Separation-type combinatorial invariants for triangulations of manifolds
In this talk, I propose a combinatorial invariant defined for simplicial complexes as an alternative way to obtain lower bounds on the size of triangulations of manifolds. The advantage of this approach lies within its simplicity: the invariants under invesitgation are defined by counting connected components and/or homological features of subcomplexes.
As an application I obtain a linear identity for triangulated 4-manifolds with a fixed f-vector, unifying previously known lower bounds on the number of faces needed to triangulate a) simply connected 4-manifolds, and b) 4-manifolds of type "boundary of a handlebody". Moreover, based on the same construction, I present a conjectured upper bound for 4-manifolds with prescribed vector of Betti-numbers.
This talk is based on joint work with Francisco Santos and Jonathan Spreer.