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 $\epsilon$-controlled homotopy equivalences for all $\epsilon>0$, 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 Poincar\'e Duality spaces. Squeezing tells us that a $PL$ Poincar\'e duality space with small enough Poincar\'e duality is a homology manifold and the main theorem tells us that a $PL$ Poincar\'e duality space $X$ is a homology manifold if and only if $X\times\mathbb{ R}$ has bounded Poincar\'e duality when measured in the open cone $O(X^+)$.
Categories: math.AT