[Paper Review] Homeomorphisms, homotopy equivalences and chain complexes

Spiros Adams-Florou|arXiv (Cornell University)|May 14, 2012

Homotopy and Cohomology in Algebraic Topology29 references3 citations

TL;DR

This paper establishes a precise algebraic and topological criterion for when a homotopy equivalence between finite-dimensional, locally finite simplicial complexes is homotopic to a homeomorphism, using controlled and bounded topology. The key result shows that a PL Poincaré duality space is a homology manifold if and only if its product with $\mathbb{R}$ admits bounded Poincaré duality over the open cone of the space, linking homotopy theory with geometric topology via algebraic subdivision and controlled chain complexes.

ABSTRACT

This thesis concerns the relationship between bounded and controlled topology and how these can be used to recognise which homotopy equivalences of reasonable topological spaces are homotopic to homeomorphisms. Let $f:X o Y$ be a simplicial map of finite-dimensional locally finite simplicial complexes. We prove that $f$ has contractible point inverses if and only if it is an $ε$-controlled homotopy equivalences for all $ε>0$, if and only if $f imes\mathrm{id}:X imes\mathbb{R} o Y imes\mathbb{R}$ is a homotopy equivalence bounded over the open cone $O(Y^+)$ of Pedersen and Weibel. This approach can be generalised to algebra using geometric categories: the $X$-controlled categories $\mathbb{A}^*(X)$, $\mathbb{A}_*(X)$ of Ranicki and Weiss together with the bounded categories $\mathcal{C}_M(\mathbb{A})$ of Pedersen and Weibel. Analogous to the barycentric subdivision of a simplicial complex, we define the algebraic subdivision of a chain complex over that simplicial complex. The main theorem of the thesis is that a chain complex $C$ is chain contractible in $\mathbb{A}(X)$ if and only if $ extit{"}C\otimes\mathbb{Z} extit{"}\in\mathbb{A}(X imes\mathbb{R})$ is boundedly chain contractible when measured in $O(X^+)$ for a functor $ extit{"}-\otimes\mathbb{Z} extit{"}$ defined appropriately using algebraic subdivision and for $\mathbb{A}=\mathbb{A}^*$ or $\mathbb{A}_*$. We prove a squeezing result: a chain complex with a small enough chain contraction has arbitrarily small chain contractions. We conclude with consequences for Poincaré Duality spaces. Squeezing tells us that a $PL$ Poincaré duality space with small enough Poincaré duality is a homology manifold and the main theorem tells us that a $PL$ Poincaré duality space $X$ is a homology manifold if and only if $X imes\mathbb{R}$ has bounded Poincaré duality when measured in the open cone $O(X^+)$.

Motivation & Objective

• To determine when a homotopy equivalence between finite-dimensional, locally finite simplicial complexes is homotopic to a homeomorphism.
• To establish a correspondence between contractible point inverses of simplicial maps and bounded homotopy equivalences over the open cone.
• To generalize controlled and bounded topology to algebraic settings using $X$-controlled and bounded categories.
• To apply the results to recognize homology manifolds within the homotopy types of Poincaré duality spaces.
• To develop an algebraic barycentric subdivision functor that preserves boundedness and enables squeezing of chain contractions.

Proposed method

• Introduces the fundamental $\epsilon$-subdivision cellulation $X'_{\epsilon}$ to relate contractible point inverses to bounded homotopy equivalences over the open cone $O(Y^+)$.
• Uses the $X$-controlled categories $\mathbb{A}^*(X)$ and $\mathbb{A}_*(X)$ of Ranicki and Weiss, and the bounded categories $\mathcal{C}_M(\mathbb{A})$ of Pedersen and Weibel.
• Defines an algebraic barycentric subdivision functor on chain complexes over simplicial complexes, generalizing topological barycentric subdivision.
• Applies a squeezing result: a chain complex with a sufficiently small chain contraction admits arbitrarily small chain contractions.
• Constructs a functor $"-\otimes\mathbb{Z}"$ using algebraic subdivision to relate chain complexes over $X$ to those over $X \times \mathbb{R}$.
• Applies the Splitting Theorem to decompose bounded chain equivalences over $X \times \mathbb{R}$ into $\epsilon$-controlled equivalences over $X$.

Experimental results

Research questions

• RQ1 When is a simplicial map with contractible point inverses homotopic to a homeomorphism?
• RQ2 How can controlled and bounded topology be used to detect when a homotopy equivalence is a homeomorphism?
• RQ3 What is the algebraic analog of barycentric subdivision for chain complexes over simplicial complexes?
• RQ4 Can bounded Poincaré duality over $X \times \mathbb{R}$ in the open cone detect whether $X$ is a homology manifold?
• RQ5 How does algebraic subdivision enable the squeezing of chain contractions to arbitrary small bounds?

Key findings

• A simplicial map $f: X \to Y$ has contractible point inverses if and only if $f \times \mathrm{id}: X \times \mathbb{R} \to Y \times \mathbb{R}$ is a homotopy equivalence bounded over the open cone $O(Y^+)$.
• A chain complex $C$ is chain contractible in $\mathbb{A}^*(X)$ or $\mathbb{A}_*(X)$ if and only if $C \otimes \mathbb{Z}$ is boundedly chain contractible in $O(X^+)$ via the algebraic subdivision functor.
• The Squeezing Theorem ensures that any chain complex with a sufficiently small chain contraction admits arbitrarily small chain contractions, enabling control over boundedness.
• A PL Poincaré duality space $X$ is a homology manifold if and only if $X \times \mathbb{R}$ has bounded Poincaré duality measured in $O(X^+)$.
• The main theorem implies that $X$ has $\epsilon$-controlled Poincaré duality for all $\epsilon > 0$ if $X \times \mathbb{R}$ has bounded Poincaré duality in $O(X^+)$.
• The construction of the algebraic barycentric subdivision functor allows for a systematic transfer of boundedness properties from $X \times \mathbb{R}$ to $X$.