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# Cambridge Journal of Mathematics

## Volume 8 (2020)

### Number 1

### An infinite-dimensional phenomenon in finite-dimensional metric topology

Pages: 95 – 147

DOI: https://dx.doi.org/10.4310/CJM.2020.v8.n1.a2

#### Authors

#### 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 the construction of cell-like maps that kill certain $\mathbb{L}$-classes. The image space in these constructions is necessarily infinite-dimensional. In dimension $\gt 5$ 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.

#### 2010 Mathematics Subject Classification

Primary 53C20, 53C23. Secondary 57N60, 57R65.

The first-named author was partially supported by the Simons Foundation.

The third-named author was partially supported by NSF grants.

Received 23 February 2017

Published 25 February 2020