2019/07/31 by Fine, Jonathan
#Combinatorics (math.CO) #FOS: Mathematics #K-Theory and Homology (math.KT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1908.00039
In 1985 Bayer and Billera defined a flag vector f(X) for every convex polytope X, and proved some fundamental properties. The flag vectors f(X) span a graded ring R=\bigoplusd≥0Rd. Here Rd is the span of the f(X) with dim X=d. It has dimension the Fibonacci number Fd+1. This paper introduces and explores the conjecture, that R has a counting basis \ei\. If true then the equation f(X) = ∑ gi(X)ei conjecturally provides a formula for the Betti numbers gi(X) of a new homology theory. As the gi(X) are linear functions of f(X), we call the new theory linear homology. Further, assuming the conjecture each gi will have a rank r≥0. The rank zero part of linear homology will be (middle perversity) intersection homology. The higher rank gi measure successively more complicated singularities. In dimension d we will have \dimRd linearly independent Betti numbers. This paper produces a basis \ei\ for R, that is conjecturally a counting basis. Warning: Conjecture withdrawn in version 2.