Reflections on closed manifolds

A local reflection r : M → M, M a closed and connected manifold, is an involution with a two-sided fixed point submanifold F (in particular, nonempty). We show that β0(F) ⩽ g(M) + 1, g(M) the genus of M (Cornea, 1989). We also show that β0(M − F) ⩽ 2. If β0(M − F) = 2 we call r a reflection. In this case we prove that r is conjugate to the reflection that exchanges the two copies of Mr in M̂r = Mr ∪∂ Mr. In particular, M is a double. Further results are obtained. These involutions arise in Dynamical Systems (Roberts and Quispel, 1986; Teixeira, 1997).