Online Talk

The toric wedge induction, developed by Choi and Park (2017), is a powerful method in toric topology, particularly useful for proving results about toric manifolds (smooth compact toric varieties) with a small Picard number. According to the fundamental theorem of toric geometry, toric manifolds correspond to complete non-singular fans. The combinatorial structure of such a fan is represented by a simplicial complex that must be a PL (piecewise-linear) sphere. The wedge operation on PL spheres  increases by one both the dimension and the number of vertices. Consequently, this operation  stabilizes the collection of PL spheres with a fixed Picard number. PL spheres that cannot be constructed as wedges of lower-dimensional PL spheres are referred to as seeds. Choi and Park proved that, for a given Picard number, there is a finite set of seeds that can encode a complete non-singular fan. This finite set provides a perfect collection of base cases for an inductive argument, where the inductive step would rely on the stability of certain properties under the wedge operation. A recent result by Choi, Jang, and V. has established this finite set of base cases for Picard number 4, which enables toric wedge induction to be applied to various problems in toric topology, such as the toric lifting problem or the classification of toric manifolds. In my presentation, I'll aim to enrich the discussion with ideas from operad and matroid theory to give additional context and connections. This is a joint work with Suyoung Choi and Hyeontae Jang.