[Paper Review] Simplicial inverse sequences in extension theory

Leonard R. Rubin, Vera Tonić|arXiv (Cornell University)|Mar 13, 2017

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This paper introduces a simplicial inverse sequence of triangulated polyhedra with simplicial bonding maps that factor through subdivisions, enabling a cell-like map π: Z → X from a metrizable compactum Z to any given compact metrizable space X. The key result is that if X is an absolute extensor for a CW-complex K₀, then so is Z, preserving extension-theoretic properties while ensuring Z arises from a well-structured simplicial system.

ABSTRACT

In extension theory, in particular in dimension theory, it is frequently useful to represent a given compact metrizable space X as the limit of an inverse sequence of compact polyhedra. We are going to show that, for the purposes of extension theory, it is possible to replace such an X by a better metrizable compactum Z. This Z will come as the limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps that factor in a certain way. There will be a cell-like map from Z to X, and we shall show that if K is a CW-complex which is an absolute extensor for X, then K is also an absolute extensor for Z.

Motivation & Objective

• To replace arbitrary compact metrizable spaces X with a simplicial inverse limit Z that preserves extension-theoretic properties.
• To construct a cell-like map π: Z → X where Z is the limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps.
• To ensure that if X τ K₀ for a CW-complex K₀, then Z τ K₀, thus preserving extension behavior.
• To develop a theory of 'adjustments' to the inverse sequence that allow control over fibers π⁻¹(x) and facilitate homotopy arguments.

Proposed method

• Construct an inverse sequence (|Tᵢ|, gᵢⁱ⁺¹) of finite polyhedra with simplicial bonding maps that factor as gᵢⁱ⁺¹ = fᵢⁱ⁺¹ ∘ φᵢ⁺¹, where φᵢ⁺¹ is a simplicial approximation to the identity on a subdivision.
• Use skeletal adjustments to modify bonding maps at selected levels, creating new inverse sequences M = (|Tⱼᵢ⁽ⁿ⁾|, hᵢⁱ⁺¹) that preserve fiber structure.
• Define fibers π⁻¹(x) as limits of sub-inverse sequences Mₓ⁺, enabling homotopy control via contractibility of key subcomplexes.
• Apply Lemma 8.2 to show that each fiber π⁻¹(x) is cell-like by verifying that bonding maps in Mₓ⁺ are null-homotopic.
• Use Theorem 6.1 to deduce that extension properties of X lift to Z, ensuring Z τ K₀ whenever X τ K₀.
• Leverage simplicial star neighborhoods and triangulation refinements to ensure compatibility and continuity across the inverse system.

Experimental results

Research questions

• RQ1 Can every compact metrizable space X be replaced by a simplicial inverse limit Z such that extension-theoretic properties are preserved under a cell-like map π: Z → X?
• RQ2 Is it possible to construct such a Z using only simplicial maps between fixed triangulations of polyhedra?
• RQ3 How can the fibers of π: Z → X be described topologically to ensure they are cell-like?
• RQ4 What role do 'adjustments' of the inverse sequence play in controlling homotopy types of fibers?
• RQ5 Under what conditions does X τ K₀ imply Z τ K₀ for a CW-complex K₀?

Key findings

• The inverse limit Z of the simplicial inverse sequence (|Tᵢ|, gᵢⁱ⁺¹) admits a cell-like map π: Z → X, with each fiber π⁻¹(x) being a compact, cell-like continuum.
• For each x ∈ X, the fiber π⁻¹(x) is homeomorphic to the inverse limit of the sub-inverse sequence Mₓ⁺ = (Bₓᵢ⁺, gᵢⁱ⁺¹|Bₓᵢ₊₁⁺), which consists of contractible subcomplexes.
• Each bonding map gᵢⁱ⁺¹|Bₓᵢ₊₁⁺: Bₓᵢ₊₁⁺ → Bₓᵢ⁺ is homotopic to a constant map, ensuring that π⁻¹(x) has trivial shape.
• The map π is surjective, as shown by the existence of points in the inverse limit over the 0-skeletons of the adjusted complexes.
• If X τ K₀ for a CW-complex K₀, then Z τ K₀, meaning that extension properties of X are preserved in Z.
• The construction ensures that all bonding maps in the adjusted inverse sequence are simplicial and factor through subdivisions, providing a canonical, structured resolution of X.