Non-separating immersions of spheres and Bing houses
In [Ann. Math. (2) 87, 29--41 (1968; Zbl 0159.23902)], \textit{A. D. Weinstein} stated the theorem below that allows us to deduce the existence of non-separating PL immersions of \(S^{n-1}\) into \(\mathbb R^{n}\) for \( n \geq 3\). Weinstein's Theorem: ``For any closed, smooth manifold \(M^{n}\) of dimension \(n \geq 2\) that is not the sphere \(S^{ 2}\), there is a Riemannian metric and a point \(p \in M\) whose conjugate locus is disjoint from the cut locus.'' This paper presents an explicit and fascinating construction of a \(3\)-dimensional Bing house example, discovered in 1950, but never published, which is an immersion of \(S^{2}\) into \(S^{3}\). Also for \(n \geq 4\), the authors construct explicit non-separating, locally flat, PL immersions of \(S^{n-1}\) into \(S^{n}\).
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