Topology of Ambient Manifolds of Regular Homeomorphisms with Codimension 1 Saddles

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. We prove that the ambient manifold of any regular $n$-homeomorphism all of whose saddle points have Morse index $n-1$ is homeomorphic to the $n$-sphere. In dimension $n=1$ a similar problem does not make sense, since the circle is the only closed $1$-manifold. Regular $2$-homeomorphisms exist on any surfaces, and all their saddle points have Morse index $1$, which implies that the obtained result does not hold in dimension $2$.