Paper detail

Homeomorphisms, homotopy equivalences and chain complexes

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 $f:X\to Y$ 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 $f\times\mathrm{id}:X\times\mathbb{R} \to Y\times\mathbb{R}$ 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 $\textit{"}C\otimes\mathbb{Z}\textit{"}\in\mathbb{A}(X\times\mathbb{R})$ is boundedly chain contractible when measured in $O(X^+)$ for a functor $\textit{"}-\otimes\mathbb{Z}\textit{"}$ 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 $X\times\mathbb{R}$ has bounded Poincaré duality when measured in the open cone $O(X^+)$.