The moduli space of thickenings

Fix $K$ a finite connected CW complex of dimension $\leq k$. An $n$-thickening of $K$ is a pair

$$(M,f)\, ,$$

in which $M$ is a compact $n$-dimensional manifold and $f\colon K \to M$ is a simple homotopy equivalence. This concept was first introduced by C.T.C. Wall approximately 40 years ago. Most of the known results about thickenings are in a range of dimensions depending on $k$, $n$ and the connectivity of $K$.

In this paper we remove the connectivity hypothesis on $K$. We define moduli space of $n$-thickenings $T_n(K)$. We also define a suspension map $E\colon T_n(K)$
$\to T_{n{+}1}(K)$ and compute its homotopy fibers in a range depending only on $n$ and $k$. We will show that these homotopy fibers can be approximated by certain section spaces whose definition depends only on the choice of a certain stable vector bundle over $K$.