Introduction

Today I submitted my preprint Linear homology in a nutshell to the Arxiv. I've spent about half my life (i.e. since 1985) working on it, so I call it my half-life problem. I hope it's now more than half-done. The abstract is below. If some wise person had suggested years ago that I investigate the ideas expressed in the abstract, I'd have made much quicker progress. (Links at foot of page.)

Note: Written after the event, and published 18 November 2019.

The best idea

The preprint's best idea is be summarized in a single sentence: Find a counting basis for the convex polytope flag vector ring. That's it. Linear homology in a nutshell. I think this statement by itself, once understood and trusted, would have saved me at least 10 years. But perhaps this opinion overestimates my wisdom, or my stupidity. Or both.

My preprint

My preprint is long (21 pages), and in places wrong, and certainly incomplete. Despite that, I think it's got a good abstract, and at least three good ideas. I'll start work soon on a shorter (and better) version.

The second best idea

That is, I think, the cone product formula $$ C(U) C(V) = C(J(U, V)) + DUV $$ where $$ J(U, V) = U C(V) + C(U) V - e_1 UV $$ is the join formula. Here $e_0 = 1$, $e_{n+1} = C(e_n)$. And $D = e_1e_1 - e_2$, the difference between a square and a triangle.

The (universal) cone product ring is the largest commutative ring, generated by cone and product, that satisfies the cone product formula. It is isomorphic to the convex polytope flag vector ring, via the cone (or pyramid) operator $C$ that both rings have.

The cone product formula arose from studying the convex polytope flag vector ring.

The third best idea

So now the problem is to find a counting basis of the cone product ring. Which is an algebraic object, hopefully the representation ring of something.

And if the cone product ring were a representation ring, satisfying Schur's lemma, then the irreducible representations would provide the counting basis. Finally, the linear homology Betti numbers of $X$ are the coefficients $\lambda_i$ in $$ f(X) = \sum \lambda_i e_i $$ where $f(X)$ is the flag vector of $X$, and the $e_i$ are the irreducible representations,

Abstract

This is the abstract, that would have saved me so many years. Please read it carefully. It's short and, I hope, weighty.

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 $\mathcal{R}=\bigoplus_{d\geq0}\mathcal{R}_d$. 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 $r\geq0$. 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 $\dim\mathcal{R}_d$ linearly independent Betti numbers.

This paper produces a basis ${e_i}$ for $\mathcal{R}$, that is conjecturally a counting basis.

Links

• J.Fine, Linear Homology in a nutshell (on the arxiv)
• J.Fine, Anything similar to cone product formula? (Math Overflow)