Cambridge Journal of Mathematics

Volume 8 (2020)

Number 1

An infinite-dimensional phenomenon in finite-dimensional metric topology

Pages: 95 – 147



Alexander N. Dranishnikov (Department of Mathematics, University of Florida, Gainesville, Fl., U.S.A.)

Steven C. Ferry (Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A.; and Department of Mathematics, Binghamton University, Vestal, New York, U.S.A.)

Shmuel Weinberger (Department of Mathematics, University of Chicago, Illinois, U.S.A.)


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