Mathematical Research Letters

Volume 28 (2021)

Number 4

Non-separating immersions of spheres and Bing houses

Pages: 1085 – 1106



Michael Freedman (Microsoft Research, Station Q; and Department of Mathematics, University of California, Santa Barbara, Calif., U.S.A.)

T. Tâm Nguễyn-Phan (Max Planck Institut für Mathematik, Bonn, Germany)


In all dimensions $n \geq 4$ we construct explicit non-separating, locally flat, $\mathsf{PL}$ immersions of $\mathbb{S}^{n-1} \looparrowright S^n$. In fact, the complement of these immersions is diffeomorphic to $\mathbb{R}^n$. These generalize the famous “house with two rooms” or “Bing house” which is an immersion $S^2 \looparrowright \mathbb{R}^3 \subset S^3$.

Received 23 January 2020

Accepted 5 February 2021

Published 22 November 2021