A proof of the spherical homeomorphism conjecture for surfaces.

Abstract

The human cerebral cortex is topologically equivalent to a sphere when it is viewed as closed at the brain stem. Due to noise and/or resolution issues, magnetic resonance imaging may see "handles" that need to be eliminated to reflect the true spherical topology. Shattuck and Leahy present an algorithm to correct such an image. The basis for their correction strategy is a conjecture, which they call the spherical homeomorphism conjecture, stating that the boundary between the foreground region and the background region is topologically spherical if certain associated foreground and background multigraphs are both graph-theoretic trees. In this paper, we prove the conjecture, and its converse, under the assumption that the foreground/background boundary is a surface.