A note on actions of the cylinder \(S^1\times \mathbb R\) (Q1862029)

It is proved that any topological action of \(S^1\times {\mathbb R}\) on a rational homology sphere of arbitrary dimension has an orbit of dimension less than two. Here rational homology sphere means a compact manifold \(M\) whose homology with rational coefficients is that of a sphere (of lower dimension in case \(M\) has nonempty boundary). It is also proved that if a connected orientable \(m\)-manifold with vanishing rational Betti numbers \(b_i\) for \(0<i<k\), where \(k=[m/2]\), admits a locally free action of \(S^1\times {\mathbb R}\), then either \(m\equiv 2\bmod 4\) and \(b_k=2\), or \(m\equiv 3\bmod 4\) and \(b_k=k+1\) or \(k+2\). In particular, \(k\) must be odd.