[Paper Review] The homotopy type of the complement of the codimension-two coordinate subspace arrangement

Jelena Grbić, Stephen Theriault|ArXiv.org|Dec 13, 2005

Advanced Combinatorial Mathematics5 references3 citations

TL;DR

This paper determines the homotopy type of the complement of a codimension-two coordinate subspace arrangement in ℂⁿ, proving it is homotopy equivalent to a wedge of spheres of dimension k+1 for k from 2 to n, with (k−1) binomial(n,k) copies per dimension. The result is established via a homotopy decomposition of the homotopy fibre of the inclusion of a wedge of ℂP∞ into a product of ℂP∞, using the Cube Lemma and iterated homotopy pushouts to refine earlier suspension-dependent decompositions to the non-suspended case.

ABSTRACT

The complement of the codimension 2 complex coordinate subspace arrangement is shown to be homotopy equivalent to a wedge of spheres.

Motivation & Objective

• To determine the precise homotopy type of the complement of a codimension-two coordinate subspace arrangement in ℂⁿ.
• To refine prior homotopy decompositions that required suspension by establishing a non-suspended decomposition.
• To prove that the homotopy fibre of the inclusion ∨ₙ ℂP∞ → ∏ₙ ℂP∞ decomposes as a wedge of spheres.
• To provide a new, accelerated proof of a key homotopy decomposition using the Cube Lemma and iterated homotopy pushouts.

Proposed method

• The authors use the Davis-Januszkiewicz space DJ(K) and its associated homotopy fibre 𝒵_K to relate the complement U(K) to a homotopy fibre sequence.
• They apply a homotopy fibration decomposition via the inclusion of the wedge into the product of n copies of ℂP∞.
• The key technique is the Cube Lemma to establish homotopy pushout properties in a diagram of spaces.
• They use iterated homotopy pushouts and the decomposition Σ(X×Y) ≃ ΣX ∨ ΣY ∨ (ΣX ∧ ΣY) to analyze the suspension of products.
• The proof relies on identifying connecting maps in fibrations as null-homotopic, enabling simplification of the fibre decomposition.
• An inductive argument on n, combined with the homotopy pushout structure, yields the final wedge decomposition of the fibre.

Experimental results

Research questions

• RQ1 What is the homotopy type of the complement of the codimension-two coordinate subspace arrangement in ℂⁿ?
• RQ2 Can the homotopy fibre of the inclusion ∨ₙ ℂP∞ → ∏ₙ ℂP∞ be decomposed without suspension?
• RQ3 How does the homotopy type of the complement relate to the Davis-Januszkiewicz space and its associated fibre?
• RQ4 What role does the Cube Lemma play in simplifying the homotopy pushout structure of the fibre?
• RQ5 Is there a non-suspended decomposition of the homotopy fibre that matches the known suspension-level results?

Key findings

• The complement of the codimension-two coordinate subspace arrangement in ℂⁿ has the homotopy type of ⋁_{k=2}^n (k−1) binom(n,k) S^{k+1}.
• The homotopy fibre of the inclusion ∨ₙ ℂP∞ → ∏ₙ ℂP∞ decomposes as a wedge of spheres, specifically Fₙ ≃ ⋁_{k=2}^n ⋁_{1≤i₁<⋯<iₖ≤n} (k−1)(ΣΩX_{i₁} ∧ ⋯ ∧ ΩX_{iₖ}).
• The decomposition holds without suspension, improving upon previous results that required at least one suspension.
• The connecting map in the fibration sequence is null-homotopic, enabling the simplification of the homotopy pushout structure.
• The proof uses the Cube Lemma to establish that the top face of a diagram is a homotopy pushout when the bottom face and sides are.
• The result provides a non-suspended, explicit homotopy decomposition of the fibre, resolving a long-standing problem in toric topology.