From the Icosahedron to Natural Triangulations of ℂP 2 and S 2×S 2

Discrete & Computational Geometry, Aug 2010

We present two constructions in this paper: (a) a 10-vertex triangulation \(\mathbb{C}P^{2}_{10}\) of the complex projective plane ℂP 2 as a subcomplex of the join of the standard sphere (\(S^{2}_{4}\)) and the standard real projective plane (\(\mathbb{R}P^{2}_{6}\), the decahedron), its automorphism group is A 4; (b) a 12-vertex triangulation (S 2×S 2)12 of S 2×S 2 with automorphism group 2S 5, the Schur double cover of the symmetric group S 5. It is obtained by generalized bistellar moves from a simplicial subdivision of the standard cell structure of S 2×S 2. Both constructions have surprising and intimate relationships with the icosahedron. It is well known that ℂP 2 has S 2×S 2 as a two-fold branched cover; we construct the triangulation \(\mathbb{C}P^{2}_{10}\) of ℂP 2 by presenting a simplicial realization of this covering map S 2×S 2→ℂP 2. The domain of this simplicial map is a simplicial subdivision of the standard cell structure of S 2×S 2, different from the triangulation alluded to in (b). This gives a new proof that Kühnel’s \(\mathbb{C}P^{2}_{9}\) triangulates ℂP 2. It is also shown that \(\mathbb{C}P^{2}_{10}\) and (S 2×S 2)12 induce the standard piecewise linear structure on ℂP 2 and S 2×S 2 respectively.

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From the Icosahedron to Natural Triangulations of ℂP 2 and S 2×S 2

Discrete Comput Geom From the Icosahedron to Natural Triangulations of CP 2 and S 2 × S 2 Bhaskar Bagchi 0 1 Basudeb Datta 0 1 0 B. Datta ( ) Department of Mathematics, Indian Institute of Science , Bangalore 560 012 , India 1 B. Bagchi Theoretical Statistics and Mathematics Unit, Indian Statistical Institute , Bangalore 560 059 , India We present two constructions in this paper: (a) a 10-vertex triangulation CP120 of the complex projective plane CP 2 as a subcomplex of the join of the standard sphere (S42) and the standard real projective plane (RP 2, the decahedron), its 6 automorphism group is A4; (b) a 12-vertex triangulation (S2 × S2)12 of S2 × S2 with automorphism group 2S5, the Schur double cover of the symmetric group S5. It is obtained by generalized bistellar moves from a simplicial subdivision of the standard cell structure of S2 × S2. Both constructions have surprising and intimate relationships with the icosahedron. It is well known that CP 2 has S2 × S2 as a two-fold branched cover; we construct the triangulation CP120 of CP 2 by presenting a simplimciaalpreisalaizsaitmiopnliocfiatlhissucbodviveirsiniognmoafpthSe2 s×taSnd2a→rd CcePll2s.tTruhcetudroemoafinS2of×thSis2,sidmifpfleirceinatl from the triangulation alluded to in (b). This gives a new proof that Kühnel's CP 2 9 triangulates CP 2. It is also shown that CP120 and (S2 × S2)12 induce the standard piecewise linear structure on CP 2 and S2 × S2 respectively. The research of B. Datta was supported by UGC grant UGC-SAP/DSA-IV. Triangulated manifolds; Complex projective plane; Product of 2-spheres; Icosahedron - 1 Introduction and Results It is well known that the minimal triangulation RP62 of the real projective plane arises naturally from the icosahedron. Indeed, it is the quotient of the boundary complex of the icosahedron by the antipodal map. In this note, we report the surprising result that there is a small triangulation (using only 10 vertices) of the complex projective plane which is also intimately related to the icosahedron. Indeed, this simplicial complex CP120 occurs as a subcomplex of the simplicial join S42 ∗ RP62. Our starting point is the beautiful fact that CP 2 is homeomorphic to the symmetrized square (S2 × S2)/Z2 of the 2-sphere, where Z2 acts by co-ordinate flip. So, letting S42 denote the 4-vertex triangulation of S2 (i.e., the boundary complex of the tetrahedron), we look for a Z2stable simplicial subdivision of the product cell complex S42 × S42, without introducing extra vertices. In order to ensure that the quotient complex (after quotienting by Z2) does triangulate the quotient space (S2 × S2)/Z2 = CP 2, the Z2-action on this simplicial subdivision must be “pure” (cf. Definition 2 and Lemma 7 in Sect. 5). It turns out that the following (S2 × S2)16 is the unique 16-vertex triangulation satisfying these requirements. Description of (S2 × S2)16 The vertices are xij , 1 ≤ i, j ≤ 4. The full automorphism group is A4 × Z2, where A4 acts on the indices and Z2 acts by xij ↔ xji . Modulo this group the facets (maximal simplices) are the following: x11x22x33x12x13, x11x22x12x14x34, x11x22x14x24x34, x11x22x21x24x31, x11x22x24x31x34. The full list of facets of (S2 × S2)16 may be obtained from these five basic facets by applying the group A4 × Z2. Under this group, the first three basic facets form orbits of length 24 each, while each of the last two forms an orbit of length 12, yielding a total of 3 × 24 + 2 × 12 = 96 facets. It may be verified that the face vector of (S2 × S2)16 is (16, 84, 216, 240, 96). Description of CP120 Quotienting the above (S2 × S2)16 by the group Z2 generated by the automorphism xij ↔ xji , we get the CP120 mentioned above. Its vertices are xij , 1 ≤ i ≤ j ≤ 4. Let α, β be the generators of the alternating group A4 given by α = (123), β = (12)(34). Then α, β act on the vertices of CP120 by: α ≡ (x11x22x33)(x23x13x12)(x24x34x14), β ≡ (x11x22)(x33x44)(x24x13)(x14x23). The following are the basic facets of CP120 modulo A4 = α, β : x11x22x33x12x13, x11x22x12x14x34, x11x22x14x24x34, x11x22x12x13x24, x11x22x13x24x34. The full list of facets of CP120 may be obtained from these five basic facets by applying the group A4. Under this group, the first three basic facets form orbits of length 12 each, while each of the last two forms an orbit of length 6, yielding a total of 3 × 12 + 2 × 6 = 48 facets. The complex CP120 is 2-neighborly and its face vector is (10, 45, 110, 120, 48). Here we prove the following: Theorem 1 There are exactly two 16-vertex simplicial complexes which (i) are simplicial subdivisions of the cell complex S42 × S42, (ii) retain the self-homeomorphism α : (x, y) → (y, x) of |S4 | × |S42| as a simplicial automorphism, and (iii) the action 2 of Z2 = α is pure (cf. Definition 2). These two complexes are isomorphic and one of them is (S2 × S2)16. Corollary 2 The complex CP120 := (S2 × S2)16/Z2 is a 10-vertex triangulation of CP 2. Its full automorphism group is A4. Let T and I denote the solid tetrahedron and the icosahedron in R3 respectively. Thus, the cell complex S42 × S42 alluded to above is a subcomplex of the boundary complex of the product polytope T × T in R6. Although we do not present the details in this paper, Theorem 1 can be strengthened (following the same line of arguments) to show that there is a unique simplicial subdivision S156 of the cell complex ∂(T × T ) which is Z2-stable with a pure Z2-action. To our utter surprise, it turns out that as an abstract simplicial complex, S156 is isomorphic to the combinatorial join S4 ∗ S122 of 2 the boundary complexes of T and I respectively. Remark 1 This last fact has the following geometric interpretation. Let T I denote the convex hull of T ∪ I, where T and I sit in two (three-dimensional) affine subspaces of R6 meeting at a point which is in the interior of both polyhedra. Then T I is a simplicial 6-polytope and the boundary complex of this polytope is combinatorially isomorphic to a simplicial subdivision of the boundary complex of T × T . This geometric result cries out for a geometric explanation; but we have none. By the construction, (S2 × S2)16 is a subcomplex of S42 ∗ S122. Since the decahedron RP62 is the quotient of S122 = ∂I by Z2, and Z2 acts trivially on S42 (the latter being the combinatorial child of the “diagonal” S2 in S2 × S2, i.e., the S42 in Fig. 1), on passing to the quotient, we find the surprising inclusion CP120 ⊆ S42 ∗ RP62. Indeed, S42 and RP62 occur as induced subcomplexes of CP120 on a complementary pair of vertex sets. Since both S42 and RP62 are classical objects, and the combinatorial join is such a well known operation on simplicial complexes, this inclusion says that CP120 was all along sitting there right before our eyes! The number 10 obtained here is not optimal. It is well known (cf. [ 2–4, 10 ]) that any triangulation of CP 2 requires at least nine vertices, and there is a unique 9-vertex triangulation CP92 of this manifold, obtained by Kühnel [ 11, 12 ]. But, our construction is natural in that it is obtained by a combinatorial mimicry of a topological construction of CP 2. It shares this naturalness with another 10-vertex triangulation, say K140, of CP 2 available in the literature, namely the “equilibrium” triangulation of Banchoff and Kühnel [ 7 ]. Here we prove the following: K140. Theorem 3 The simplicial complex CP120 is bistellar equivalent to both CP92 and Corollary 4 (a) Kühnel’s 9-vertex simplicial complex CP 2 triangulates CP 2. 9 (b) Both CP120 and CP92 induce the standard pl-structure on CP 2. Of course, in principle these ideas generalize to arbitrary dimensions. In general, the d -dimensional complex projective space CP d is the symmetric d -th power of S2, i.e., the quotient of (S2)d by the symmetric group Sd acting by co-ordinate permutations. Unfortunately, even in the next case d = 3, it is not possible to subdivide the cell complex S42 × S42 × S42 into a simplicial complex, with a pure S3-action, without adding more vertices. Indeed, we found that we need to add 60 more vertices to obtain an (S2 × S2 × S2)124. On quotienting, we obtain a CP330—again with full automorphism group A4. The details are so complicated that we decided to postpone publication. We are presently trying to see if one can apply bistellar moves to this CP330 to reduce the number of vertices. It is known that any triangulation of CP 3 requires at least 17 vertices (cf. [ 2 ]). After we submitted a preliminary version of this paper to arXiv:1004.3157v1, 2010, Ulrich Brehm [ 9 ] communicated to us that he had the idea of obtaining CP120 as a quotient of a 16-vertex S2 × S2 in the 1980’s; however, he never published the details. We obtain a second simplicial subdivision (S2 × S2)16 of S42 × S42. Description of (S2 × S2)16 This is a second simplicial subdivision of the cell complex S42 × S42. It has the same vertex set and automorphism group A4. Modulo the group A4, its basic facets are: x11x12x13x21x31, x11x12x14x21x31, x11x13x14x21x31, x12x13x23x31x32, x12x14x21x24x31, x12x14x24x31x34, x12x21x24x31x32, x12x24x31x32x34. Each facets is in an orbit of length 12, yielding a total of 8 × 12 = 96 facets. The complex (S2 × S2)16 has the same face vector as (S2 × S2)16, namely, (16, 84, 216, 240, 96). We perform a finite sequence of generalized bistellar moves on (S2 × S2)16 and obtain the following 12-vertex triangulation (S2 × S2)12 of S2 × S2. Description of (S2 × S2)12 The vertices are xij , 1 ≤ i = j ≤ 4. Its automorphism group 2S5 is generated by the two automorphisms h = (x12x14x21x24x31) × (x13x42x43x32x34) and g = (x12x21x24x42x14x41x43x34x13x31x32x23). Modulo this group, (S2 × S2)12 is generated by the following two basic facets: x12x14x21x24x31, x12x13x14x21x31. The first basic facet is in an orbit of size 12, while the second is in an orbit of size 60, yielding a total of 72 facets. Its face vector is (12, 60, 160, 180, 72). Theorem 5 The simplicial complex (S2 × S2)12 is a triangulation of S2 × S2. Its full automorphism group is 2S5, the non-split extension of Z2 by S5. The complex (S2 × S2)12 has many remarkable properties. Its automorphism group is transitive on its vertices and edges. All its vertices have degree 10 and all its edges have degree 8. Indeed, the link of each edge is isomorphic to the 2-sphere S82 obtained from the boundary complex of the octahedron by starring two vertices in a pair of opposite faces. Also, all triangles of (S2 × S2)12 are of degree 3 or 5. The automorphism group is transitive on its triangles of each degree. The degree 3 triangles constitute a weak pseudomanifold whose strong components are two icosahedra. Thus, we find a pair I1, I2 of icosahedra sitting canonically inside the 2-skeleton of (S2 × S2)12. These two icosahedra are “antimorphic” in the sense that the identity map is an antimorphism between them (cf. Definition 1 below). The structure of (S2 × S2)12 is completely described in terms of this antimorphic pair of icosahedra. The full automorphism group 2S5 of (S2 × S2)12 is a double cover of the common automorphism group of these two icosahedra. Again, the number 12 here is not optimal. In [ 12 ], Kühnel and Laßmann have shown that any triangulation of S2 × S2 needs at least 11 vertices, and in [ 14 ], Lutz finds (via computer search) several 11-vertex triangulations of S2 × S2, all with trivial automorphism groups. Surprisingly, even though (S2 × S2)12 is not minimal, it does not admit any proper bistellar moves. Thus, there is no straightforward way to obtain a minimal triangulation of S2 × S2 starting from (S2 × S2)12. In [ 17 ], Sparla proved two remarkable inequalities on the Euler characteristic χ of a combinatorial 4-manifold M satisfying certain conditions. His first result is that if there is a centrally symmetric simplicial polytope P of dimension d ≥ 6 such that M ⊆ ∂ P and skel2(M ) = skel2(P ), then 10(χ − 2) ≥ 43 (d−31)/2 . Equality holds here if and only if P is a cross polytope (i.e., dual of a hypercube). His second result is: if M has 2d vertices and admits a fixed point free involution then 10(χ − 2) ≤ 43 (d−31)/2 . Equality holds if and only if M also satisfies the hypothesis of the first result for a cross polytope P . Notice that, in view of the Dehn– Sommerville equations, equality in either inequality determines the face vector of M in terms of d alone. To obtain an example of equality (in both results) with d = 6, Sparla searched for (and found) a 4-manifold with the predicted face vector under the assumption of an automorphism group A5 × Z2. To determine the topological type of the resulting 12-vertex 4-manifold, he had to compute its intersection form and then appeal to Freedman’s classification of simply connected smooth 4-manifolds. We believe that our approach to Sparla’s complex not only elucidates its true genesis, but also reveals its rich combinatorial structure and contributes to an elementary determination of its topological type. Note, however, that Sparla’s approach reveals yet another remarkable property of (S2 × S2)12. It provides a tight rectilinear embedding of S2 × S2 in R6. Remark 2 If X is a triangulated 4-manifold on at most 12 vertices, then its vertex links are homology 3-spheres on at most 11 vertices, and hence (cf. [ 5 ]) are combinatorial spheres. Thus all triangulated 4-manifolds on at most 12 vertices are combinatorial manifolds. (More generally, this argument yields: All triangulated d -manifolds on at most d + 8 vertices are combinatorial manifolds.) In particular, both CP120 and (S2 × S2)12 are combinatorial manifolds. Actually, an old result of Bing [ 8 ] says that all the vertex links of any triangulated 4-manifold are simply connected triangulated 3-manifolds. Therefore, in view of Perelman’s theorem (Poincaré conjecture) [ 15 ], all triangulated 4-manifolds are combinatorial manifolds, irrespective of the number of vertices. Remark 3 In [ 1 ], Akhmedov and Park have shown that S2 × S2 has countably infinite number of distinct smooth structures. Since there is an one to one correspondence between the smooth structures and pl-structures on a 4-manifold (cf. [16, p. 167]), it follows that S2 × S2 has infinitely many distinct pl-structures. Since (S2 × S2)16 and (S2 × S2)16 are simplicial subdivisions of S42 × S42, it follows that the pl-structures given by (S2 × S2)16 and (S2 × S2)16 are standard. Again, (S2 × S2)12 is combinatorially equivalent to (S2 × S2)16 (cf. Remark 5) and hence gives the same pl-structure as (S2 × S2)16. So, all the triangulations of S2 × S2 discussed here give the standard pl-structure on S2 × S2. 2 Preliminaries All simplicial complexes considered here are finite and the empty set is a simplex (of dimension −1) of every simplicial complex. We now recall some definitions here. For a finite set V with d + 2 (d ≥ 0) elements, the set ∂ V (respectively, V¯ ) of all the proper (resp. all the) subsets of V is a simplicial complex and triangulates the d -sphere Sd (resp. the (d + 1)-ball). The complex ∂ V is called the standard d sphere and is also denoted by Sdd+2(V ) (or simply by Sd the standard (d + 1)-ball and is also denoted by Dddd+++212()V. T)h(eorcosmimpplelyx Vb¯yisDcdda++ll21e)d. (Generally, we write X = Xnd to indicate that X has n vertices and dimension d .) For simplicial complexes X, Y with disjoint vertex sets, their join X ∗ Y is the simplicial complex whose simplices are all the disjoint unions A ∪ B with A ∈ X, B ∈ Y . If σ is a simplex of a simplicial complex X then the link of σ in X, denoted by lkX(σ ), is the simplicial complex whose simplices are the simplices τ of X such that τ ∩ σ = ∅ and σ ∪ τ is a simplex of X. The number of vertices in the link of σ is called the degree of σ . Also, the star of σ , denoted by starX(σ ) or star(σ ), is the subcomplex σ¯ ∗ lkX(σ ) of X. For a simplicial complex X, |X| denotes the geometric carrier. It may be described as the subspace of [ 0, 1 ]V (X) (where V (X) is the vertex set of X) consisting of all functions f : V (X) → [ 0, 1 ] satisfying (i) Support(f ) ∈ X and (ii) x∈V (X) f (x) = 1. If a space Y is homeomorphic to |X| then we say that X triangulates Y . If |X| is a topological manifold (respectively, d -sphere) then X is called a triangulated manifold (resp. triangulated d -sphere). If |X| is a pl manifold (with the pl-structure induced by X) then X is called a combinatorial manifold. For 1 ≤ d ≤ 4, X is a combinatorial d -manifold if and only if the vertex links are triangulated (d − 1)-spheres. The face vector of a d -dimensional simplicial complex is the vector (f0, f1, . . . , fd ), where fi is the number of i-dimensional simplices in the complex. If X is a d -dimensional pure simplicial complex (i.e., every maximal simplex is d -dimensional) and D, D are triangulations of the d -ball such that (i) ∂D = ∂D = D ∩ X, and (ii) D ⊆ X, then the simplicial complex X := (X \ D) ∪ D˜ is said to be obtained from X by a generalized bistellar move (GBM) with respect to the pair (D, D). Clearly, in this case, X and X triangulate the same topological space and if u is a vertex in ∂D then lkX(u) is obtained from lkX(u) by a GBM (cf. [ 6 ]). In particular, let A be a simplex of X whose link in X is a standard sphere ∂B. Suppose also that B ∈ X. Then, we may perform the GBM with respect to the pair of balls (A ∗ ∂B, B ∗ ∂A). Such an operation is called a bistellar move, and will be denoted by A → B. Also, if C is any simplex of X and x is a new symbol, then we may perform the GBM on X with respect to the pair (C¯ ∗ lkX(C), (x¯ ∗ ∂C) ∗ lkX(C)). The resulting simplicial complex X is said to be obtained from X by starring the vertex x in the simplex C. In case C is a facet, this is a bistellar move—the only sort of bistellar move which increases the number of vertices. All other kinds of bistellar moves are said to be proper. Two pure simplicial complexes are called bistellar equivalent if one is obtained from the other by a finite sequence of bistellar moves. If X is obtained from Y by the bistellar move A → B then the complex Z obtained from Y by starring a new vertex u in B is a subdivision of both X and Y . This implies that bistellar equivalent complexes induce same pl-structure on their common geometric carrier. The group Z2 acts on S2 × S2 by co-ordinate flip. The following proposition is well known to algebraic geometers (cf. [ 13 ]): Proposition 6 The quotient space (S2 × S2)/Z2 is homeomorphic to the complex projective plane CP 2. 3 Relations with the Icosahedron Emergence of the Icosahedron Let T0 be the tetrahedron with vertex set V = {x1, x2, x3, x4}. Then, viewed abstractly, the boundary complex of the product polytope T0 × T0 has vertex set V × V , and faces A × B, where A and B range over all the subsets of V . The product cell complex for S42 × S42 = (∂T0) × (∂T0) is the subcomplex consisting of cells A × B, where A and B range over all the proper subsets of V . We use the notation xij to denote the vertex (xi , xj ) of T0 × T0. For i = j , k = l, xij xkl forms an edge of T0 × T0 if and only if it is one of the solid edges of the icosahedron in Fig. 1. (This picture is a Schlegel diagram obtained by projecting the boundary of the icosahedron on one of its faces. Thus, there is only one “hidden” face (namely, x41x42x43) in the picture. What is important for us is the label given to the vertices.) Notice that the broken edges in the icosahedron are precisely the edges xij xkl where {i, j, k, l} is an even permutation of {1, 2, 3, 4}. To obtain the appropriate triangulation of S2 × S2, we join xii to all vertices for all i and also introduce the broken edges of the icosahedron. Thus viewed, one sees the simplicial subdivision (S2 × S2)16 of the cell complex (∂T0) × (∂T0) as a subcomplex of (∂T ) ∗ (∂I), where T is the tetrahedron with vertex set {xii : 1 ≤ i ≤ 4} and I is the icosahedron depicted in Fig. 1. Notice also that the Z2-action xij ↔ xji fixes the vertices of T and acts on I as the antipodal map. Thus, going modulo Z2, we find CP120 as a subcomplex of the 5-dimensional simplicial complex S42 ∗ RP62, where S42 is the 4-vertex 2-sphere given by the boundary complex of T and RP62 is the (minimal) triangulation of the real projective plane (with vertices of the same name being identified) given in Fig. 1. From our nomenclature for the vertices, the inclusion CP120 ⊆ S42 ∗ RP62 is obvious, as is the fact that (∂T ) ∗ (∂I) is a simplicial subdivision of the boundary complex of T0 × T0. i Finally, note that Δi = {xij : j = i} and Δ = {xji : j = i} are triangles of the icosahedron, and {Δ1, Δ2, Δ3, Δ4} and {Δ1, Δ2, Δ3, Δ4} are antipodal quadruples (consisting of triangles) partitioning the vertex set of the icosahedron. It is easy to see that there are exactly five such pairs in the icosahedron, and the automorphism group A5 × Z2 of I acts transitively on them. The stabilizer of each such pair is A4 × Z2, and A4 acts regularly on the vertex set of I. Our choice of nomenclature for the vertices of I amounts to choosing one such antipodal pair of quadruples. This is because we have Δi ∩ Δj = ∅ if i = j and = {xij } if i = j . Viewed dually, one sees Kepler’s regular tetrahedra embedded in the dodecahedron. Namely, the centers of Δi , 1 ≤ i ≤ 4 (as well as of Δi , 1 ≤ i ≤ 4) are the vertices of a regular tetrahedron inscribed in the dual dodecahedron. The 12-vertex triangulation (S2 × S2)12 of S2 × S2 is obtained from (S2 × S2)16 by a sequence of bistellar moves (cf. proof of Theorem 5). However, its most elegant description requires the introduction of the following definition. Definition 1 Let I1 and I2 be two copies of the icosahedron. A bijection f : V (I1) → V (I2) is said to be an antimorphism if, for all vertices x, y of I1, we have (a) x and y are at distance one in I1 if and only if f (x) and f (y) are at distance two in I2, and (b) x and y are at distance two in I1 if and only if f (x) and f (y) are at distance one in I2. (It follows that x and y are at distance 3 (antipodal) in I1 if and only if f (x) and f (y) are at distance 3 (antipodal) in I2.) Here distance refers to the usual graphical distance on the respective edge graph. In case V (I1) = V (I2) and the identity map is an antimorphism between I1 and I2, then we say that I1 and I2 are antimorphic. Thus, the two icosahedra in Fig. 2 below are antimorphic (the map, taking each vertex of the left icosahedron in Fig. 2 to the vertex of the same name in the right icosahedron, is an antimorphism). Another description of (S2 × S2)12 Take an antimorphic pair of icosahedra, say I1 and I2 (with common vertex set V ). It turns out that I1 and I2 have the identical automorphism group A5 × Z2 (not merely isomorphic; cf. Lemma 9 below). Also, there is a bijection ϕ from the triangles of I1 to the triangles of I2 such that for each triangle Δ = abc of I1, ϕ(Δ) = ij k is the only triangle of I2 for which aij , bj k and cik are triangles of I2 (cf. Lemma 9). Now, the vertex set of (S2 × S2)12 is V (= V (I1) = V (I2)) and it has two types of facets. (i) For each vertex x, the neighbors of x in I1 form facets. (ii) For each triangle Δ of I1 and each vertex y in Δ = ϕ(Δ), (Δ ∪ Δ ) \ {y} is a facet. Thus (S2 × S2)12 has 12 facets of the first type and 20 × 3 = 60 facets of the second type. From the description, it is clear that the common automorphism group A5 × Z2 of I1 and I2 is an automorphism group of (S2 × S2)12. It turns out that its full automorphism group is 2S5 generated by the two automorphisms g = (x12x21x24x42x14x41x43x34x13x31x32x23) and h = (x12x14x21x24x31)(x13x42x43x32x34). The automorphism g interchanges I1 and I2. Remark 4 It should be emphasized that the existence of an antimorphic pair of icosahedra (exploited in the above construction of (S2 × S2)12) is a minor miracle, and only an empirically verified fact. Its deeper geometric significance, if any, remains to be understood. 4 A Self-dual CW Decomposition of CP 2 Here we have taken the cell complex ∂T0 × ∂T0, and triangulated it to obtain the simplicial complex (S2 × S2)16 and finally quotiented this simplicial complex by Z2 to obtain CP120. This procedure reflects our obsession with simplicial complexes. However, one may straightaway quotient the cell complex by Z2 to obtain a (nonregular) CW decomposition of CP 2. This CW complex is self-dual in the sense that its face-vector (10, 24, 31, 24, 10) exhibits a curious palindromic symmetry. We proceed to describe it in some details. Consider the Z2-action on R6 ≡ R3 × R3 given by (x, y) ↔ (y, x). Let η: R6 → R6/Z2 be the quotient map. We know that η(S2 × S2) = CP 2. We give a CW decomposition W of the space η(∂T0 × ∂T0). For 0 ≤ i ≤ 4, let W i denote the set of i-cells in W . For i = 2 the i-cells in W are the images (under the map η) of i-cells in ∂T0 × ∂T0. A 2-cell in W is the image of a 2-cell F in ∂T0 × ∂T0 which is not of the form E × E for some edge E in ∂T0. More explicitly W 0 = V CP120 , W 1 = η(E) : E is an edge of ∂T0 × ∂T0 , W 2 = η |xij xikxil | : 1 ≤ j < k < l ≤ 4, 1 ≤ i ≤ 4 ∪ η |xi xj | × |xkxl | : i < j, k < l and either i < k or i = k and j < l , W 3 = η(A) : A is a 3-cell of ∂T0 × ∂T0 W 4 = η(B) : B is a 4-cell of ∂T0 × ∂T0 . and Then, W 1 contains 24 cells, W 2 contains 16 + 15 = 31 cells, W 3 contains 4 × 6 = 24 cells and W 4 contains 10 cells. Clearly, each 1-cell in W is regular (i.e., homeomorphic to a closed interval). Since all the 2-cells are homeomorphic images of the corresponding 2-cells in ∂T0 × ∂T0, it follows that all the 2-cells in W are regular. For 0 ≤ i ≤ 4, let Xi = β∈W 0∪···∪W i β. Then ∂α ⊆ Xi−1 if α ∈ W i for i = 3. Let γ be a 3-cell in W . If γ = η(|xi xj xk| × |xi xj |), i < j < k, then γ is obtained from |xi xj xk| × |xi xj | by identifying |xii xjj xij | with |xii xjj xji | (by the identification given by xij ↔ xji ). Thus, γ is a regular 3-cell and ∂γ = η(|xi xk| × |xi xj |) ∪ η(|xj xk| × |xi xj |) ∪ η(|xii xji xki |) ∪ η(|xij xjj xkj |). (Now, it is clear why we do not have to take η(|xi xj | × |xi xj |) in W 2. In fact, η(|xi xj | × |xi xj |) is inside of γ .) Therefore, ∂γ ⊆ X2. Same things are true if γ = η(|xi xj xk| × |xi xk|) or η(|xi xj xk| × |xj xk|). On the other hand, if γ = η(F × E), where E is an edge and F is a 2-simplex and E ⊆ F , then γ is homeomorphic to F × E and hence is a regular 3-cell. In this case, it follows from the definition of W 2 that ∂γ ⊆ X2. Thus W is a CW complex. If σ is a 4-cell in W then, either σ = η(|xi xj xk| × |xi xj xk|), for some i < j < k or σ = η(|xi xj xk| × |xi xj xl |), where {i, j, k, l} is an even permutation of {1, 2, 3, 4}. In the first case, σ is homeomorphic to |xii xjj xkkxij xik| ∪ |xii xjj xkkxij xjk| ∪ |xii xjj xkkxikxjk| and hence σ is a regular 4-cell. In the second case, σ is obtained from |xi xj xk| × |xi xj xl | by identifying |xii xjj xij | with |xii xjj xji | (by the identification given by xij ↔ xji ). So, σ is not a regular cell. Thus W 4 contains four regular 4-cells and six singular 4-cells. Since each cell in W is the quotient of a cell in S42 × S42, (S2 × S2)16 is a simplicial subdivision of S42 × S42 and CP120 is the quotient of (S2 × S2)16, it follows that CP120 is a simplicial subdivision of W . 5 Proofs Definition 2 Let G be a group of simplicial automorphisms of a simplicial complex X with vertex set V (X). We shall say that the action of G on X is pure if it satisfies: (a) whenever u, v are distinct vertices from the same G-orbit, uv is a non-edge of X, and (b) for each G-orbit θ ⊆ V (X) and each α ∈ X, the stabilizer Gα of α in G acts transitively on θ ∩ V (lkX(α)). Lemma 7 Let G be a group of simplicial automorphisms of a simplicial complex X. Let q: V (X) → V (X)/G denote the quotient map, and X/G := {q(α) : α ∈ X}. If the action of G on X is pure then X/G is a simplicial complex which triangulates |X|/G (where the action of G on V (X) is extended to an action of G on |X| piecewise linearly, i.e., affinely on the geometric carrier of each simplex). That is, we have |X/G| = |X|/G. Proof The condition (a) ensures that the quotient map q is one-one on each simplex of X. The simplicial map q: X → X/G induces a piecewise linear continuous map |q| from |X| onto |X/G|. Claim The fibers of q: X → X/G are precisely the G-orbits on simplices of X (that is, if α, α ∈ X are such that q(α) = q(α ) then there exists g ∈ G such that g(α) = α ). We prove the claim by induction on k = dim(α) = dim(α ). The claim is trivial for k = −1. So, assume k ≥ 0, and the claim is true for all smaller dimensions. Choose a simplex β ⊆ α of dimension k − 1, and let β ⊆ α be such that q(β ) = q(β). By induction hypothesis, β and β are in the same G-orbit. Therefore, applying a suitable element of G, we may assume, without loss of generality, that β = β. Let α = β ∪ {x}, α = β ∪ {x }. Then q(x) = q(x ), i.e., x and x are in the same G-orbit. Now, by assumption (b), there is a g ∈ Gβ such that g(x) = x . Then g(α) = α . This proves the claim. In the presence of condition (a), the claim ensures that the fibers of |q| are precisely the G-orbits on points of |X|. Hence |q| induces the required homeomorphism between |X|/G and |X/G|. Up to isomorphism, there are exactly two 6-vertex 2-spheres, namely, S1 and S2 given in Fig. 3. We need the following lemma to prove Theorem 1. Lemma 8 Let C be the triangular prism given in Fig. 3(b) (i.e., C is the product of a 2-simplex and an edge). Up to isomorphism, there exists a unique 6-vertex simplicial subdivision C of C. The facets (tetrahedra) in C are a1b1b2b3, a1a2b2b3, a1a2a3b3. Moreover, ∂C is isomorphic to S2 of Fig. 3(a) and determines C uniquely. Proof Let C be a 6-vertex subdivision of C. Then there exists a 3-simplex σ in C which contains the 2-simplex b1b2b3. Without loss of generality, we may assume that σ = a1b1b2b3. Then C is the union of σ and the pyramid P given in Fig. 4. Since we are not allowed to introduce new vertices, clearly the rectangular base of P must be triangulated using two triangles, in one of two isomorphic ways, and the remaining tetrahedra in C must have the apex of P as a vertex and one of these two triangles as base. Thus, without loss of generality, P = a1a2b2b3 ∪ a1a2a3b3. This proves the first part. The last part follows from the fact that the facets of C are the maximal cliques in the 1-skeleton of ∂C. Proof of Theorem 1 Let X be a 16-vertex simplicial subdivision of S42 × S42 satisfying (i), (ii) and (iii). For i = j , consider the 2-cell xi xj × xi xj . By (iii), xij xji can not be an edge in X. This implies that xii xjj , xii xji xjj , xii xij xjj ∈ X and xi xj × xi xj = xii xji xjj ∪ xii xij xjj (cf. Fig. 5(a)). For i, j, k distinct, consider the 2-cell xi xj × xi xk . Since X satisfies (iii), both xij and xji cannot be in lkX(xik). Now, xikxij is an edge in the cell complex S42 × S42 and hence is an edge in X. Thus, xikxji can not be an edge in X. This implies that xii xjk , xii xji xjk , xii xikxjk ∈ X and xi xj × xi xk = xii xji xjk ∪ xii xikxjk (cf. Fig. 5(b)). Consider the 2-cell x1x3 × x2x4. Clearly, x1x3 × x2x4 = x12x32x34 ∪ x12x14x34 or = x12x32x14 ∪ x32x14x34. Case 1. x1x3 × x2x4 = x12x32x34 ∪ x12x14x34 (cf. Fig. 5(c)). So, x12x34 ∈ X. Then, by (ii), x21x43 ∈ X and, by (iii), x12x43, x21x34 ∈ X. This implies that x2x3 × x1x4 = x21x31x24 ∪ x31x24x34 (cf. Fig. 5(d)). So, x31x24 ∈ X. Then, by (ii), x13x42 ∈ X and, by (iii), x13x24, x31x42 ∈ X. This implies that x1x2 × x3x4 = x13x23x14 ∪ x23x14x24 (cf. Fig. 5(e)). So, x14x23 ∈ X. Then, by (ii), x41x32 ∈ X and, by (iii), x14x32, x41x23 ∈ X. These give the 2-skeleton of X. Observe that we have already 84 edges as mentioned in the construction of (S2 × S2)16 and, since X satisfies (iii), all the 36 remaining 2-sets are non-edges in X. Observe that any 3-cell in S42 × S42 is the product of a 2-simplex and an edge. For i, j, k distinct, consider the 3-cell xi xj xk × xi xj . Since xii xjj , xii xki and xjj xki are edges, by Lemma 8, xi xj xk × xi xj = xii xij xkj xjj ∪ xii xki xkj xjj ∪ xii xki xji xjj is the unique subdivision of xi xj xk × xi xj (cf. Fig. 6(a)). Similarly, xi xj × xi xj xk = xii xji xjkxjj ∪ xii xikxjkxjj ∪ xii xikxij xjj is the unique subdivision of xi xj × xi xj xk (cf. Fig. 6(b)). For i, j, k, l distinct, consider the 3-cell xi xj xk × xi xl . Here xii xjl and xii xkl are edges. By interchanging j and k (if required) we may assume that {i, j, k, l} is an even permutation of {1, 2, 3, 4}. Then xki xjl is an edge and hence, by Lemma 8, xi xj xk × xi xl = xii xil xkl xjl ∪ xii xki xkl xjl ∪ xii xki xji xjl is the unique subdivision of xi xj xk × xi xl (cf. Fig. 6(c)). Similarly, for the 3-cell xi xl × xi xj xk , we may assume that {i, j, k, l} is an even permutation of {1, 2, 3, 4}. Then xikxlj is an edge and hence, by Lemma 8, xi xl × xi xj xk = xii xli xlkxlj ∪ xii xikxlkxlj ∪ xii xikxij xlj is the unique subdivision of xi xl × xi xj xk (cf. Fig. 6(d)). These give the 3-skeleton of X. For i, j, k distinct, consider the 4-cell A = xi xj xk × xi xj xk . The boundary ∂A of A consists of six 3-cells. From above, it follows that S1({xii , xjj , xkk}) ∗ C6 ⊆ X is the subdivision of ∂A, where C6 is the 6-cycle C6(xij , xik, xjk, xji , xki , xkj ). Let D ⊆ X be the subdivision of A. Then, D is a 9-vertex 4-ball with boundary ∂D = S1({xii , xjj , xkk}) ∗ C6. Clearly, C6 is an induced subcomplex of X. Therefore, each 4-simplex in B must contain xii xjj xkk . Thus, xii xjj xkk is a simplex in D \ ∂D. Therefore, lkD(xii xjj xkk) is a cycle and hence = C6. These imply that D = xii xjj xkk ∗ C6. Now, consider the 4-cell B = xi xj xk × xi xj xl , where i, j, k, l are distinct. By interchanging i and j (if required) we may assume that {i, j, k, l} is an even permutation of {1, 2, 3, 4}. The boundary ∂B of B consists of six 3-cells. From above, it follows that the subdivision of ∂B in X is a 9-vertex triangulated 3-sphere and 1 obtained from S3 ({xii , xjj , xkl }) × C5 by starring the vertex xji in the 3-simplex α := xii xjj xjl xki , where C5 is the 5-cycle C5(xij , xil , xjl , xki , xkj ). Since xji xij , xji xil , xji xkj and xji xkl are non-edges, it follows that σ := xii xjj xji xjl xki is the only possible 4-simplex containing xji inside B. So, σ ∈ X. Then B = σ ∪ P , where 1 P is a 4-cell such that P ∩ σ = α and S3 ({xii , xjj , xkl }) ∗ C5 ⊆ X is the subdivision of ∂P in X (i.e., P is the 4-cell whose geometric carrier is (|B| \ |σ |) ∪ |α|). 1 Let Q be the simplicial subdivision of P in X. So, ∂Q = S3 ({xii , xjj , xkl }) ∗ C5. Since C5 is induced in X, it follows that any 4-simplex in Q must contain xii xjj xkl . Since xii xjj xkl ∈ Q \ ∂Q, lkQ(xii xjj xkl ) is a cycle and hence = C5. These imply that Q = xii xjj xkk ∗ C5. Then B = (xii xjj xkl ∗ C5) ∪ σ¯ . Now, we have subdivided all the 4-cells in S42 × S42. It is routine to check that the resulting simplicial complex X is identical with the complex (S2 × S2)16 defined in Sect. 1. Case 2. x1x3 × x2x4 = x12x32x14 ∪ x32x14x34. By the same method as in Case 1, one can show that X is uniquely determined and is isomorphic to (S2 × S2)16 via the map f given by the transposition (1, 2) on the suffixes, i.e., f ≡ (x11x22)(x13x23)(x14x24)(x31x32)(x41x42). This completes the proof. Proof of Corollary 2 From Proposition 6, Lemma 7 and Theorem 1, it is immediate that CP120 triangulates CP 2. Since the automorphism group A4 = α, β of (S2 × S2)16 commutes with Z2, it descends to an automorphism group A4 = α¯ , β¯ of CP120. We need to show that there are no other automorphisms. It is easy to check that the four vertices xii , 1 ≤ i ≤ 4, are the only ones with 2-neighborly links. Therefore, the full automorphism group must fix this set of four vertices. Since A4 is 2-transitive on this 4-set, it suffices to show that there is no nontrivial automorphism γ fixing both x11 and x22. Suppose the contrary. Then γ is a non-trivial automorphism of lk(x11x22). But lk(x11x22) is the 8-vertex triangulated 2-sphere given in Fig. 7. From the picture, it is apparent that lk(x11x22) has only one non-trivial automorphism, namely (x13, x24)(x14, x23)(x33, x44). Therefore, γ = (x13, x24)(x14, x23)(x33, x44) and hence γ fixes the 3-simplex x11x33x44x34. Then γ must either fix or interchange the two vertices x13 and x14 in the link of this 3-simplex, a contradiction. This completes the proof. Proof of Theorem 3 Consider the following sequence of bistellar moves on CP120 (performed one after the other): (i) x22x33x44 → x23x24x34, (iii) x11x22x44 → x12x14x24, (v) x22x34x44 → x13x23x24, (vii) x12x22x44 → x13x14x24, (ix) x22x44 → x13x14x23x24. (ii) x11x33x44 → x13x14x34, (iv) x14x33x44 → x12x13x34, (vi) x23x33x44 → x12x24x34, (viii) x33x44 → x12x13x24x34, At the end of these moves, we get a 10-vertex triangulation K of CP 2. On K we perform the following sequence of bistellar moves one after another. (x) x11x24x44 → x12x14x23, (xii) x11x44 → x12x14x23x34, (xiv) x44x14 → x34x12x13x23, (xi) x11x13x44 → x14x23x34, (xiii) x44x14x24 → x12x13x23, (xv) x44 → x24x34x12x13x23. (Note that the last three bistellar moves together is same as the GBM with respect to (x44 ∗ S3 ({x14, x24, x34}) ∗ S3 ({x12, x13, x23}), S31({x14, x24, x34}) ∗ x12x13x23).) The 1 1 last bistellar move deletes the vertex x44 and hence obtain a 9-vertex triangulation L of CP 2. (Observe that A1 = {x11, x23, x24}, A2 = {x14, x33, x12}, A3 = {x34, x22, x13} is an amicable partition of L whose layer is of first type (cf. [ 4 ]).) Let CP92 be as described in [ 11 ] with vertex set {1, 2, . . . , 9}. Consider the map ϕ: L → CP92 given by: ϕ(x11) = 1, ϕ(x23) = 2, ϕ(x24) = 3, ϕ(x34) = 4, ϕ(x22) = 5, ϕ(x13) = 6, ϕ(x14) = 7, ϕ(x33) = 8, ϕ(x12) = 9. It is easy to see that ϕ is an isomorphism. Thus, CP120 is bistellar equivalent to CP 2. 9 Now, on K we perform the following sequence of bistellar moves: (xvi) x11x22x33 → x12x13x23, (xviii) x22x33x13 → x12x14x23, (xvii) x22x33x24 → x14x23x34, (xix) x22x33 → x12x14x23x34. We obtain a 10-vertex triangulation M of CP 2. Let K140 be as described in [ 7 ] with vertex set {X, Y, Z, 0, 1, . . . , 6}. Consider the map ψ : M → K140 given by ψ (x33) = X, ψ (x22) = Y , ψ (x44) = Z, ψ (x11) = 0, ψ (x13) = 1, ψ (x12) = 2, ψ (x23) = 3, ψ (x14) = 4, ψ (x34) = 5, ψ (x24) = 6. It is easy to see that ψ is an isomorphism. Thus, CP120 is bistellar equivalent to K140. This completes the proof. Proof of Corollary 4 Part (a) follows from Corollary 2 and Theorem 3. In [ 7 ], explicit coordinates for simplices of K140 in the Fubini–Study metric were given. This shows that the induced pl-structure on CP 2 by K140 is the standard one. Part (b) now follows from Theorem 3. Lemma 9 Let I1 and I2 be an antimorphic pair of icosahedra. Then we have: (a) Aut(I1) = Aut(I2) = A5 × Z2. (b) For each triangle Δ of I1, there is a unique triangle Δ of I2 such that each of the three triangles of I2 sharing an edge with Δ has its third vertex in Δ. Further, the map ϕ: Δ → Δ is a bijection from the triangles of I1 to the triangles of I2. There is a similarly defined bijection ψ from the triangles of I2 to the triangles of I1, and (c) Every isomorphism f : I1 → I2 intertwines ϕ and ψ . (Warning: The maps ϕ and ψ are not induced by any vertex-to-vertex map!) Proof Recall that I1 and I2 have the same vertex set and the same pairs of antipodal vertices. Thus, they have the same antipodal map (sending each vertex x to its antipode x¯ ). Now, the full automorphism group of the icosahedron is generated by its rotation group A5 and the antipodal map. So, to prove Part (a), it suffices to show that I1 and I2 share the same rotation group. For each pair x, x¯ of antipodes, Ii has a i rotation symmetry αx,x¯ which fixes x and x¯ and rotates the remaining vertices along the 5-cycles lkIi (x) and lkIi (x¯). The rotation group of Ii is generated by these automorphisms of order five. But, lkI2 (x) (respectively, lkI2 (x¯)) is the graph theoretic 2 complement of the pentagon lkI1 (x¯) (respectively, lkI1 (x)). Therefore, αx,x¯ is the square of αx1,x¯ . This proves Part (a). Notice that if f1, f2: I1 → I2 are two antimorphisms, then f1 ◦ f2−1 ∈ Aut(I2) and f2−1 ◦ f1 ∈ Aut(I1). Thus, the antimorphism is unique up to right multiplication by elements of Aut(I1) (or left multiplication by elements of Aut(I2)). Therefore, there is no loss of generality in taking the antimorphic pair of icosahedra as the one given in Fig. 2. Since the common automorphism group is transitive on the triangles of I1 (and of I2), it is enough to look at the triangle Δ = x12x13x14 of I1. From Fig. 2, we see that the links in I2 of two vertices of Δ have exactly two vertices in common. Namely, we have V (lkI2 (x12)) ∩ V (lkI2 (x13)) = {x21, x32}, V (lkI2 (x12)) ∩ V (lkI2 (x14)) = {x24, x41}, V (lkI2 (x13)) ∩ V (lkI2 (x14)) = {x31, x43}. Therefore, any triangle Δ of I2 satisfying the requirement must be contained in the vertex set {x21, x32, x24, x41, x31, x43}. But one sees that this set of six vertices contains a unique triangle in I2, namely Δ = x21x31x41. Thus the map ϕ: Δ → Δ is well defined. Similarly, there is a well defined map ψ from the triangles of I2 to the triangles of I1. The map ψ ◦ ϕ is the antipodal map on the triangles of I1 to themselves. Similarly, ϕ ◦ ψ is the antipodal map on triangles of I2. Hence ϕ (as well as ψ ) is a bijection. This proves Part (b). To prove Part (c), let f be any isomorphism from I1 to I2. Since I1 and I2 are antimorphic, it is immediate that f also defines an isomorphism from I2 to I1. Let Δ be any triangle of I1 and let Δ = ϕ(Δ). By definition, there are three triangles Δ1, Δ2, Δ3 of I2 each of which shares a vertex with Δ and an edge with Δ . Then f (Δ) and f (Δ ) are triangles of I2 and I1, respectively. Also, f (Δ1), f (Δ2), f (Δ3) are three triangles of I1 each of which shares a vertex with f (Δ) and an edge with f (Δ ). Therefore, by definition of ψ , ψ (f (Δ)) = f (Δ ) = f (ϕ(Δ)). Proof of Theorem 5 As in the proof of Theorem 1, one may verify that (S2 × S2)16 is a simplicial subdivision of S42 × S42, and hence it triangulates S2 × S2. We apply the following sequence of bistellar moves to (S2 × S2)16 to create a second 16-vertex triangulation (S2 × S2)16 of S2 × S2: Since this set of bistellar moves is stable under the automorphism group A4 of (S2 × S2)16, it follows that (S2 × S2)16 inherits the group A4. Also, both complexes have lk(x11) = S31({x12, x13, x14}) ∗ S31({x21, x31, x41}). However, while (S2 × S2)16 has both x12x13x14 and x21x31x41 as triangles, we have chosen the bistellar moves judiciously to ensure that (S2 × S2)16 does not have the triangle x12x13x14. Therefore, we may apply the following four GBM’s (one after the other) to (S2 × S2)16 to delete the four vertices xii , 1 ≤ i ≤ 4: st(x11), D32 {x12, x13, x14} ∗ S31 {x21, x31, x41} , st(x22), D32 {x21, x23, x24} ∗ S31 {x12, x32, x42} , st(x33), D32 {x31, x32, x34} ∗ S31 {x13, x23, x43} , st(x44), D32 {x41, x42, x43} ∗ S31 {x14, x24, x34} . The resulting complex X is therefore a 12-vertex triangulation of S2 × S2. So, to confirm the first statement of this theorem, it suffices to show that X is isomorphic to the complex (S2 × S2)12 described in Sect. 3. Indeed, with the antimorphic pair of icosahedra (and their vertex names) as in Fig. 2, we shall show that we actually have X = (S2 × S2)12. Notice that X inherits the automorphism group A4 from (S2 × S2)16, and modulo this group, the following six are basic facets of X: x12x14x21x24x31, x12x13x14x21x31, x12x23x31x13x32, x12x31x34x14x24, x24x31x32x12x21, x24x31x32x12x41. Each basic facet is in an A4-orbit of size 12, yielding a total of 6 × 12 = 72 facets of X. Since (S2 × S2)12 also has 72 facets and since the group A4 (acting on subscripts) is a subgroup of the automorphism group A5 × Z2 of (S2 × S2)12, it suffices to observe that all six basic facets of X listed above are also facets of (S2 × S2)12. Indeed, the first facet x12x14x21x24x31 is in (S2 × S2)12 since these five vertices are the neighbors of x23 in I1 (and of x41 in I2). In each of the remaining five basic facets of X, the first three vertices constitute a triangle Δ of I1 with the last two vertices in the corresponding triangle Δ = ϕ(Δ) of I2 (cf. Lemma 9). (For instance, Δ = x12x13x14 is a triangle of I1, with corresponding triangle Δ = x21x31x41 of I2. Therefore, the second basic facet of X is a facet of (S2 × S2)12.) This shows that (S2 × S2)12 = X, so that (S2 × S2)12 triangulates S2 × S2. To compute the full automorphism group of (S2 × S2)12, notice that it has exactly 40 triangles of degree 3 (the rest are of degree 5), namely the twenty triangles of I1 and the twenty triangles of I2. Consider the graph whose vertices are these forty triangles, two of them being adjacent if and only if they share an edge. This graph has exactly two connected components, of size 20 each, namely the triangles of I1 and I2. This shows that any automorphism f of (S2 × S2)12 either fixes both I1 and I2 or interchanges them. So, Aut(I1) = Aut(I2) = A5 × Z2 is a subgroup of index at most two in the full automorphism group of (S2 × S2)12. Let f : I1 → I2 be any isomorphism. Since I1 and I2 are antimorphic, it is immediate that f is also an isomorphism from I2 to I1. Since the five neighbors in I1 of any vertex are also the neighbors in I2 of the antipodal vertex, it is immediate that f maps each of the 12 facets of the first kind in (S2 × S2)12 to a facet of the same kind. Also, for any triangle Δ of I1, the construction of (S2 × S2)12 shows that lk(Δ) = S31(ϕ(Δ)), and also, for any triangle Δ of I2, lk(Δ ) = S31(ψ (Δ )). Since f intertwines ϕ and ψ (Lemma 9), we also have lk(f (Δ)) = S31(ψ (f (Δ))) = S31(f (ϕ(Δ))) = f (S31(ϕ(Δ))) = f (lk(Δ)). Similarly, for any triangle Δ of I2, lk(f (Δ )) = f (lk(Δ )). Thus, f also maps all sixty facets of the second type in (S2 × S2)12 to facets of the same type. Thus, any isomorphism between I1 and I2 is also an automorphism of (S2 × S2)12. Therefore, the full automorphism group G of (S2 × S2)12 has H = A5 × Z2 as an index two subgroup. Thus, G is of order 240. Indeed, G consists of the 120 common automorphisms of I1 and I2, and the 120 isomorphisms between I1 and I2. In particular, take g = (x12x21x24x42x14x41x43x34x13x31x32x23), which is an isomorphism between I1 and I2. Note that g6 is the common antipodal map of I1 and I2, hence it is in the center of G. Thus, G/ g6 is the extension of A5 by the involution α = g (mod g6). But A5 has only one non-trivial extension by an involution, namely S5. So, G is an extension of a central involution by S5. It can not be the split extension S5 × Z2 since this has no element of order 12. Therefore, G is the unique non-split extension 2S5 of Z2 by S5. Remark 5 If the link of a vertex u in a triangulated 4-manifold X is S31({x, y, z}) ∗ S31({a, b, c}) and xyz is not a simplex in X then the GBM (stX(u), D32({x, y, z}) ∗ 1 S3 ({a, b, c}) is equivalent to the sequence of the following three bistellar moves: uab → xyz, ua → cxyz, u → bcxyz. Thus, from the proof of Theorem 5, (S2 × S2)12 can be obtained from (S2 × S2)16 by a sequence of bistellar moves only. Acknowledgements The authors thank the anonymous referee for many useful comments which led to substantial improvements in the presentation of this paper. The authors are thankful to Siddhartha Gadgil, Frank Lutz, Wolfgang Kühnel and Alberto Verjovsky for useful conversations and references. The authors thank Ipsita Datta for her help in the proof of Theorem 3. 1. Akhmedov , A. , Park , B.D. : Exotic smooth structures on S2 × S2 , 12 pp. ( 2010 ). arXiv: 1005 .3346v5 2. Arnoux , P. , Marin , A. : The Kühnel triangulation of the complex projective plane from the view-point of complex crystallography (part II) . Mem. Fac. Sci. Kyushu Univ. Ser. A 45 , 167 - 244 ( 1991 ) 3. Bagchi , B. , Datta , B. : On Kühnel's 9-vertex complex projective plane . Geom. Dedic . 50 , 1 - 13 ( 1994 ) 4. Bagchi , B. , Datta , B. : A short proof of the uniqueness of Kühnel's 9-vertex complex projective plane . Adv. Geom. 1 , 157 - 163 ( 2001 ) 5. Bagchi , B. , Datta , B. : Combinatorial triangulations of homology spheres . Discrete Math. 305 , 1 - 17 ( 2005 ) 6. Bagchi , B. , Datta , B. : Lower bound theorem for normal pseudomanifolds . Exp. Math . 26 , 327 - 351 ( 2008 ) 7. Banchoff , T.F. , Kühnel , W. : Equilibrium triangulations of the complex projective plane . Geom. Dedic . 44 , 413 - 433 ( 1992 ) 8. Bing , R.H.: Some aspects of the topology of 3-manifolds related to the Poincaré conjecture . In: Saaty, T.L . (ed.) Lectures on Modern Mathematics , vol. II, pp. 93 - 128 . Wiley, New York ( 1964 ). Chap. 3 9. Brehm , U. : Private communication ( 2010 ) 10. Brehm , U. , Kühnel , W. : Combinatorial manifolds with few vertices . Topology 26 , 465 - 473 ( 1987 ) 11. Kühnel , W. , Banchoff , T.F. : The 9-vertex complex projective plane . Math. Intell . 5 ( 3 ), 11 - 22 ( 1983 ) 12. Kühnel , W. , Laßmann , G. : The unique 3-neighbourly 4-manifold with few vertices . J. Comb. Theory Ser. A 35 , 173 - 184 ( 1983 ) 13. Lawson , T. : Splitting S4 on RP 2 via the branched cover of CP 2 over S4 . Proc. Am. Math. Soc . 86 , 328 - 330 ( 1982 ) 14. Lutz , F.H. : Triangulated Manifolds with Few Vertices and Vertex-Transitive Group Actions . Shaker Verlag, Aachen ( 1999 ). Thesis (D 83, TU Berlin) 15. Perelman , G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, 7 pp . ( 2003 ). arXiv:math/0307245v1 16. Saveliev , N.: Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant . Walter de Gruyter, Berlin ( 1999 ) 17. Sparla , E. : An upper and a lower bound theorem for combinatorial 4-manifolds . Discrete Comput. Geom . 19 , 575 - 593 ( 1998 )


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Bhaskar Bagchi, Basudeb Datta. From the Icosahedron to Natural Triangulations of ℂP 2 and S 2×S 2, Discrete & Computational Geometry, 2010, 542-560, DOI: 10.1007/s00454-010-9281-0