Basudeb Datta (Indian Institute of Science)

Title: Stacked and tight triangulations of manifolds

Abstract: Walkup constructed a class of triangulated d-manifolds from a stacked d-sphere by adding successive handles. Now, we know that Walkup's class of manifolds are same as the class of stacked manifolds. (A stacked triangulated manifold is the boundary of triangulated manifold whose co-dimension 2 spaces are on the boundary.) Tight triangulated manifolds are generalisations of neighbourly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. A triangulation of a closed 2-manifold is tight (with respect to a field F) if and only if it is F-orientable and neighbourly. Recently, it is shown that a triangulation of a closed 3-manifold is tight (with respect to a field F) if and only if it is F-orientable, neighbourly and stacked. Thus every tight triangulation of closed 3-manifold is strongly minimal. Except few, all the known tight triangulated manifolds are stacked. From some recent works, we know more on tight triangulations. In this talk, we present a survey on the works done on stacked and tight triangulation.