Algebraic subdivision in simplicially controlled categories

Abstract

We generalise the notion of subdivision of a finite-dimensional locally finite simplicial complex $X$ to geometric algebra, namely to the simplicially controlled categories ${\mathbb{A}}^{\ast }(X)$, ${\mathbb{A}}_{\ast }(X)$ of Ranicki and Weiss. We prove a squeezing result: a bounded chain equivalence of sufficiently algebraically subdivided chain complexes can be squeezed to a simplicially controlled chain equivalence of the unsubdivided chain complexes. Giving $X\times \mathbb{R}$ a bounded triangulation measured in the open cone $O(X_{+})$ we use algebraic subdivision to define a functor $"-\otimes \mathbb{Z}":\mathbb{B}(\mathbb{A}(X))\to \mathbb{B}(\mathbb{A}(X\times \mathbb{R}))$ that corresponds to tensoring with the simplicial chain complex of $\mathbb{Z}$ and algebraically subdividing to be bounded over $O(X_{+})$. We show that $C\simeq 0\in \mathbb{B}(\mathbb{A}(X))$ if and only if $"C\otimes \mathbb{Z}"$ is boundedly chain contractible over $O(X_{+})$. These results have applications to Poincaré duality and homology manifold detection as a finite-dimensional locally finite simplicial complex $X$ is a homology manifold if and only if it has $X$-controlled Poincaré duality. We prove a Poincaré duality squeezing theorem that such a space $X$ with sufficiently controlled Poincaré duality must have $X$-controlled Poincaré duality and we prove a Poincaré duality splitting theorem with the consequence that $X$ is a homology manifold if and only if $X\times \mathbb{R}$ has bounded Poincaré duality over $O(X_{+})$.

35 pages, 5 figures

References in corpus (1)

• A controlled local-global theorem for simplicial complexes