On the First Cohomology Group of a Minimal Set
Ippei Ishii · Tokyo Journal of Mathematics · 1978
It is one of the most important problems in the theory of topological dynamics to determine what space can be a minimal set under a con- tinuous flow.For example, it has been conjectured that there is no minimal flow on the 3-sphere $S^{3}$ .In this paper, we shall study the first cohomology of minimal sets.It is known that the space on which an almost periodic minimal flow or a distal minimal flow exists has a non-trivial first cohomology group.However the "almost periodicity" and the "distality" are both destroyed by a time-change, while the "minimality" is invariant by a time-change.The method for calculating the first cohomology of minial sets which is exhibited in this paper is quite independent of the parametrization by the time.In \S 3 we will establish a method for calculating the first cohomology of a minimal set from certain O-th cohomology groups.As an application of the consequence of \S 3, we can get a method for deciding the first cohomology of a minimal set which forms a 3-dimensional manifold (\S 4, Theorems 1 and 2).And in \S 5 we will investigate on l-cycles of a 3- dimensional minimal set.\S 1 and \S 2 are preliminaries.Higher dimen- sional cases can be treated by the same way, but it seems to be impos- sible to prove the non-triviality of the first cohomology of a minimal set by our method in the case of higher dimensional manifolds.Hence we do not treat the higher dimensional case in this paper.In the case of 3-manifolds, our method seems to be useful for the proof of the non- triviality of the first cohomology of a minimal set.In the case when the minimal set is a two dimensional manifold, using our method, we can decide the first cohomology of it completely.But it is well-known that the only two dimensional manifold admitting a minimal flow on it is the 2-torus.Therefore the results for two