Cohomology and asymptotic stability of 1-dimensional continua

Abstract

We prove that a 1-dimensional continuum carrying a flow without singular points is homeomorphic to the unit circle if its first Cech cohomology group with integer coefficients is isomorphic to ℤ. As an application of this we obtain that an asymptotically stable invariant 1-dimensional continuum of a flow on a locally compact ANR, which does not contain singular points, must be a periodic orbit.