We know that the polyhedra corresponding to the Platonic solids are equivelar. In this article we have classified completely all the simplicial equivelar polyhedra on ≤ 11 vertices. There are exactly 27 such polyhedra. For each n ≥ −4, we have classified all the ( p, q) such that there exists an equivelar polyhedron of type { p, q} and of Euler characteristic n. We have also constructed five types of equivelar polyhedra of Euler characteristic −2m, for each m ≥ 2.

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices.In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists an n-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinct n-vertex weakly regular triangulations of the torus for each n ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for each m ≥ 2. For 12 ≤ n ≤ 15, we have classified all the n-vertex degree-regular triangulations of the torus and the Klein bottle. There are exactly 19 such triangulations, 12 of which are triangulations of the torus and remaining 7 are triangulations of the Klein bottle. Among the last 7, only one is weakly regular.Proposition 1 . There are exactly 77 weakly regular combinatorial 2-manifolds on at most 15 vertices; 42 of these are orientable and 35 are non-orientable. Among these 77 combinatorial 2-manifolds, 20 are of Euler characteristic 0. These 20 are T

A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinatorial 2-manifold of Euler characteristic −2 must contain 12 vertices.In

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We introduce the k-stellated spheres and consider the class W k (d) of triangulated d-manifolds all whose vertex links are k-stellated, and its subclass W * k (d) consisting of the (k + 1)-neighbourly members of W k (d). We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of its Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-vector of any 2-neighbourly member of W k (d) for d ≥ 2k. As one consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, as well as determine the integral homology type of members of W * k (d) for d ≥ 2k + 2. As another application, we prove that, when d = 2k + 1, all members of W * k (d) are tight. We also characterize the tight members of W * k (2k + 1) in terms of their k th Betti numbers. These results more or less answer a recent question of Effenberger, and also provide a uniform and conceptual tightness proof for all except two of the known tight triangulated manifolds.We also prove a lower bound theorem for triangulated manifolds in which the members of W 1 (d) provide the equality case. This generalises a result (the d = 4 case) due to Walkup and Kühnel. As a consequence, it is shown that every tight member of W 1 (d) is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kühnel and Lutz asserting that tight triangulated manifolds should be strongly minimal. (2010): 57Q15, 52B05. Mathematics Subject ClassificationDefinition 1.1. For 0 ≤ k ≤ d + 1, a d-dimensional simplicial complex X is said to be k-stellated if X may be obtained from S d d+2 by a finite sequence of bistellar moves, each of index < k. By convention, S d d+2 is the only 0-stellated simplicial complex of dimension d. Clearly, for 0 ≤ k ≤ l ≤ d + 1, k-stellated implies l-stellated. All k-stellated simplicial complexes are combinatorial spheres. By Pachner's theorem [25], the (d + 1)-stellated dspheres are precisely the combinatorial d-spheres.We also recall : Definition 1.2. For 0 ≤ k ≤ d + 1, a triangulated (d + 1)-dimensional ball B is said to be k-stacked if all the faces of B of codimension (at least) k + 1 lie in its boundary; i.e., if

For integers d 2 and ε = 0 or 1, let S 1,d−1 (ε) denote the sphere product S 1 × S d−1 if ε = 0 and the twisted sphere product S 1 S d−1 if ε = 1. The main results of this paper are: (a) if d ≡ ε (mod 2) then S 1,d−1 (ε) has a unique minimal triangulation using 2d + 3 vertices, and (b) if d ≡ 1 − ε (mod 2) then S 1,d−1 (ε) has minimal triangulations (not unique) using 2d + 4 vertices. In this context, a minimal triangulation of a manifold is a triangulation using the least possible number of vertices. The second result confirms a recent conjecture of Lutz. The first result provides the first known infinite family of closed manifolds (other than spheres) for which the minimal triangulation is unique. Actually, we show that while S 1,d−1 (ε) has at most one (2d + 3)-vertex triangulation (one if d ≡ ε (mod 2), zero otherwise), in sharp contrast, the number of non-isomorphic (2d + 4)-vertex triangulations of these d-manifolds grows exponentially with d for either choice of ε. The result in (a), as well as the minimality part in (b), is a consequence of the following result: (c) for d 3, there is a unique (2d + 3)-vertex simplicial complex which triangulates a non-simply connected closed manifold of dimension d. This amazing simplicial complex was first constructed by Kühnel in 1986. Generalizing a 1987 result of Brehm and Kühnel, we prove that (d) any triangulation of a non-simply connected closed d-manifold requires at least 2d + 3 vertices. The result (c) completely describes the case of equality in (d). The proofs rest on the Lower Bound Theorem for normal pseudomanifolds and on a combinatorial version of Alexander duality.

In this paper we present a self-contained combinatorial proof of the lower bound theorem for normal pseudomanifolds, including a treatment of the cases of equality in this theorem. We also discuss McMullen and Walkup's generalised lower bound conjecture for triangulated spheres in the context of the lower bound theorem. Finally, we pose a new lower bound conjecture for non-simply connected triangulated manifolds. MSC 2000: 57Q15; 57R05.

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