Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Algebraic subdivision in simplicially controlled categories   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 $O(X_+)$.

18
The categories of this paper are: econ
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat