Cell-like maps and topological structure groups on manifolds

Details Cell-like maps and topological structure groups on manifolds Cell-like maps and topological structure groups on manifolds

2006-10-31

arxiv.org/abs/math/0611004v1

Mathematics

Geometric Topology

Scientific paper

We show that there are homotopy equivalences $h:N\to M$ between closed manifolds which are induced by cell-like maps $p:N\to X$ and $q:M\to X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of cell-like maps that kill certain $\bL$-classes. In dimension $>5$ we identify all such homotopy equivalences to $M$ with a torsion subgroup $\CS^{CE}(M)$ of the topological structure group $\CS(M)$. In the case of simply connected $M$ with finite $\pi_2(M)$, the subgroup $\CS^{CE}(M)$ coincides with the odd torsion in $\CS(M)$. For general $M$, the group $\CS^{CE}(M)$ admits a description in terms of the second stage of the Postnikov tower of $M$. As an application, we show that there exist a contractibility function $\rho$ and a precompact subset (\mathcal{C}) of Gromov-Hausdorff space such that for every $\epsilon>0$ there are nonhomeomorphic Riemannian manifolds with contractibility function $\rho$ which lie within $\epsilon$ of each other in (\mathcal{C}).

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