Speaker: Rustam Sadykov  (joint work with O. Saeki)

A smooth map between manifolds is said to be image simple if its restriction to its singular point set is a topological embedding. Consider the parity of the number of connected components of the singular point set for image simple fold maps from a closed manifold of dimension ≥2 to a surface. It is known that this parity is a homotopy invariant when the source manifold is of even dimension and the target surface is orientable. I will use generalized Dehn twists and open book decompositions to show that for an arbitrary image simple fold map from a closed odd-dimensional manifold of dimension ≥3  to a (possibly non-orientable) surface, this parity is not a homotopy invariant.