Rational homology of the order complex of zero sets of homogeneous quadratic polynomial systems in $R^3$

TL;DR

This paper investigates the rational homology of the order complex formed by zero sets of quadratic polynomial systems in three-dimensional real space, revealing it has the same rational homology as a 13-sphere.

Contribution

It provides a detailed analysis of the rational homology of the order complex associated with quadratic form zero sets in $R^3$, a novel topological insight.

Findings

The order complex's rational homology is equivalent to that of a 13-sphere.

The study offers new topological understanding of algebraic subsets in $RP^2$.

It advances the knowledge of the topology of algebraic sets defined by quadratic forms.

Abstract

The naturally topologized order complex of proper algebraic subsets in $RP^2$, defined by systems of quadratic forms, has rational homology of $S^{13}$