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- W3186561483 abstract "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 $mathcal{R}=bigoplus_{dgeq0}mathcal{R}_d$. Here $mathcal{R}_d$ is the span of the $f(X)$ with $dim X=d$. It has dimension the Fibonacci number $F_{d+1}$. This paper introduces and explores the conjecture, that $mathcal{R}$ has a counting basis ${e_i}$. If true then the equation $f(X) = sum g_i(X)e_i$ conjecturally provides a formula for the Betti numbers $g_i(X)$ of a new homology theory. As the $g_i(X)$ are linear functions of $f(X)$, we call the new theory linear homology. Further, assuming the conjecture each $g_i$ will have a rank $rgeq0$. The rank zero part of linear homology will be (middle perversity) intersection homology. The higher rank $g_i$ measure successively more complicated singularities. In dimension $d$ we will have $dimmathcal{R}_d$ linearly independent Betti numbers. This paper produces a basis ${e_i}$ for $mathcal{R}$, that is conjecturally a counting basis. Warning: Conjecture withdrawn in version 2." @default.
- W3186561483 created "2021-08-02" @default.
- W3186561483 creator A5063096109 @default.
- W3186561483 date "2019-07-31" @default.
- W3186561483 modified "2023-09-27" @default.
- W3186561483 title "Some stumbling first steps towards linear homology in a nutshell" @default.
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