  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 $O(X_+)$.
