Boundary measurement and sign variation in real projective space

TL;DR

This paper introduces two new generalizations of the totally nonnegative Grassmannian in real projective space, analyzing their topology and triangulations, and revealing their homotopy equivalence to smaller-dimensional real projective spaces.

Contribution

It defines two novel generalizations of the totally nonnegative Grassmannian in real projective space and characterizes their topological and combinatorial properties.

Findings

Spaces are PL manifolds with boundary homotopy equivalent to smaller real projective spaces.

Boundary measurement spaces admit Cohen-Macaulay triangulations.

The two generalizations are based on sign variation and boundary measurement methods.

Abstract

We define two generalizations of the totally nonnegative Grassmannian and determine their topology in the case of real projective space. We find the spaces to be PL manifolds with boundary which are homotopy equivalent to another real projective space of smaller dimension. One generalization makes use of sign variation while the other uses boundary measurement. Spaces arising from boundary measurement are shown to admit Cohen-Macaulay triangulations.