Topology of ambient manifolds of regular homeomorphisms with codimension one saddles

In this paper we consider regular homeomorphisms on topological $n$-manifolds (not necessarily orientable), which are a generalization of Morse-Smale diffeomorphisms. By a regular homeomorphism we mean a homeomorphism of a topological $n$-manifold ($n\geq 3$) whose chain recurrent set is finite and hyperbolic (in the topological sense). The hyperbolic structure of periodic points allows us to classify them according to their Morse indices (the dimension of the unstable manifold). In this case, the points of extreme indices are called nodal and the rest are called saddle points. The authors prove that the ambient manifold of any regular $n$-homeomomorphism, all saddle points of which have Morse index $n-1$, is homeomorphic to the $n$-sphere. In dimension $n=1$ the analogous problem does not make sense, since the circle is the only closed $n$-manifold. Regular 2-homeomomorphisms exist on any surfaces and all their saddle points have Morse index 1, whence it follows that the obtained result is not true in dimension 2.