Abstracts of the Research Papers

1. Characterisation of Fueter mappings and their Jacobians : We continue the study of a class of real analytic mappings (christened Fueter mappings) from open subsets of IR ⁿ into IR ⁿ. This class has connections with spaces of holomorphic functions and functions of quaternionic and octonionic variables. We characterise such mappings by a system of partial differential equations and determine conditions for their K-quasiconformality in the sense of Ahlfors. We also charcterise the Jacobian matrices of these mappings. The Jacobian matrices form a family of subgroups of GL(n, IR) (which are mutually isomorphic) parametrised by certain projective spaces.
2. Characterisation theorems for compact hypercomplex manifolds : We have defined and studied some pseudogroups of diffeomorphisms which generalise the complex analytic pseugroups. A 4-dimensional (or 8-dimensional) manifold moddelled on these `Fueter pseudogroups' turns out to be a quaternionic (respectively octonionic) manifold.
   We characterise compact Fueter manifolds as being products of compact Riemann surface with appropriate dimensional spheres. It then trnspires that a connected compact quaternionic (IH) (respectively O) manifold X, minus a finite number of circles (its `real set'), is the orientation double covering of the product Y ×P ², (respectively Y ×P ⁶), where Y is a connected surface equipped with a canonical confrmal structure and P ⁿ is n-dimensional real projective space.
   A corollary is that the only simply-connected compact manifolds which can allow IH (respectively O) structure are S ⁴ and S ² ×S ² (respectively S ⁸ and S ²×S ⁶).
   Previous authors, for example Marchiava and Salamon, have studied very closely-related classes of manifolds by differential geometric methods. Our techniques in this paper are function theoretic and topological.
3. Zero-sets of quaternionic and octonionic analytic functions with central coefficients : We prove that the zero set of any quaternionic (or octonionic) analytic function f with central (that is, real) coefficients is the disjoint union of codimension two spheres in IR ⁴ or IR ⁸ (respectively) and certain purely real points. In particular, for polynomials with real coefficients, the complete root-set is geometrically characterisable from the lay-out of the roots in the complex plane. The root-set becomes the union of a finite number of codimension 2 Euclidean spheres together with a finite number of real points. We also find the preimages f^{- 1}(A) for any quaternion (or octonion) A.
   We demonstrate that this surprising phenomenon of complete spheres being part of the solution set is very markedly a special `real' phenomenon. For example, the quaternionic or octonionic Nth roots of any non-real quaternion (respectively octonion) turn out to be precisely N distinct points. All this allows us to do some interesting topology for self-maps of spheres.
4. Non-existence of almost complex structures on products of even dimensional spheres : In this paper we prove the following theorem: S ^2p ×S ^2q allows an almost complex structure if and only if (p, q) = (1, 1), (1, 2), (2, 1),(1,3), (3, 1), (3, 3).
5. Fueter and hypercomplex structures on smooth manifolds : Fueter and hypercomplex structures are compared with f-structure and quaternion structure on smooth manifolds. It is also shown that a Fueter structure on a smooth manifold defines a foliation with a canonical complex structure on leaves.
6. (1, 2)-symplectic structures, nearly Kähler structures and S ⁶ : The exact relations between the Hermitian, the Symplectic and the Nearly Kähler Structures are established.
   The standard complex structure J_s on SO(2 n + 1)/U(n) is shown to be the same as the almost complex structure J₁ (one of the canonical almost complex structure on SO(2 n + 1)/U(n) considered as a twistor space over (S ^{2 n}, g₀)). A corollary is that S ⁶ does not allow any complex structure orthogonal to the standard metric g₀.
   On an almost complex manifold with an arbitary metric a (1,2)-tensor A is defined. If A is closed as a vector valued 2-form, then it is proved that constant sectional curvature implies zero curvature. This is related to Hsiung's work, which is discussed at the end.
7. On Kühnel's 9-vertex complex projective plane : We present an elementary combinatorial proof of the existence and uniqueness of the 9-vertex triangulation of C P². The original proof of existence, due to Kühnel, as well as the original proof of uniqueness, due to Kühnel and Labmann, were based on extensive computer search. Recently Arnoux and Marin have used cohomology theory to present a computer-free proof. Our proof has the advantage of displaying a canonical copy of the affine plane over the three-element field inside this complex in terms of which the entire complex has a very neat and short description. This explicates the full automorphism group of the Kühnel complex as a subgroup of the automorphism group of this affine plane. Our method also brings out the rich combinatorial structure inside this complex.
8. Combinatorial manifolds with complementarity : A simplicial complex is said to satisfy complementarity if exactly one of each complementary pair of nonempty vertex-sets constitutes a face of the complex.
   We show that if a d-dimensional combinatorial manifold M with n vertices satisfies complementarity then d = 0, 2, 4, 8 or 16 with n = 3d/2 + 3 and | M | is a "manifold like a projective plane". Arnoux and Marin had earlier proved the converse statement.
9. A discrete isoperimetric problem : We prove that the perimeter of any convex n-gons of diameter 1 is at most 2 n sin(p/2 n). Equality is attained here if and only if n has an odd factor. In the latter case, there are (up to congruence) only finitely many extremal n-gons. In fact, the convex n-gons of diameter 1 and perimeter 2 nsin(p/2 n) are in bijective correspondence with the solutions of a diophantine problem.
10. A structure theorem for pseudomanifolds : We introduce the notion of primitive pseudomanifolds and prove that all pseudomanifolds (without boundary) are built out of the primitive ones by a canonical procedure. This theory is used to explicitly determine and count all the pseudomanifolds of dimension d > 0 on at most d + 4 vertices. As a consequence, it turns out that their geometric realisations are either spheres or iterated suspensions of the real projective plane.
11. Pseudomanifolds with complementarity : A simplicial complex is said to satisfy complementarity if exactly one of each complementary pair of nonempty vertex-sets constitutes a simplex of the complex. In this article we show that if there exists a n-vertex d-dimensional pseudomanifold M with complementarity and either n < d + 7 or d < 7 then d = 0, 2, 4 or 6 with n = 3 d/2 + 3. We also show that if M is a d-dimensional pseudomanifold with complementarity and the number of vertices in M is < d + 6 then M is either a set of three points or the unique 6-vertex real projective plane or the unique 9-vertex complex projective plane.
12. Two dimensional weak pseudomanifolds on seven vertices : In this article we have explicitly determined all the 2-dimensional weak pseudomanifolds on 7 vertices. We have proved that there are (up to isomorphism) 13 such weak pseudomanifolds. The geometric carriers of them are 6 topological spaces, three of which are non-manifolds.
13. A short proof of the uniqueness of Kühnel's 9-vertex complex projective plane : We introduce the notion of amicable partitions for combinatorial manifolds with complementarity. We prove that any 4-dimensional combinatorial manifold X₉⁴ satisfying complementarity has an amicable partition and any amicable partition determines X₉⁴ up to isomorphism. This gives a short proof of the uniqueness of Kühnel's 9-vertex complex projective plane.
14. Equivelar polyhedra with few vertices : 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 < 12 vertices. There are exactly 27 such polyhedra. For each n > - 5, 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 - 2 m, for each m > 1.
15. Two dimensional weak pseudomanifolds on eight vertices : We explicitly determine all the two-dimensional weak pseudomanifolds on 8 vertices. We prove that there are (up to isomorphism) exactly 95 such weak pseudomanifolds, 44 of which are combinatorial 2-manifolds. These 95 weak pseudomanifolds triangulate 16 topological spaces. As a consequence, we prove that there are exactly three 8-vertex two-dimensional orientable pseudomanifolds which allow degree three maps to the 4-vertex 2-sphere.
16. The edge-minimal polyhedral maps of Euler characteristic - 8 : In 1990, a {5, 5}-equivelar polyhedral map of Euler characteristic -8 was constructed. In this article we prove that {5, 5}-equivelar polyheral map of Euler characteristic -8 is unique. As a consequence, we get that the minimum number of edges in a non-orientable polyhedral maps of Euler characteristic -8 is > 40. We have also constructed {5, 5}-equivelar polyhedral map of Euler characteristic -2m for each m > 3.
17. Non-existence of 6-dimensional pseudomanifolds with complementarity : In a previous paper the second author showed that if M is a pseudomanifold with complementarity other than the 6-vertex real projective plane and the 9-vertex complex projective plane, then M must have dimension > 5, and - in case of equality - M must have exactly 12 vertices. In this paper we prove that such a 6-dimensional pseudomanifold does not exist. On the way to proving our main result we also prove that all combinatorial triangulations of the 4-sphere with at most 10 vertices are combinatorial 4-spheres.
18. Degree-regular triangulations of torus and Klein bottle : 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 1999, Lutz has classified all the weakly regular triangulations on at most 15 vertices. In 2001, 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 > 8. We have constructed two distinct n-vertex weakly regular triangulations of the torus for each n > 11 and a (4m +2)-vertex weakly regular triangulation of the Klein bottle for each m > 1. For 11 < n < 16, 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.
19. A note on the existence of {k, k}-equivelar polyhedral maps: A polyhedral map is called {p, q}-equivelar if each face has p edges and each vertex belongs to q faces. In [12], it was shown that there exist infinitely many geometrically realizable {p, q}-equivelar polyhedral maps if q > p = 4, p > q = 4 or q − 3 > p = 3. It was shown in [6] that there exist infinitely many {4, 4}- and {3, 6}-equivelar polyhedral maps. In [1], it was shown that {5, 5}- and {6, 6}-equivelar polyhedral maps exist. In this note, examples are constructed, to show that infinitely many self dual {k, k}-equivelar polyhedral maps exist for each k > 4. Also vertex-minimal non-singular {p, p}-pattern are constructed for all odd primes p.
20. Combinatorial triangulations of homology spheres : Let M be an n-vertex combinatorial triangulation of a Z₂-homology d-sphere. In this paper we prove that if n < d + 9 then M must be a combinatorial sphere. Further, if n = d + 9 and M is not a combinatorial sphere then M can not admit any proper bistellar move. Existence of a 12-vertex triangulation of the lens space L(3, 1) shows that the first result is sharp in dimension three.
    In the course of the proof we also show that any Z₂-acyclic simplicial complex on < 8 vertices is necessarily collapsible. This result is best possible since there exist 8-vertex triangulations of the Dunce Hat which are not collapsible.
21. Degree-regular triangulations of the double-torus : 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 1982, McMullen et al. constructed a 12-vertex geometrically realized triangulation of the double-torus in IR³. As an abstract simplicial complex, this triangulation is a weakly regular combinatorial 2-manifold. In 1999, Lutz showed that there are exactly three weakly regular orientable combinatorial 2-manifolds of Euler characteristic -2. In this article, we classify all the orientable degree-regular combinatorial 2-manifolds of Euler characteristic -2. There are exactly six such combinatorial 2-manifolds. This classifies all the orientable equivelar polyhedral maps of Euler characteristic -2.
22. Uniqueness of Walkup's 9-vertex 3-dimensional Klein bottle : Via a computer search, Altshuler and Steinberg found that there are 1296 +1 combinatorial 3-manifolds on nine vertices, of which only one is non-sphere. This exceptional 3-manifold K³₉ triangulates the twisted S ²-bundle over S ¹. It was first constructed by Walkup. In this paper, we present a computer-free proof of the uniqueness of this non-sphere combinatorial 3-manifold. As opposed to the computer-generated proof, ours does not require wading through all the 9-vertex 3-spheres. As a preliminary result, we also show that any 9-vertex combinatorial 3-manifold is equivalent by proper bistellar moves to a 9-vertex neighbourly 3-manifold.
23. Minimal triangulations of sphere bundles over the circle : For integers d > 1 and e = 0 or 1, let S^{1, d -1}(e) denote the sphere product S¹×S^{d - 1} if e = 0 and the twisted sphere product S¹ ×_-  S^d-1 if e = 1. The main results of this paper are : (a) if d º e (mod 2) then S^{1, d - 1}(e) has a unique minimal triangulation using 2d+3 vertices, and (b) if d º 1 - e (mod 2) then S^{1, d -1}(e) 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}(e) has at most one (2d +3)-vertex triangulation (one if d º e (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 e. The result in (a), as well as the minimality part in (b), is a consequence of the following result : (c) for d > 2, 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.
24. Three dimensional pseudomanifolds on eight vertices : A normal pseudomanifold is a pseudomanifold in which the links of simplices are also pseudomanifolds. So, a normal 2-pseudomanifold triangulates a connected closed 2-manifold. But, normal d-pseudomanifolds form a broader class than triangulations of connected closed d-manifolds for d > 2. Here, we classify all the 8-vertex neighbourly normal 3-pseudomanifolds. This gives a classification of all the 8-vertex normal 3-pseudomanifolds. As a preliminary result, we show that any 8-vertex 3-pseudomanifold is equivalent by proper bistellar moves to an 8-vertex neighbourly 3-pseudomanifold. This result is best possible since there exists a 9-vertex non-neighbourly 3-pseudomanifold (B³₉ in Example 7 below) which does not allow any proper bistellar move.
25. Lower bound theorem for normal pseudomanifolds : 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.
26. On Walkup's class K(d) and a minimal triangulation of (S³ ×_-  S¹)^#3 : For d > 1, Walkup's class K(d) consists of the d-dimensional simplicial complexes all whose vertex-links are stacked (d-1)-spheres. Kalai showed that for d > 3, all connected members of K(d) are obtained from stacked d-spheres by finitely many elementary handle additions. According to a result of Walkup, the face vector of any triangulated 4-manifold X with Euler characteristic c satisfies f₁ > 5f₀ - 15c/2-1, with equality only for X in K(4). Kühnel observed that this implies f₀(f₀ - 11) > -15c-1, with equality only for 2-neighborly members of K(4). Kühnel also asked if there is a triangulated 4-manifold with f₀ = 15, c = -4 (attaining equality in his lower bound). In this paper, guided by Kalai's theorem, we show that indeed there is such a triangulation. It triangulates the connected sum of three copies of the twisted sphere product S³ ×_-  S¹. Because of Kühnel's inequality, the given triangulation of this manifold is a vertex-minimal triangulation. By a recent result of Effenberger, the triangulation constructed here is tight. Apart from the neighborly 2-manifolds and the infinite family of (2d+ 3)-vertex sphere products S ^d-1 ×S¹ (twisted for d odd), only fourteen tight triangulated manifolds were known so far. The present construction yields a new member of this sporadic family. We also present a self-complete proof of Kalai's result.
27. From the icosahedron to natural triangulations of CP² and S² ×S² : We present two constructions in this paper : (a) A 10-vertex triangulation CP ²₁₀ of the complex projective plane CP ² as a subcomplex of the join of the standard sphere (S ²₄) and the standard real projective plane (RP ²₆, the decahedron), its automorphism group is A₄; (b) a 12-vertex triangulation (S ² ×S ²)₁₂ of S ² ×S ² 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 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 CP ²₁₀ 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 CP ²₉ triangulates CP ². It is also shown that C P ²₁₀ and (S ² ×S ²)₁₂ induce the standard piecewise linear structure on CP ² and S ²×S ² respectively.
28. A triangulation of CP³ as symmetric cube of S² : The symmetric group Sym(d) acts on the Cartesian product (S²)^d by coordinate permutation, and the quotient space (S²)^d/Sym(d) is homeomorphic to the complex projective space C P^d. We used the case d=2 of this fact to construct a 10-vertex triangulation of CP² earlier. In this paper, we have constructed an 124-vertex simplicial subdivision (S²)³₁₂₄ of the 64-vertex standard cellulation (S²₄)³ of (S²)³, such that the Sym(3)-action on this cellulation naturally extends to an action on (S²)³₁₂₄. Further, the Sym(3)-action on (S²)³₁₂₄ is "good", so that the quotient simplicial complex (S²)³₁₂₄/Sym(3) is a 30-vertex triangulation CP³₃₀ of C P³. In other words, we have constructed a simplicial realization (S²)³₁₂₄ ® C P³₃₀ of the branched covering (S²)³ ® C P³.
29. On polytopal upper bound spheres : Generalizing a result (the case k = 1) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension 2k + 1 belongs to the generalized Walkup class K_k(2k + 1), i.e., all its vertex links are k-stacked spheres. This is surprising since it is far from obvious that the vertex links of polytopal upper bound spheres should have any special combinatorial structure.
    It has been conjectured that for d ¹ 2k + 1, all (k + 1)-neighborly members of the class K_k(d) are tight. The result of this paper shows that the hypothesis d ¹ 2k +1 is essential for every value of k > 0.
30. Higher dimensional analogues of the map colouring problem : After a brief discussion of the history of the problem, we propose a generalization of the map coloring problem to higher dimensions.
31. On k-stellated and k-stacked spheres : We introduce the class S_k(d) of k-stellated (combinatorial) spheres of dimension d (-1 < k < d+2) and compare and contrast it with the class S_k(d) (-1 < k < d+1) of k-stacked homology d-spheres. We have S₁(d) = S₁(d), and S_k(d) Í S_k(d) for d > 2k-2. However, for each k > 1 there are k-stacked spheres which are not k-stellated. For d < 2k -1, the existence of k-stellated spheres which are not k-stacked remains an open question.
    We also consider the class W_k(d) (and K_k(d)) of simplicial complexes all whose vertex-links belong to S_k(d-1) (respectively, S_k(d-1)). Thus, W_k(d) Í K_k(d) for d > 2k-1, while W₁(d) = K₁(d). Let [`(K)]<sub>k</sub>(d) denote the class of d-dimensional complexes all whose vertex-links are k-stacked balls. We show that for d > 2k+1, there is a natural bijection M®[`M] from K_k(d) onto [`(K)]<sub>k</sub>(d+1) which is the inverse to the boundary map ¶:[`(K)]_k(d+1) ® K_k(d).
    Finally, we complement the tightness results of our recent paper [5] by showing that, for any field F, an F-orientable (k+1)-neighborly member of W_k(2k+1) is F-tight if and only if it is k-stacked.
32. An infinite family of tight triangulations of manifolds : We give explicit construction of vertex-transitive tight triangulations of d-manifolds for d > 1. More explicitly, for each d > 1, we construct two (d²+5d+5)-vertex neighborly triangulated d-manifolds whose vertex-links are stacked spheres. The only other non-trivial series of such tight triangulated manifolds currently known is the series of non-simply connected triangulated d-manifolds with 2d+3 vertices constructed by Kühnel. The manifolds we construct are strongly minimal. For d > 2, they are also tight neighborly as defined by Lutz, Sulanke and Swartz. Like Kühnel's complexes, our manifolds are orientable in even dimensions and non-orientable in odd dimensions.
33. On stellated spheres and a tightness criterion for combinatorial manifolds : 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 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-1. As a 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+1. As another application, we prove that, when d ¹ 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 homology manifolds in which the members of W₁(d) provide the equality case. This generalizes a result (the d=4 case) due to Walkup and Kühnel. As a consequence, it is shown that every tight member of W₁(d) is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kühnel and Lutz asserting that tight homology manifolds should be strongly minimal.
34. Minimal crystallizations of 3-manifolds : We have introduced the weight of a group which has a presentation with number of relations is at most the number of generators. We have shown that the number of facets of any contracted pseudotriangulation of a connected closed 3-manifold M is at least the weight of the fundamental group of M. This lower bound is sharp for the 3-manifolds R P³, L(3,1), L(5,2), S¹×S¹ ×S¹, S² ×S¹, S¹ ×_-  S² and S³/Q₈, where Q₈ is the quaternion group. Moreover, there is a unique such facet minimal pseudotriangulation in each of these seven cases. We have also constructed contracted pseudotriangulations of L(kq-1,q) with 4(q+k-1) facets for q > 2, k > 1 and L(kq+1,q) with 4(q+k) facets for q > 3 , k > 0. By a recent result of Swartz, our pseudotriangulations of L(kq+1, q) are facet minimal when kq+1 are even. In 1979, Gagliardi found presentations of the fundamental group of a manifold M in terms of a contracted pseudotriangulation of M. Our construction is the converse of this, namely, given a presentation of the fundamental group of a 3-manifold M, we construct a contracted pseudotriangulation of M. So, our construction of a contracted pseudotriangulation of a 3-manifold M is based on a presentation of the fundamental group of M and it is computer-free.
35. Separation index of graphs and stacked 2-spheres : In 1987, Kalai proved that stacked spheres of dimension d > 2 are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result does not extend to dimension d=2. In this article, we give a characterisation of stacked 2-spheres using what we call the separation index. Namely, we show that the separation index of a triangulated 2-sphere is maximal if and only if it is stacked. In addition, we prove that, amongst all n-vertex triangulated 2-spheres, the separation index is minimised by some n-vertex flag sphere for n > 5.
    Furthermore, we apply this characterisation of stacked 2-spheres to settle the outstanding 3-dimensional case of the Lutz-Sulanke-Swartz conjecture that "tight-neighbourly triangulated manifolds are tight". For dimension d > 3, the conjecture has already been proved by Effenberger following a result of Novik and Swartz.
36. Tight triangulations of closed 3-manifolds : A triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known for higher dimensional manifolds. In this paper, we prove that a triangulation of a closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighbourly, orientable and stacked. In consequence, the Kühnel-Lutz conjecture is valid in dimension three for fields of odd characteristic.
    Next let F be a field of characteristic two. It is known that, in this case, any neighbourly and stacked triangulation of a closed 3-manifold is F-tight. For closed, triangulated 3-manifolds with at most 71 vertices or with first Betti number at most 188, we show that the converse is true. But the possibility of the existence of an F-tight, non-stacked triangulation on a larger number of vertices remains open. We prove the following upper bound theorem on such triangulations. If an F-tight triangulation of a closed 3-manifold has n vertices and first Betti number b₁, then (n-4)(617n- 3861) < 15444b₁ +1. Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.
37. A characterization of tightly triangulated 3-manifolds : For a field F, the notion of F-tightness of simplicial complexes was introduced by Kühnel. Kühnel and Lutz conjectured that F-tight triangulations of a closed manifold are the most economic of all possible triangulations of the manifold.
    The boundary of a triangle is the only F-tight triangulation of a closed 1-manifold. A triangulation of a closed 2-manifold is F-tight if and only if it is F-orientable and neighbourly. In this paper we prove that a triangulation of a closed 3-manifold is F-tight if and only if it is F-orientable, neighbourly and stacked. In consequence, the Kühnel-Lutz conjecture is valid in dimension < 4.
38. A construction principle for tight and minimal triangulations of manifolds : Tight triangulations are exotic objects in combinatorial topology. A triangulation is tight if all its piecewise linear embeddings into euclidean space are as convex as allowed by the topology of the underlying manifold. Tight triangulations are conjectured to be strongly minimal, and proven to be so for dimension two. However, in spite of substantial theoretical results about such triangulations, there are precious few examples. In fact, apart from dimension two, we do not know if there are infinitely many of them in any given dimension.
    In this paper, we present a computer friendly combinatorial scheme to obtain tight triangulations, and present new examples in dimensions three, four and five. Furthermore, we describe a family of tight triangulated d-manifolds, with 2^d-1 [d / 2] ! [(d-1) / 2]! homeomorphic but isomorphically distinct members for each dimension d > 1. While we still do not know if there are infinitely many tight triangulations in a fixed dimension, d > 2, it does look like there are abundantly many.
39. Minimal triangulations of manifolds : Finding vertex-minimal triangulations of closed manifolds is a very difficult problem. Except for spheres and two series of manifolds, vertex-minimal triangulations are known for only few manifolds of dimension more than 2 (see the table given at the end of Section 5). In this article, we present a brief survey on the works done in last 30 years on the following: (i) Finding the minimal number of vertices required to triangulate a given pl manifold. (ii) Given positive integers n and d, construction of n-vertex triangulations of different d-dimensional pl manifolds. (iii) Classifications of all the triangulations of a given pl manifold with same number of vertices.
    In Section 1, we have given all the definitions which are required for the remaining part of this article. A reader can start from Section 2 and come back to Section 1 as and when required. In Section 2, we have presented a very brief history of triangulations of manifolds. In Section 3, we have presented examples of several vertex-minimal triangulations. In Section 4, we have presented some interesting results on triangulations of manifolds. In particular, we have stated the Lower Bound Theorem and the Upper Bound Theorem. In Section 5, we have stated several results on minimal triangulations without proofs. Proofs are available in the references mentioned there. We have also presented some open problems/conjectures in Sections 3 and 5.
40. Efficient algorithms to decide tightness : Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is "as convex as possible", given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time algorithm, but more efficient methods to decide tightness are only known in the trivial setting of triangulated surfaces.
    In this article, we present a new polynomial time procedure to decide tightness for triangulations of 3-manifolds - a problem which previously was thought to be hard. In addition, for the more difficult problem of deciding tightness of 4-dimensional combinatorial manifolds, we describe an algorithm that is fixed parameter tractable in the treewidth of the 1-skeletons of the vertex links. Finally, we show that simpler treewidth parameters are not viable: for all non-trivial inputs, we show that the treewidths of both the 1-skeleton and the dual graph must grow too quickly for a standard treewidth-based algorithm to remain tractable.
41. Minimal triangulation, complementarity and projective planes : Brehm and Kühnel proved that if M^d is a combinatorial d-manifold with 3d/2 + 3 vertices and |M^d| is not homeomorphic to S ^d then the combinatorial Morse number of M^d is 3 and hence d is in {0, 2, 4, 8, 16} and |M^d| is a manifold like a projective plane in the sense of Eells and Kuiper.
    We discuss the existence and uniqueness of such combinatorial manifolds. We also present the following result: "Let M^d_n be a combinatorial d-manifold with n vertices. M^d_n satisfies complementarity if and only if d is in { 0, 2, 4, 8, 16 } with n = 3d/2 + 3 and |M^d_n| is a manifold like a projective plane".
42. Tight triangulations of some 4-manifolds : Walkup's class K(d) consists of the d-dimensional simplicial complexes all whose vertex links are stacked (d-1)-spheres. According to a result of Walkup, the face vector of any triangulated 4-manifold X with Euler characteristic c satisfies f₁ > 5f₀ - 15c/2-1, with equality only for X in K(4). Kühnel observed that this implies f₀(f₀ - 11) > -15c-1, with equality only for 2-neighborly members of K(4). For n = 6, 11 and 15, there are triangulated 4-manifolds with f₀=n and f₀(f₀ - 11) = -15c. In this article, we present triangulated 4-manifolds with f₀ = 21, 26 and 41 which satisfy f₀(f₀ - 11) = -15c. All these triangulated manifolds are tight and strongly minimal.
43. On stacked triangulated manifolds : We prove two results on stacked triangulated manifolds in this paper: (a) every stacked triangulation of a connected manifold with or without boundary is obtained from a simplex or the boundary of a simplex by certain combinatorial operations; (b) in dimension d > 3, if D is a tight connected closed homology d-manifold whose ith homology vanishes for 1 < i < d-1, then D is a stacked triangulation of a manifold. These results give affirmative answers to questions posed by Novik and Swartz and by Effenberger.
44. Tight and stacked triangulations of manifolds : Tight triangulated manifolds are generalisations of neighborly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. Except few, all the known tight triangulated manifolds are stacked. It is known that locally stacked tight triangulated manifolds are strongly minimal. Except for three infinite series and neighborly surfaces, very few tight triangulated manifolds are known. From some recent works, we know more on tight triangulation. In this article, we present a survey on the works done on tight triangulation. In Section 2, we state some known results on tight triangulations. In Section 3, we present all the known tight triangulated manifolds. Details are available in the references mentioned there. In Section 1, we present some essential definitions.
45. Equilibrium triangulations of some quasitoric 4-manifolds : Quasitoric manifolds, introduced by M. Davis and T. Januskiewicz in 1991, are topological generalizations of smooth complex projective spaces. In 1992, Banchoff and Kühnel constructed a 10-vertex equilibrium triangulations of CP². We generalize this construction for quasitoric manifolds and construct some equilibrium triangulations of 4-dimensional quasitoric manifolds. In some cases, our constructions give vertex minimal equilibrium triangulations.
46. Minimal contact triangulations of 3-manifolds : In this paper, we explore minimal contact triangulations on contact 3-manifolds. We give many explicit examples of contact triangulations that are close to minimal ones. The main results of this article say that on any closed oriented 3-manifold the number of vertices for minimal contact triangulations for overtwisted contact structures grows at most linearly with respect to the relative d³ invariant. We conjecture that this bound is optimal. We also discuss contact triangulations for a certain family of overtwisted contact structures on 3-torus.
47. Semi-equivelar and vertex-transitive maps on the torus : A vertex- transitive map X is a map on a closed surface on which the automorphism group Aut(X) acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. We show that there are eleven types of semi-equivelar maps on the torus. Three of these are equivelar maps. It is known that two of the three types of equivelar maps on the torus are always vertex-transitive. We show that this is true for the remaining one type of equivelar map and one other type of semi-equivelar maps, namely, if X is a semi-equivelar map of type [6³] or [3³, 4²] then X is vertex-transitive. We also show, by presenting examples, that this result is not true for the remaining seven types of semi-equivelar maps. There are ten types of semi-equivelar maps on the Klein bottle. We present examples in each of the ten types which are not vertex-transitive.
48. The Pachner graph of 2-spheres : It is well-known that the Pachner graph of triangulated n-vertex 2-spheres is connected, i.e., any two triangulated n-vertex 2-spheres are connected by a sequence of edge flips. In this article, we study various induced subgraphs of this graph. In particular, we prove that the subgraph induced by the set of n-vertex flag 2-spheres distinct from the double cone is still connected. In contrast, we show that the subgraph induced by the n-vertex stacked spheres has at least as many connected components as there are trees on [(n-5)/3] nodes of maximum degree < 5.