Reflections on closed manifolds (Q1295243)

It is known that a local reflection \(r:M\to M\), \(M\) being a closed and connected manifold, is an involution with a two-sided fixed point submanifold \(F\) (in particular, nonempty). The author proves that \(\beta_0(F)\leq g(M)+1\), \(g(M)\) being the genus of \(M\) [\textit{O. Cornea}, Stud. Cercet. Mat. 41, No. 3, 169-178 (1989; Zbl 0694.57015)]. He also proves that \(\beta_0(M-F) \leq 2\), and in the case \(\beta_0(M-F)=2\), he calls \(r\) a reflection. In this case it is proved that \(r\) is conjugate to the reflection that exchanges the two copies of \(M/r\) in \(M/r= M/r\cup_\delta M/r\). Further results are also obtained.