Geometry · combinatorics · topology

A hands-on guide to Goodwillie polytopes.

Explore the geometry behind a family of self-dual polytopes arising in the study of free loop spaces and Koszul duality.M. Rivera and D. Tolosa, Cyclic homology of categorical coalgebras and the free loop space, arXiv:2403.08116

Start with the definition

The precise object

Definition of the Goodwillie polytope.

Fix an integer n ≥ 1 and put J = {2, …, n}. When n ≥ 3, choose

0 < ε_n < min{½, 1/(n − 2)}.

For n = 1 or 2, no choice of ε_n is needed: there are respectively no added points or one added point whose coordinates do not involve ε_n. We write these parameter-independent polytopes as G₁ and G₂.

Cube points

For t ∈ {0,1} and A ⊆ J, let C_t,A ∈ ℝⁿ whose coordinates are given by

(C_t,A)₁ = t,(C_t,A)_i =1, if i ∈ A,0, if i ∉ A,for i ∈ J.

Added points

For r ∈ J and B ⊆ J ∖ {r}, let u^ε_r,B ∈ ℝⁿ whose coordinates are given by

(u^ε_r,B)₁= (r − 1)/n,(u^ε_r,B)_r = −1,(u^ε_r,B)_i =1 − ε, if i ∈ B,ε, if i ∉ B,for i ∈ J ∖ {r}.

Convex hull

G^ε_n:= conv({C_t,A : t ∈ {0,1}, A ⊆ J}∪{u^ε_r,B : r ∈ J, B ⊆ J ∖ {r}}).

In plain language: start with every vertex of the unit n-cube, add one labeled family of points beyond selected cube facets, and take the smallest convex polytope containing them all.

Interactive chapters

Begin with the shape.

01Inflating a cubeFollow the geometric idea that opens a cube into G₃.02Low dimensionsCompare G₁, G₂, G₃, and exact face counts through n = 30.03Prismatic decompositionPull apart the four maximal cells of G₃.04Self-dualityPair every vertex with its corresponding facet.05Goodwillie to freehedronCollapse G₃ onto F₃ through the deformation retraction—and reverse it.06Incidence matrixRead self-duality as a symmetric 12 × 12 matrix.07Epsilon laboratoryMove ε from 0 to 5 and watch the hull cross exact events.08The facets of G₄Explore G₄ one genuine three-dimensional facet at a time.