Toric wedge induction and application to the toric lifting problem (Mathieu Vallée)

Developed by Choi and Park (2017), the toric wedge induction is a powerful tool for proving results in toric topology for toric manifolds (smooth compact toric varieties) of small Picard number. In fact, the fundamental theorem of toric geometry states that toric manifolds are characterized by complete non-singular fans. The combinatorial structure of these fans is encoded by a simplicial complex which must be a PL sphere. The wedge operation on simplicial complexes increments both the dimension and the number of vertices and stabilizes the set of PL spheres of fixed Picard number. The PL spheres which are not obtained as the wedge of lower dimensional ones are called the seeds. The result of Choi and Park is that the number of seeds of fixed Picard number which can encode a complete non-singular fan is finite. Therefore this finite set is a good candidate for being the set of base cases of an induction. The inductive step is represented by the stability of a property after a wedge operation.

In particular, a recent result of Choi, Jang and V. provides this finite set of base cases for Picard number 4. I will explain how to apply the wedge induction for the lifting problem in toric topology.