On Shellability, Cohen-Macauleyness and the Homotopy Type of Boolean Representable Simplicial Complexes

יום א', 15/11/2015 - 14:00
Boolean representable simplicial complexes are a generalization of matroids based on 
independence over the Boolean semiring. The work was pioneered by work of Izhakian and 
Rowen at Bar-Ilan on notions of rank and independence over semirings. It was later defined 
and initially developed by Izhakian and Rhodes and later Silva.
In this talk it is proved that fundamental groups of boolean representable simplicial complexes
are free and the rank is determined by the number and nature of the connected
components of their graph of flats for dimension ≥ 2. In the case of dimension 2, it
is shown that boolean representable simplicial complexes have the homotopy type of
a wedge of spheres of dimensions 1 and 2. Also in the case of dimension 2, necessary
and sufficient conditions for shellability and being sequentially Cohen-Macaulay are
determined. These notions are equivalent in dimension 2, but despite having the appropriate homotopy type, 
not all 2 dimensional boolean representable simplicial complexes are shellable. 
Complexity bounds are provided for all the algorithms involved.
All terms will be defined. This is joint work of the speaker, John Rhodes and Pedro Silva.