Geometric Combinatorics of Polynomials II: Polynomials and Cell Structures

TL;DR

This paper constructs and analyzes geometric cell complexes representing spaces of monic centered complex polynomials with specified critical value regions, revealing their topological and metric structures and their relation to braid groups.

Contribution

It introduces new cell complexes for polynomial spaces with fixed critical value regions and establishes their topological and metric properties, linking them to braid group classifying spaces.

Findings

The branched rectangle complex is homeomorphic to a 2n-dimensional ball.

The branched annulus complex is a quotient of the rectangle complex and compactifies polynomial spaces.

The dual braid complex embeds as a spine, proving homotopy equivalence with polynomial space classifying spaces.

Abstract

This article introduces a finite piecewise Euclidean cell complex homeomorphic to the space of monic centered complex polynomials of degree $d$ whose critical values lie in a fixed closed rectangular region. We call this the branched rectangle complex since its points are indexed by marked $d$-sheeted planar branched covers of the fixed rectangle. The vertices of the cell structure are indexed by the combinatorial "basketballs" studied by Martin, Savitt and Singer. Structurally, the branched rectangle complex is a full subcomplex of a direct product of two copies of the order complex of the noncrossing partition lattice. Topologically, it is homeomorphic to the closed $2n$-dimensional ball where $n=d-1$. Metrically, the simplices in each factor are orthoschemes. It can also be viewed as a compactification of the space of all monic centered complex polynomials of degree $d$. We also introduce a finite piecewise Euclidean cell complex homeomorphic to the space of monic centered complex polynomials of degree $d$ whose critical values lie in a fixed closed annular region. We call this the branched annulus complex since its points are indexed by marked $d$-sheeted planar branched covers of the fixed annulus.It can be constructed from the branched rectangle complex as a cellular quotient by isometric face identifications. And it can be viewed as a compactification of the space of all monic centered complex polynomials of degree $d$ with distinct roots. Finally, the branched annulus complex deformation retracts to the branched circle complex, which we identify with the dual braid complex. Our explicit embedding of the dual braid complex as a spine for the space of polynomials with distinct roots provides a direct proof that these two classifying spaces for the braid group are homotopy equivalent.