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A weakly neighborly polyhedral map (w.n.p. map) is a two-dimensional cell-complex which decomposes a closed 2-manifold without boundary, such that for every two vertices there is a 2-cell containing them. We prove that there are just four w.n.p, maps with Euler characteristic -1 and we describe them.

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Mathematics Subject ClassO~cation (1991): 57Q15, 52B70, 53C42 Introduction and main theoremIt is well known that any simplicial decomposition of the real projective plane RP 2 must have at least 6 vertices, and that the 2-fold quotient of the icosahedron provides the unique 6-vertex triangulation RP62. By construction it is invariant under the action of the icosahedral group As. Moreover it has the following properties:(1) It has the smallest number of vertices among all simplicial 2-manifolds which are not PL homeomorphic to the sphere,(2) any two vertices are contained in a common edge. This property is also called 2-neighborliness.Regarding NP62 as a subcomplex of the 5-dimensional simplex A 5 we have Skl(A 5) C RP~ C Sk2(Z~5),(3) NP 2 C z55 C E 5 is a tight polyhedral embedding with the maximal possible essential codimension. Recall that there is a tight algebraic embedding ~.p2 C S 4 C E 5 known as the Veronese surface (and these two embeddings are essentially the only tight embeddings RP 2 --+ E 5, see [KP]). A submanifold M C E n is called tight if for any half space h c E n the induced homomorphism H,(M N h) ~ H,(M) is injective where H. denotes the ordinary simplicial or singular homology with coefficients in Z2. A d-dimensional combinatorial manifold M d is a simplicial decomposition of a topological manifold such that the link of each vertex is a combinatorial (d-1)-sphere. M is called k -neighborly if any k-tuple of vertices spans a (k -1)-dimensional simplex of M, or equivalently, if M contains the full (k -1)-dimensional skeleton of the (n -1)-simplex A '~-I where n is the number of vertices. 168 U. Brehm and W. KtihnelThe corresponding result for the complex projective plane CP 2 says that any combinatorial triangulation of CP 2 must have at least 9 vertices, and that there is exactly one such triangulation CP~ (up to relabeling). It is invariant under the action of a 2-fold extension of the Heisenberg group over Z3, see [KB, KL]. Moreover it has the following analogous properties:(1) It has the smallest number of vertices among all combinatorial 4-manifolds which are not PL homeomorphic to the sphere (see [BG, BK2]),(2) any three vertices are contained in a common triangle (3-neighborliness), thus Sk2(A s) C CP~ C Sk4(As),(3) CP92 C A 8 C E 8 is a tight polyhedral embedding with the maximal possible essential codimension. It corresponds to the tight algebraic embedding CP 2 c S 7 c E s, see [KB, Kui2] (compare [Kuil, Theorem 15] for the uniqueness of the tight algebraic embedding).The analogy between RP 2 and CP~ and the correspondence with the algebraic standard embeddings RP 2 ---, E 5, CP 2 --+ E s suggests to consider also the cases of the quaternionic projective plane HP 2 and of the Cayley plane CaP 2. The tight standard embeddings HP 2 --* E 14, CaP 2 --~ E 26 (see [T, TK]) suggest n = 15 or n = 27 for the number n of vertices of such a combinatorial triangulation. Similarly the tightness suggests the 5-neighborliness for a possible HP25 and the 9

We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, determines a triangle in the complex and quaternion projective spaces CP" and HP" (n ~> 2) uniquely up to isometry. We give inequalities describing the exact range of the four basic invariants. We express the angular invariants of a triangle with our basic invariants, giving a new completely elementary proof of the laws of trigonometry. As a corollary we derive a large number of congruence theorems. Finally we get, in exactly the same way, the corresponding results for triangles in the complex and quaternion hyperbolic spaces CH" and HH" (n >~ 2).We imagine the vertices [x], I-y], I-z] of a triangle as being connected by the (uniquely existing) shortest paths between them, called sides (Figure 1).Since our metric is an inner metric we have that the length of the side Geometriae Dedicata 33: [59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76] 1990.