I) of Lemma 4.5, we have |g | = 5 4 (24 - 2k-1 ). If g is the generator b of the second Z2 -subgroup of Aut(F(P3 × I)) and g , then (b1 ) = (b2 ). By the argument of Case 1 in (I) of Lemma 4.5, we also have |g | = 5 4 (24 - 2k-1 ). If g is the automorphism ab of Aut(F(P3 × I)), similarly we have |g | = 2 4 (24 - 2k-1 ). If g is other automorphism and isn't unit element of Aut(F(P3 × I)), then g is empty. Thus, from Lemma 4.5, we have Eo (P3 × I) = 1 (5 + 5 + 2 + 8) 48 (24 - 2k-1 ) = 8400. In the similar way, we have that there are 28 equivariant homeomorphism classes of orientable over P3 . Acknowledgments. This work is supported by the National Natural Science Foundation of China (No. 11201126 and 11371018), SRFDP (No. 20121303110004), the Basic Science and Technological Frontier Project of Henan (No. 122300410414 and No. 132300410432) and the research program for scientific technology of Henan province (No. 13A110540). Finally, the author would like to thank the referee for the invaluable comments. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of the Alexandru Ioan Cuza University - Mathematics de Gruyter

# Small Covers Over the Product of 3-Sided Prism with n -Simplex

, Volume 60 (2) – Nov 24, 2014
16 pages

/lp/de-gruyter/small-covers-over-the-product-of-3-sided-prism-with-n-simplex-ynSU2XW0t1
Publisher
de Gruyter
ISSN
1221-8421
eISSN
1221-8421
DOI
10.2478/aicu-2013-0034
Publisher site
See Article on Publisher Site

### Abstract

In this paper, the author calculates the number of equivariant homeomorphism classes of (orientable) over the product of 3-sided prism with n-simplex. Mathematics Subject Classification 2010: 57S10, 57S25, 52B11. Key words: small cover, equivariant homeomorphism, polytope. 1. Introduction A small cover, defined by Davis and Januszkiewicz in [9], is a smooth closed manifold M n with a locally standard (Z2 )n -action such that its orbit space is a simple convex polytope. For instance, the real projective space RP n with a natural (Z2 )n -action is a small cover over an n-simplex. This gives a direct connection between equivariant topology and combinatorics and makes it possible to study the topology of through the combinatorial structure of quotient spaces. ¨ In [13], Lu and Masuda showed that the equivariant homeomorphism class of a small cover over a simple convex polytope P n agrees with the equivalence class of its corresponding (Z2 )n -coloring under the action of automorphism group of face poset of P n . This holds for orientable by the orientability condition in [15] (see Theoerem 4.4). But there aren't general formulas to calculate the number of equivariant homeomorphism classes of (orientable) over an arbitrary simple convex polytope. In recent years, several studies have attempted to enumerate the number of equivalence classes of all over a specific polytope. Garrison and Scott [10] used a computer program to calculate the number of homeomorphism classes of all over a dodecahedron. In 2008, Choi [6] determined the number of equivariant homeomorphism classes of over cubes. Prisms are an interesting class of 3-dimensional ¨ polytopes. Cao and Lu [4] classified over prisms up to homeomorphism and calculated the number of homeomorphism classes of small ¨ covers over prisms. Cai, Chen and Lu [3] calculated the number of equivariant homeomorphism classes of over prisms. As a generalization, Wang and Chen [17] determined the number of equivariant homeomorphism classes of over products of a polygon with a simplex. Furthermore, Chen and Wang [5] calculated the number of D-J equivalence classes of all orientable over products of a prism with a simplex. ¨ The smallest interesting prism is the 3-sided prism. Lu and Yu [14] showed that one can obtain each 3-dimensional small cover from the small cover S 1 × RP 2 over the 3-sided prism and the small cover RP 3 over the 3-simplex by using cut and paste strategies in the sense of six equivariant operations. Let M be the set of equivariant unoriented cobordism classes of all 3-dimensional . Then M is generated by the classes of S 1 ×RP 2 over the 3-sided prism and RP 3 over the 3-simplex ([12, 14]). Much further research on over the 3-sided prism has been carried on ([11, 16]). An example of a cohomologically non-rigid polytope was obtained from a 3-sided prism by iterating the operation of vertex cut twice ([8]). Products of polytopes are another interesting class of polytopes and more complicated than one might think ([18]). And over products of simple convex polytopes have become an important search object ([5, 6, 7, 17]). Motivated by these, the author calculates the number of equivariant homeomorphism classes of over the product of 3-sided prism with n-simplex in this paper. From [15], there exist orientable over every simple convex 3-polytope and there exist non-orientable over every simple convex 3-polytope, except the 3-simplex. An orientable 3-dimensional small cover corresponds to a 4-colored simple convex 3-polytope and the existence of an orientable small cover over every simple convex 3-polytope is closely related to the four color theorem (see [2] for the four color theorem). In [7], Choi calculated the number of D-J equivalence classes of orientable over cubes. The number of equivariant homeomorphism classes of orientable over the product of 3-sided prism with n-simplex is also calculated in this paper. Let P3 , n be a 3-sided prism (i.e., the product of a 3-gon and the interval I) and an n-simplex respectively. The main results of this paper are stated as follows: Theorem 1.1. Let E(P3 × n ) be the number of equivariant homeomorphism classes of over P3 × n . When n 3, E(P3 × n ) = (5 · 4n + 9 · 2n+1 + 33) n+3 (2n+3 - 2k-1 ) . 12(n + 1)! Up to equivariant homeomorphism, there are 12846400 over P3 × 2 , 47460 over P3 × I and 98 over P3 . Theorem 1.2. Let Eo (P3 × n ) be the number of equivariant homeomorphism classes of orientable over P3 × n . When n 3 and n is odd, (5 · 4n-1 + 2n + 6) n+3 (2n+3 - 2k-1 ) . Eo (P3 × ) = 12(n + 1)! When n 4 and n is even, Eo (P3 × n ) = (2n+1 + 6) n+3 (2n+3 - 2k-1 ) . 12(n + 1)! Up to equivariant homeomorphism, there are 972160 orientable over P3 × 2 , 8400 orientable over P3 × I and 28 orientable over P3 . The paper is organized as follows. In Section 2, we review the basic theory about and calculate the automorphism group of face poset of P3 × n . In Section 3, we determine the number of all (Z2 )n+3 colorings on P3 × n and prove Theorem 1.1. In Section 4, we calculate the number of all orientable colorings on P3 × n and prove Theorem 1.2. 2. Preliminaries A convex polytope P n of dimension n is said to be simple if every vertex of P n is the intersection of exactly n facets (i.e. faces of dimension (n - 1)) (see [18]). An n-dimensional smooth closed manifold M n is said to be a small cover if it admits a smooth (Z2 )n -action such that the action is locally isomorphic to a standard action of (Z2 )n on Rn and the orbit space M n /(Z2 )n is a simple convex polytope of dimension n. Let P n be a simple convex polytope of dimension n and F(P n ) = {F1 , . . ., F } be the set of facets of P n . Suppose that : M n P n is a small cover over P n . Then there are connected submanifolds -1 (F1 ), . . ., -1 (F ). Each submanifold -1 (Fi ) is fixed pointwise by a Z2 -subgroup Z2 (Fi ) of (Z2 )n , so that each facet Fi corresponds to the Z2 -subgroup Z2 (Fi ). Obviously, the Z2 -subgroup Z2 (Fi ) actually agrees with an element i in (Z2 )n as a vector space. For each face F of codimension u, since P n is simple, there are u facets Fi1 , . . . , Fiu such that F = Fi1 . . .Fiu . Then, the corresponding submanifolds -1 (Fi1 ), . . . , -1 (Fiu ) intersect transversally in the (n - u)-dimensional submanifold -1 (F ), and the isotropy subgroup Z2 (F ) of -1 (F ) is a subtorus of rank u and is generated by Z2 (Fi1 ), . . . , Z2 (Fiu ) (or is determined by i1 , . . . , iu in (Z2 )n ) (see [9]). Consider a map : F(P n ) - (Z2 )n which satisfies the nonsingularity condition: {(Fi1 ), . . . , (Fin )} is a basis of (Z2 )n whenever the intersection Fi1 . . . Fin is non-empty. We call a characteristic function. If we regard each nonzero vector of (Z2 )n as being a color, then the characteristic function means that each facet is colored by a color. Here we also call a (Z2 )n -coloring on P n . In fact, Davis and Januszkiewicz [9] gave a reconstruction process of a small cover by using a (Z2 )n -coloring : F(P n ) - (Z2 )n . Let Z2 (Fi ) be the subgroup of (Z2 )n generated by (Fi ). Given a point p P n , by F (p) we denote the minimal face containing p in its relative interior. Assume F (p) = Fi1 . . . Fiu and Z2 (F (p)) = u Z2 (Fij ). Note that Z2 (F (p)) is j=1 a u-dimensional subgroup of (Z2 )n . Let M () denote P n × (Z2 )n / , where (p, g) (q, h) if p = q and g-1 h Z2 (F (p)). The free action of (Z2 )n on P n × (Z2 )n descends to an action on M () with quotient P n . Thus M () is a small cover over P n . Two M1 and M2 over P n are said to be weakly equivariantly homeomorphic if there is an automorphism : (Z2 )n (Z2 )n and a homeomorphism f : M1 M2 such that f (t · x) = (t) · f (x) for every t (Z2 )n and x M1 . If is an identity, then M1 and M2 are equivariantly homeomorphic. Following [9], two M1 and M2 over P n are said to be Davis-Januszkiewicz equivalent (or simply, D-J equivalent) if there is a weakly equivariant homeomorphism f : M1 M2 covering the identity on P n . By (P n ) we denote the set of all (Z2 )n -colorings on P n . Then we have: Theorem 2.1. All over P n are given by {M ()| (P n )}, i.e. for each small cover M n over P n , there is a (Z2 )n -coloring with an equivariant homeomorphism M () - M n covering the identity on P n . Remark 2.2. Generally speaking, we can't make sure that there always exist over a simple convex polytope P n when n 4. For example, see Nonexample 1.22 of [9]. However, the Four Color Theorem makes sure that there always exist over every 3-dimensional simple convex polytope, so there exist over P3 . Since RP n is a small cover over n , there exist over P3 × n . There is a natural action of GL(n, Z2 ) on (P n ) defined by the correspondence - , and the action on (P n ) is free. Without loss of generality, we assume that F1 , . . . , Fn of F(P n ) meet at one vertex p of P n . Let e1 , . . . , en be the standard basis of (Z2 )n . Write A(P n ) = { (P n )|(Fi ) = ei for i = 1, . . . , n}. In fact, A(P n ) is the orbit space of (P n ) under the action of GL(n, Z2 ). Then we have Lemma 2.3. |(P n )| = |A(P n )| × |GL(n, Z2 )|. n n k-1 ([1]). Two Note that |GL(n, Z2 )| = 2 - 2 M (1 ) and M (2 ) over P n are D-J equivalent if and only if there is GL(n, Z2 ) such that 1 = 2 . So the number of D-J equivalence classes of over P n is |A(P n )|. Let P n be a simple convex polytope of dimension n. All faces of P n form a poset (i.e. a partially ordered set by inclusion). An automorphism of F(P n ) is a bijection from F(P n ) to itself which preserves the poset structure of all faces of P n , and by Aut(F(P n )) we denote the group of automorphisms of F(P n ). One can define the right action of Aut(F(P n )) on (P n ) by × h - h, where (P n ) and h Aut(F(P n )). The following theorem is well known (see [13]). Theorem 2.4. Two over an n-dimensional simple convex polytope P n are equivariantly homeomorphic if and only if there is h Aut(F(P n )) such that 1 = 2 h, where 1 and 2 are their corresponding (Z2 )n -colorings on P n . So the number of orbits of (P n ) under the action of Aut(F(P n )) is just the number of equivariant homeomorphism classes of over P n . Thus, we are going to count the orbits. Burnside Lemma is very useful in the enumeration of the number of orbits. Burnside Lemma. Let G be a finite group acting on a set X. Then 1 the number of orbits of X under the action of G equals |G| gG |Xg |, where Xg = {x X|gx = x}. Burnside Lemma suggests that we need to understand the structure of Aut(F(P n )) in order to determine the number of the orbits of (P n ) under the action of Aut(F(P n )). We shall particularly be concerned with the case in which the simple convex polytope is P3 × n . To be convenient, we introduce the following marks. By s and s we 2 1 denote the top and bottom facets of P3 respectively, and by a , a , a we 1 2 3 denote three sided facets of P3 in their general order. For n-simplex n , n n by b , b , . . . , b 1 2 n+1 we denote all facets of . Set F = {s1 = s1 × , s2 = s × n , ai = a × n |1 i 3}, F = {bj = P3 × b |1 j n + 1}. Then 2 i j F(P3 × n ) = F F . Lemma 2.5. Let P3 , n be a 3-sided prism and an n-simplex respectively. The automorphism group Aut(F(P3 × n )) is isomorphic to S3 × (Z2 )3 , n = 1, 2 × (Z )2 , (S ) n = 2, 2 3 S3 × Z2 × Sn+1 , n 3, where Sn+1 is the symmetric group of rank n + 1. Proof. First, we show that the automorphism group Aut(F(P3 )) is isomorphic to S3 × Z2 . Let three sided facets of P3 interchange and the top and bottom facets stay unchanged. Then these automorphisms form a group S3 . Let the top and bottom facets interchange and three sided facets of P3 stay unchanged. These automorphisms form a group Z2 . Since any one of all sided facets cannot be mapped to the top facet or bottom facet under the automorphisms of F(P3 ), Aut(F(P3 )) is isomorphic to S3 × Z2 . When n 3, the facets of F and F are mapped to F and F respectively under the automorphisms of Aut(F(P3 × n )). Since Aut(F(P3 )) is isomorphic to S3 × Z2 and Aut(F(n )) is isomorphic to Sn+1 , Aut( F(P3 × n )) is isomorphic to S3 × Z2 × Sn+1 . Next, we consider the case in which n=2. Below we show that the automorphism group Aut(F(2 × 2 )) is isomorphic to S3 × S3 × Z2 . 2 is a 3-gon. By c , c , c we denote three edges of 2 in their general order. 1 2 3 Let F1 = {ci = c × 2 |1 i 3}, F1 = {dj = 2 × c |1 j 3}. Then i j F(2 × 2 ) = F1 F1 . There are automorphisms of F(2 × 2 ) under which the facets of F1 and F1 are mapped to F1 and F1 respectively. These automorphisms form a group S3 × S3 . We choose an automorphism f such that f (ci ) = di for 1 i 3 and f (dj ) = cj for 1 j 3. Let Z2 = {f, 1}. Then we get a new group S3 × S3 × Z2 , each of which is an automorphism under which the facets in F1 and F1 are mapped to F1 and F1 or to F1 and F1 respectively. Since other bijections from F(2 × 2 ) to itself don't preserve the poset structure of all faces of 2 × 2 , Aut(F(2 × 2 )) is isomorphic to S3 × S3 × Z2 . When n=2, P3 ×n is 2 ×I ×2 . Since Aut(F(2 ×2 )) is isomorphic to S3 × S3 × Z2 and Aut(F(I)) is isomorphic to Z2 , Aut(F(P3 × 2 )) is isomorphic to S3 × S3 × Z2 × Z2 . Finally, we consider the case in which n=1. Below we show that the automorphism group Aut(F(I 2 )) is isomorphic to (Z2 )3 . Aut(F(I 2 )) contains a Z2 -subgroup since there is one automorphism for the permutation of the two pairs of opposite facets of I 2 = I × I. All elements of Aut(F(I 2 )) can be written in a simple form as follows: ae1 be2 · u, where e1 , e2 Z2 , with reflections a, b and u the former Z2 -subgroup. Thus, Aut(F(I 2 )) is isomorphic to (Z2 )3 . When n = 1, P3 × n is 2 × I 2 . Since the automorphism group Aut(F(2 )) is isomorphic to S3 and Aut(F(I 2 )) is isomorphic to (Z2 )3 , Aut(F(P3 × I)) is isomorphic to S3 × (Z2 )3 . 3. The number of over P3 × n First, we give a criterion for a map : F(P3 × n ) - (Z2 )n+3 to be a characteristic function. The nonsingularity condition of the characteristic function means the following: (1) {(s1 ), (a1 ), (a2 ), (b1 ), . . . , (bn )} is a basis of (Z2 )n+3 . (2) (bn+1 ) satisfies that {(bn+1 ), (bk1 ), . . . , (bkn-1 ), (s1 ), (a1 ), (a2 )} is a basis of (Z2 )n+3 , where k1 < k2 < . . . < kn-1 and k1 , . . . , kn-1 {1, 2, . . . , n}. (3) (a3 ) satisfies that {(a3 ), (s1 ), (al1 ), (bh1 ), . . . , (bhn )} is a basis of (Z2 )n+3 , where l1 {1, 2}, h1 < h2 < . . . < hn and h1 , . . . , hn {1, 2, . . . , n + 1}. (4) (s2 ) satisfies that {(s2 ), (aj1 ), (aj2 ), (bh1 ), . . . , (bhn )} is a basis of (Z2 )n+3 , where j1 < j2 , j1 , j2 {1, 2, 3}, h1 < h2 < . . . < hn and h1 , . . . , hn {1, 2, . . . , n + 1}. Next, we calculate the number of (Z2 )n+3 -colorings on P3 × n . n Lemma 3.1. When n 1, the number of (Z2 )n+3 -colorings over P3 × is |(P3 × n )| = (5 · 4n + 7 · 2n+1 + 21) n+3 (2n+3 - 2k-1 ). Proof. Let e1 , e2 , . . . , en+3 be the standard basis of (Z2 )n+3 , then (Z2 )n+3 contains 2n+3 - 1 nonzero elements (or 2n+3 - 1 colors). We choose s1 , a1 , a2 from F and b1 , . . . , bn from F , then s1 , a1 , a2 , b1 , . . . , bn meet at one vertex of P3 × n . Then A(P3 × n ) = { (P3 × n )|(s1 ) = e1 , (a1 ) = e2 , (a2 ) = e3 , (bi ) = ei+3 , 1 i n}. Then, by Lemma 2.3, we have that n+3 |(P3 × )| = |A(P3 × )|×|GL(n+3, Z2 )| = (2n+3 -2k-1 )|A(P3 ×n )|. Write A0 (P3 × n ) = { A(P3 × n )|(bn+1 ) = e4 + . . . + en+3 }, A1 (P3 × n ) = { A(P3 × n )|(bn+1 ) = e4 + . . . + en+3 + eh1 , 1 h1 3}, A2 (P3 × ) = { A(P3 × n )|(bn+1 ) = e4 + . . . + en+3 + ek1 + ek2 , 1 k1 < k2 3}, A3 (P3 × ) = { A(P3 × n )|(bn+1 ) = e4 + . . . + en+3 + e1 + e2 + e3 }. By the definition of (Z2 )n+3 -colorings, we have that 3 n n |A(P3 × n )| = i=0 |Ai (P3 × n )|. Then, our argument proceeds as follows: (I) Calculation of |A0 (P3 × n )|. In this case, (a3 ) = e2 + e3 + em1 . . . + eml , where 1 m1 < . . . < ml n + 3, m1 = 2, m1 = 3, . . . , ml = 2, ml = 3 and 0 l n + 1. When (a3 ) = e2 + e3 + em1 + . . . + eml , 4 m1 < . . . < ml n + 3 and 0 l n, (s2 ) = e1 + en1 + . . . + enj , where 2 n1 < . . . < nj n + 3 and 0 j n+2. When (a3 ) = e2 +e3 +e1 +em1 +. . .+eml , 4 m1 < . . . < ml n+3 and 0 l n, (s2 ) = e1 + eg1 + . . . + egk , where 4 g1 < . . . < gk n + 3 and 0 k n. Thus, we have |A0 (P3 × n )| = 5 · 4n . (II) Calculation of |A1 (P3 × n )|. The argument is divided into two cases. Case 1. (bn+1 ) = e4 + . . . + en+3 + e1 In this case, (a3 ) = e2 + e3 + em1 . . . + eml , where 1 m1 < . . . < ml n + 3, m1 = 2, m1 = 3, . . . , ml = 2, ml = 3 and 0 l n + 1. When (a3 ) = e2 + e3 , (s2 ) = e1 , e1 + e2 , e1 + e3 , e1 + e2 + e3 . When (a3 ) = e2 + e3 + em1 . . . + eml , 1 m1 < . . . < ml n + 3, m1 = 2, m1 = 3, . . . , ml = 2, ml = 3 and 1 l n + 1, (s2 ) = e1 . Case 2. (bn+1 ) = e4 + . . . + en+3 + e2 or e4 + . . . + en+3 + e3 In this case, no matter which value of (bn+1 ) is chosen, we have (a3 ) = e2 + e3 , e2 + e3 + e1 . When (a3 ) = e2 + e3 , (s2 ) = e1 + en1 + . . . + enj , 2 n1 < . . . < nj n + 3 and 0 j n + 2. When (a3 ) = e2 + e3 + e1 , (s2 ) = e1 . Thus, we have |A1 (P3 × n )| = 5 · 2n+1 + 5. (III) Calculation of |A2 (P3 × n )|. The argument is also divided into two cases. Case 1. (bn+1 ) = e4 + . . . + en+3 + e1 + e2 or e4 + . . . + en+3 + e1 + e3 In this case, no matter which value of (bn+1 ) is chosen, we have (a3 ) = e2 + e3 , e2 + e3 + e1 . When (a3 ) = e2 + e3 , (s2 ) = e1 , e1 + e2 , e1 + e3 , e1 + e2 + e3 . When (a3 ) = e2 + e3 + e1 , (s2 ) = e1 . Case 2. (bn+1 ) = e4 + . . . + en+3 + e2 + e3 In this case, (a3 ) = e2 + e3 , e2 + e3 + e1 . When (a3 ) = e2 + e3 , (s2 ) = e1 + en1 + . . . + enj , 2 n1 < . . . < nj n + 3 and 0 j n + 2. When (a3 ) = e2 + e3 + e1 , (s2 ) = e1 . Thus, we have |A2 (P3 × n )| = 2n+2 + 11. (IV) Calculation of |A3 (P3 × n )|. In this case, (a3 ) = e2 + e3 , e2 + e3 + e1 . When (a3 ) = e2 + e3 , (s2 ) = e1 , e1 + e2 , e1 + e3 , e1 + e2 + e3 . When (a3 ) = e2 + e3 + e1 , (s2 ) = e1 . Thus, we have |A3 (P3 × n )| = 5. Remark 3.2. From Lemma 3.1, the number of D-J equivalence classes of over P3 × n is 5 · 4n + 7 · 2n+1 + 21. The proof of Theorem 1.1. When n 3, from Theorem 2.4, Burnside Lemma and Lemma 2.5, we have that E(P3 × n ) = 1 12(n + 1)! |g |, gAut(F (P3 ×n )) where g = { (P3 × n )| = g}. When n 3, the automorphism group Aut(F(P3 ×n )) is isomorphic to S3 ×Z2 ×Sn+1 . If g is the generator of the Z2 -subgroup of Aut(F(P3 × n )) and g , then (s1 ) = (s2 ). By the argument of Lemma 3.1, we have |g | = (2n+2 + 12) n+3 (2n+3 - 2k-1 ). If g is other automorphism and isn't unit element of Aut(F(P3 × n )), by the nonsingularity condition of characteristic functions, we have that g is empty. Thus, from Lemma 3.1, when n 3, E(P3 × n ) = 1 (2n+2 + 12 + 5 · 4n + 7 · 2n+1 + 21)· 12(n + 1)! n+3 (2n+3 - 2k-1 ) (5 · 4n + 9 · 2n+1 + 33) n+3 (2n+3 - 2k-1 ) . 12(n + 1)! When n=2, similarly we have E(P3 × 2 ) = 12846400. When n=1, n is the interval I. Aut(F(P3 × I)) is isomorphic to S3 × Z2 ×Z2 ×Z2 . If g is the generator a of the first Z2 -subgroup of Aut(F(P3 ×I)) and g , then (s1 ) = (s2 ). By the argument of Lemma 3.1, we have |g | = 20 4 (24 - 2k-1 ). If g is the generator b of the second Z2 subgroup of Aut(F(P3 × I)) and g , then (b1 ) = (b2 ). By the argument of (I) of Lemma 3.1, we also have |g | = 20 4 (24 - 2k-1 ). If g is the automorphism ab of Aut(F(P3 × I)), similarly we have |g | = 4 4 (24 - 2k-1 ). If g is other automorphism and isn't unit element of Aut(F(P3 × I)), by the nonsingularity condition of characteristic functions, we have that g is empty. Thus, from Lemma 3.1, we have E(P3 × I) = 1 (20 + 20 + 4 + 69) 48 (24 - 2k-1 ) = 47460. In the similar way, we have that there are 98 equivariant homeomorphism classes of over P3 . 4. The number of orientable over P3 × n Nakayama and Nishimura [15] found an orientability condition for a small cover. Theorem 4.1. For a basis {e1 , . . . , en } of (Z2 )n , a homomorphism : (Z2 )n - Z2 = {0, 1} is defined by (ei ) = 1(i = 1, . . . , n). A small cover M () over a simple convex polytope P n is orientable if and only if there exists a basis {e1 , . . . , en } of (Z2 )n such that the image of is {1}. We call a (Z2 )n -coloring which satisfies the orientability condition in Theorem 4.1 an orientable coloring of P n . We can know the existence of orientable over P3 × n by the existence of orientable colorings and determine the number of equivariant homeomorphism classes. By O(P n ) we denote the set of all orientable colorings on P n . There is a natural action of GL(n, Z2 ) on O(P n ) defined by the correspondence - , and the action on O(P n ) is free. Assume that F1 , . . . , Fn of F(P n ) meet at one vertex p of P n . Let e1 , . . . , en be the standard basis of (Z2 )n . Write B(P n ) = { O(P n )|(Fi ) = ei for i = 1, . . . , n}. It is easy to check that B(P n ) is the orbit space of O(P n ) under the action of GL(n, Z2 ). Remark 4.2. In fact, we have B(P n ) = { O(P n )|(Fi ) = ei for i = 1, . . . , n, and for n + 1 j , (Fj ) = ej1 + ej2 + . . . + ej2hj +1 , where 1 j1 < j2 < . . . < j2hj +1 n}. Below we show that (Fj ) = ej1 + ej2 + . . . + ej2hj +1 for n + 1 j . If O(P n ), there exists a basis {e , . . . , e } of (Z2 )n such that for 1 i , (Fi ) = e 1 + . . . + e 2f +1 , n 1 i i i where 1 i1 < . . . < i2fi +1 n. Since (Fi ) = ei for i = 1, . . . , n, we have ei = e 1 + . . . + e 2f +1 . So we obtain that for n + 1 j , there aren't i i i j1 , . . . , j2k such that (Fj ) = ej1 + . . . + ej2k , where 1 j1 < . . . < j2k n. Since B(P n ) is the orbit space of O(P n ), we have Lemma 4.3. |O(P n )| = |B(P n )| × |GL(n, Z2 )|. One can define the right action of Aut(F(P n )) on O(P n ) by × h - h, where O(P n ) and h Aut(F(P n )). By improving the classifying result on in [13], we have Theorem 4.4. Two orientable over an n-dimensional simple convex polytope P n are equivariantly homeomorphic if and only if there is h Aut(F(P n )) such that 1 = 2 h, where 1 and 2 are their corresponding orientable colorings on P n . Proof. We know Theorem 4.4 is true by combining Lemma 5.4 in [13] with Theorem 4.1. By Theorem 4.4, the number of orbits of O(P n ) under the action of Aut(F(P n )) is the number of equivariant homeomorphism classes of orientable over P n . So we also are going to count the orbits. Lemma 4.5. When n 1, the number of orientable colorings on P3 × n is n+3 (5 · 4n-1 + 3) (2n+3 - 2k-1 ), n odd, |O(P3 × n )| = n+3 n (2 + 3) (2n+3 - 2k-1 ), n even. Proof. Let e1 , e2 , . . . , en+3 be the standard basis of (Z2 )n+3 . Then B(P3 × n ) = { O(P3 × n )|(s1 ) = e1 , (a1 ) = e2 , (a2 ) = e3 , (bi ) = ei+3 , 1 i n}. Then, by Lemma 4.3, we have that |O(P3 × n )| = |B(P3 × n )| × |GL(n + 3, Z2 )| = n+3 (2n+3 - 2k-1 )|B(P3 × n )|. The calculation of |B(P3 × n )| is divided into two cases: (I) n odd, (II) n even. (I) n odd Write: B0 (P3 × n ) = { B(P3 × n )|(bn+1 ) = e4 + . . . + en+3 }, B1 (P3 × n ) = { B(P3 × n )|(bn+1 ) = e4 + . . . + en+3 + ek1 + ek2 , 1 k1 < k2 3}. By the definition of B(P n ), we have that |B(P3 × n )| = |B0 (P3 × n )| + |B1 (P3 × n )|. Then, our argument proceeds as follows. Case 1. Calculation of |B0 (P3 × n )|. In this case, (a3 ) = e2 + e3 + em1 . . . + eml , where 1 m1 < . . . < ml n + 3, m1 = 2, m1 = 3, . . . , ml = 2, ml = 3, l odd and 1 l n + 1. When (a3 ) = e2 + e3 + em1 + . . . + eml , 4 m1 < . . . < ml n + 3, l odd and 1 l n, (s2 ) = e1 + en1 + . . . + enj , 2 n1 < . . . < nj n + 3, j even and 0 j n + 2. When (a3 ) = e2 + e3 + e1 + em1 + . . . + eml , 4 m1 < . . . < ml n + 3, l even and 0 l n, (s2 ) = e1 + eg1 + . . . + egk , 4 g1 < . . . < gk n + 3, k even and 0 k n. Thus, we have |B0 (P3 × n )| = 5 · 4n-1 . Case 2. Calculation of |B1 (P3 × n )|. In this case, no matter which value of (bn+1 ) is chosen, we have (a3 ) = e2 + e3 + e1 , (s2 ) = e1 . Thus, we have |B1 (P3 × n )| = 3. (II) n even Write B0 (P3 × n ) = { B(P3 × n )|(bn+1 ) = e4 + . . . + en+3 + eh1 , 1 h1 3}, B1 (P3 ×n ) = { B(P3 × n )|(bn+1 )=e4 + . . . +en+3 +e1 + e2 + e3 }. n )| By the definition of B(P n ), we have that |B(P3 × n )| = |B0 (P3 × (P × n )|. Then, our argument proceeds as follows. + |B1 3 Case 1. Calculation of |B0 (P3 × n )|. The argument is divided into two cases: (1) (bn+1 ) = e4 + . . . + en+3 + e1 . In this case, (a3 ) = e2 + e3 + em1 . . . + eml , where 1 m1 < . . . < ml n + 3, m1 = 2, m1 = 3, . . . , ml = 2, ml = 3, l odd and 1 l n + 1, (s2 ) = e1 . (2) (bn+1 ) = e4 + . . . + en+3 + e2 or e4 + . . . + en+3 + e3 . In this case, no matter which value of (bn+1 ) is chosen, we have (a3 ) = e2 + e3 + e1 , (s2 ) = e1 . Thus, we have |B0 (P3 × n )| = 2n + 2. Case 2. Calculation of |B1 (P3 × n )|. In this case, (a3 ) = e2 + e3 + e1 , (s2 ) = e1 . Thus, we have |B1 (P3 × n )| = 1. The proof of Theorem 1.2. When n 3, from Theorem 4.4, Burnside Lemma and Lemma 2.5, we have that Eo (P3 × n ) = 1 12(n + 1)! |g |, gAut(F (P3 ×n )) where g = { O(P3 × n )| = g}. When n 3 and n is odd, the automorphism group Aut(F(P3 × n )) is isomorphic to S3 × Z2 × Sn+1 . If g is the generator of the Z2 -subgroup of Aut(F(P3 × n )) and g , then (s1 ) = (s2 ). By the argument of (I) of Lemma 4.5, we have |g | = (2n + 3) n+3 (2n+3 - 2k-1 ). If g is other automorphism and isn't unit element of Aut(F(P3 × n )), then g is empty. Thus, from Lemma 4.5, when n 3 and n is odd, Eo (P3 × n ) = = 1 (2n + 3 + 5 · 4n-1 + 3) 12(n + 1)! (5 · 4n-1 + 2n + 6) 12(n + 1)! n+3 n+3 (2 n+3 (2n+3 - 2k-1 ) 2k-1 ) When n 4 and n is even, using the above method we have Eo (P3 × n ) = (2n+1 + 6) n+3 (2n+3 - 2k-1 ) . 12(n + 1)! When n=2, similarly we have Eo (P3 × 2 ) = 972160. When n=1, Aut(F(P3 × n )) is isomorphic to S3 × Z2 × Z2 × Z2 . If g is the generator a of the first Z2 -subgroup of Aut(F(P3 × I)) and g , then (s1 ) = (s2 ). By the argument of (I) of Lemma 4.5, we have |g | = 5 4 (24 - 2k-1 ). If g is the generator b of the second Z2 -subgroup of Aut(F(P3 × I)) and g , then (b1 ) = (b2 ). By the argument of Case 1 in (I) of Lemma 4.5, we also have |g | = 5 4 (24 - 2k-1 ). If g is the automorphism ab of Aut(F(P3 × I)), similarly we have |g | = 2 4 (24 - 2k-1 ). If g is other automorphism and isn't unit element of Aut(F(P3 × I)), then g is empty. Thus, from Lemma 4.5, we have Eo (P3 × I) = 1 (5 + 5 + 2 + 8) 48 (24 - 2k-1 ) = 8400. In the similar way, we have that there are 28 equivariant homeomorphism classes of orientable over P3 . Acknowledgments. This work is supported by the National Natural Science Foundation of China (No. 11201126 and 11371018), SRFDP (No. 20121303110004), the Basic Science and Technological Frontier Project of Henan (No. 122300410414 and No. 132300410432) and the research program for scientific technology of Henan province (No. 13A110540). Finally, the author would like to thank the referee for the invaluable comments.

### Journal

Annals of the Alexandru Ioan Cuza University - Mathematicsde Gruyter

Published: Nov 24, 2014