# Topological structure group and cell-like maps

The structure set $\mathcal S^{CAT}(X)$ of the Poincare complex $X$ measures the number of distinct $CAT$-manifolds in the simple homotopy class of $X$ where $CAT$ is the category.

Contrary to the DIFF, in the case of topological manifolds $\mathcal S^{TOP}(M)$ is a group.

We define a subset $\mathcal S^{CE}(M)\subset \mathcal S^{TOP}(M)$ generated by homotopy equivalences $h:N\to M$ that come as homotopy lifts of $g$ in the diagram $N\stackrel{g}\rightarrow X \stackrel{f}\leftarrow M$

where $f$ and $g$ are cell-like maps.

We give a complete description of $\mathcal S^{CE}(M)$. In particular we prove the following

**THEOREM.** *For any manifold $M$ the set $\mathcal S^{CE}(M)$ is a group.*

For a simply connected manifold $M$ with finite $\pi_2(M)$ the group $\mathcal S^{CE}(M)$ equals the odd torsion subgroup of $\mathcal S^{TOP}(M)$.

As a corollary we construct two smooth nonhomeomorphic manifolds that admit cell-like maps with the same image.

We use this result to construct exotic convergence of Reimannian manifolds in the Gromov-Hausdorff moduli space of manifolds with fixed contractibility function.

This is a joint work with Steve Ferry and Shmuel Weinberger.