Minimally separating sets, mediatrices, and Brillouin spaces

The topology of surface mediatrices

“…On S 2 then, there is only one type of mediatrix up to homeomorphism. Since mediatrices on Riemannian manifolds are minimally separating, this classification on S 2 follows immediately from the Jordan Curve separation theorem, which should not surprise us since a key ingredient used in our proof (see [7]) was Lefschetz Duality, a very general version of the Jordan separation theorem.…”

“…Proof. By Theorem 1.21, L is a finite closed 1-simplex, and by Corollary 4.4 of [7], we have H 1 (L; Z 2 ) = Z 2 . Together these imply that L is homeomorphic to S 1 .…”

“…We can now isolate the regions in which L pq lies, at least locally. First of all, L pq will not intersect geodesics from x to p or from xinL pq to q, except at x, by the following lemma, proved in [7]: Lemma 1.3. Let p, q ∈ M with p = q.…”

“…We can now use the results from Section 1, combined with results from [7] to turn our attention to the classification of mediatrices on a torus up to homeomorphism. We first note as an aside, however, that combining the results from Section 1 with those from [7], we can now readily classify mediatrices on spheres up to homeomorphism. Proposition 2.1.…”

“…In [7], some topological restrictions placed on L pq by the topology of M were found. In this paper, we focus on the particular case in which M is a 2-manifold to determine what can said about L in that case.…”