## Homeomorphisms, homotopy equivalences and chain complexes

##### Abstract

This thesis concerns the relationship between bounded and controlled topology and in
particular how these can be used to recognise which homotopy equivalences of reasonable
topological spaces are homotopic to homeomorphisms.
Let f : X → Y be a simplicial map of finite-dimensional locally finite simplicial
complexes. Our first result is that f has contractible point inverses if and only if it is an ε-
controlled homotopy equivalences for all ε > 0, if and only if f × id : X × R → Y × R is a
homotopy equivalence bounded over the open cone O(Y +) of Pedersen and Weibel. The most
difficult part, the passage from contractible point inverses to bounded over O(Y +) is proven
using a new construction for a finite dimensional locally finite simplicial complex X, which
we call the fundamental ε-subdivision cellulation X'ε.
This whole approach can be generalised to algebra using geometric categories. In the
second part of the thesis we again work over a finite-dimensional locally finite simplicial
complex X, and use the X-controlled categories A*(X), A*(X) of Ranicki and Weiss (1990)
together with the bounded categories CM(A) of Pedersen and Weibel (1989). Analogous
to the barycentric subdivision of a simplicial complex, we define the algebraic barycentric
subdivision of a chain complex over that simplicial complex. The main theorem of the
thesis is then that a chain complex C is chain contractible in
(
A*(X)
A*(X)
if and only if
“C ¤ Z” 2
(A*(X × R)
A*(X × R)
is boundedly chain contractible when measured in O(X+) for a
functor “ −
Z” defined appropriately using algebraic subdivision. In the process we prove
a squeezing result: a chain complex with a sufficiently small chain contraction has arbitrarily
small chain contractions.
The last part of the thesis draws some consequences for recognising homology manifolds
in the homotopy types of Poincare Duality spaces. Squeezing tells us that a PL Poincare
duality space with sufficiently controlled Poincare duality is necessarily a homology manifold
and the main theorem tells us that a PL Poincare duality space X is a homology manifold if
and only if X × R has bounded Poincare duality when measured in the open cone O(X+).