Algebraic subdivision in simplicially controlled categories

TL;DR

This paper extends subdivision concepts to algebraic categories related to simplicial complexes, proving a squeezing theorem and applying it to Poincaré duality and homology manifold detection.

Contribution

It introduces algebraic subdivision in simplicially controlled categories and establishes a squeezing theorem linking algebraic and geometric chain equivalences.

Findings

Bounded chain equivalences can be squeezed to simplicially controlled chain equivalences.

A functor relating chain complexes over X and X×R is defined via algebraic subdivision.

Controlled Poincaré duality characterizes homology manifolds.

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 $\mathbb{A}^*(X)$, $\mathbb{A}_*(X)$ 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 $\mathrm{"}-\otimes\mathbb{Z}\mathrm{"}:\mathbb{B}(\mathbb{A}(X))\to \mathbb{B}(\mathbb{A}(X\times\mathbb{R}))$ 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 $\mathrm{"}C\otimes\mathbb{Z}\mathrm{"}$ is boundedly chain contractible over $O(X_+)$. These results have applications to Poincar\'e 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 Poincar\'e duality. We prove a Poincar\'e duality squeezing theorem that such a space $X$ with sufficiently controlled Poincar\'e duality must have $X$-controlled Poincar\'e duality and we prove a Poincar\'e duality splitting theorem with the consequence that $X$ is a homology manifold if and only if $X\times\mathbb{R}$ has bounded Poincar\'e duality over $O(X_+)$.