On the number of components of folds of image simple fold maps

A smooth map between manifolds is said to be \emph{image simple} if its restriction to its singular point set is a topological embedding. We study the parity of the number of connected components of the singular point set for image simple fold maps from a closed manifold of dimension $\ge 2$ 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. In this paper we show that for an arbitrary image simple fold map from a closed odd-dimensional manifold of dimension $\ge 3$ to a (possibly non-orientable) surface, this parity is not a homotopy invariant. We give two constructive proofs: one uses techniques from open book decompositions and round fold maps, and the other uses allowable moves. The constructed homotopies add one new component to the singular point set. We also give a new proof of the homotopy invariance of the parity of the number of connected components of the singular point set for image simple fold maps from a closed even-dimensional manifold to an orientable surface together with an example showing that this does not hold for maps into non-orientable surfaces in general.