An infinite-dimensional phenomenon in finite-dimensional metric topology

A. Dranishnikov; S. Ferry; S. Weinberger

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An infinite-dimensional phenomenon in finite-dimensional metric topology

A. Dranishnikov·S. Ferry·S. Weinberger

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Abstract

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 $\mathbb L$-classes. The image space in these constructions is necessarily infinite-dimensional. In dimension $>6$ we classify all such homotopy equivalences. As an application, we show that such homotopy equivalences are realized by deformations of Riemannian manifolds in Gromov-Hausdorff space preserving a contractibility function.

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A. Dranishnikov, S. Ferry, S. Weinberger. 2006-10-31. An infinite-dimensional phenomenon in finite-dimensional metric topology. https://arxiv.org/abs/math/0611004

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