[Paper Review] Convex Equipartitions via Equivariant Obstruction Theory

Pavle V. M. Blagojević, Günter M. Ziegler|arXiv (Cornell University)|Feb 24, 2012

Advanced Combinatorial Mathematics32 references4 citations

TL;DR

This paper resolves the Nandakumar-Ramana Rao conjecture for prime power values of n by proving the non-existence of certain equivariant maps using equivariant obstruction theory. It constructs a regular cell complex model for the configuration space F(ℝᵈ,n), showing that an 𝔖ₙ-equivariant map to the sphere S(Wₙ⊕(d−1)) exists if and only if n is not a prime power, thereby establishing that every convex d-dimensional body can be partitioned into n convex pieces of equal volume and surface area when n is a prime power.

ABSTRACT

We describe a regular cell complex model for the configuration space F(\R^d,n). Based on this, we use Equivariant Obstruction Theory to prove the prime power case of the conjecture by Nandakumar and Ramana Rao that every polygon can be partitioned into n convex parts of equal area and perimeter.

Motivation & Objective

• To resolve the Nandakumar-Ramana Rao conjecture on convex equipartitions of polygons into n pieces of equal area and perimeter.
• To establish the non-existence of 𝔖ₙ-equivariant maps from F(ℝ²,n) to S(Wₙ) for prime power n.
• To develop a regular cell complex model for F(ℝᵈ,n) that is an 𝔖ₙ-equivariant strong deformation retract.
• To apply equivariant obstruction theory to prove that such maps exist if and only if n is not a prime power.
• To generalize Borsuk–Ulam-type theorems to cyclic groups of arbitrary order using equivariant cohomology.

Proposed method

• Constructs a regular 𝔖ₙ-equivariant cell complex model 𝔛(d,n) of dimension (d−1)(n−1) that strongly deformation retracts onto F(ℝᵈ,n).
• Uses the Fadell–Husseini index and equivariant cohomology to analyze the obstruction to the existence of 𝔖ₙ-equivariant maps to S(Wₙ⊕(d−1)).
• Applies Dold’s theorem on free group actions to show that a (d−1)(n−1)-connected free ℤ/nℤ-space cannot admit an equivariant map to a (d−1)(n−1)-dimensional free ℤ/nℤ-space.
• Employs transfer and Sylow subgroup arguments to strengthen the non-existence result to p-Sylow subgroups of 𝔖ₙ.
• Leverages the fact that the top Stiefel–Whitney class of the bundle Wₙ → F(ℝᵈ,n) ×_{𝔖ₙ} Wₙ → F(ℝᵈ,n)/𝔖ₙ is non-trivial when n is a prime power.
• Uses the Borel–Moore homology and closed support cohomology to compute the obstruction class in H^{M}_{𝔖ₙ}(𝔛(d,n); π_{M−1}(S(Wₙ⊕(d−1)))) with coefficients in ℤ/pℤ when n = pᵏ.

Experimental results

Research questions

• RQ1 Does there exist an 𝔖ₙ-equivariant map from F(ℝ²,n) to S(Wₙ) for all n ≥ 2?
• RQ2 For which values of n does the Nandakumar-Ramana Rao conjecture on equal area and perimeter partitions hold?
• RQ3 Can the non-existence of equivariant maps to S(Wₙ⊕(d−1)) be established via obstruction theory for prime power n?
• RQ4 Is the cell complex model 𝔛(d,n) a strong deformation retract of F(ℝᵈ,n) with the same equivariant homotopy type?
• RQ5 Does the Borsuk–Ulam-type theorem extend from cyclic groups of prime order to arbitrary cyclic groups of order n?

Key findings

• An 𝔖ₙ-equivariant map F(ℝᵈ,n) → S(Wₙ⊕(d−1)) exists if and only if n is not a prime power.
• The top equivariant cohomology group H^{M}_{𝔖ₙ}(𝔛(d,n); π_{M−1}(S(Wₙ⊕(d−1)))) is isomorphic to ℤ/pℤ when n = pᵏ is a prime power, and trivial otherwise.
• The cell complex 𝔛(d,n) is a regular 𝔖ₙ-equivariant (d−1)(n−1)-dimensional CW complex with n! vertices and n! facets.
• For prime powers n = pᵏ, there is no 𝔖ₙ-equivariant map to S(Wₙ⊕(d−1)), and this holds even for p-Sylow subgroups of 𝔖ₙ.
• The result implies that every convex d-dimensional body admits a partition into n convex pieces of equal volume and surface area when n is a prime power.
• A generalized Borsuk–Ulam theorem is proven: if X is a free (d−1)(n−1)-connected ℤ/nℤ-space and f: X → ℝᵈ is continuous, then f(x) = f(a·x) for some x and a ≠ 0 in ℤ/nℤ.