Spaces of Maps With Only Fold Singularities

We explore the topology of smooth maps between manifolds that exhibit only fold singularities, commonly known as S-immersions. We focus particularly on extending the study of S-immersions to codimension 1 settings, where source and target manifolds differ in dimension by one. This involves adapting the notions of taut and flexible maps, originally formulated for equidimensional cases, to this new context. Flexible maps allow singularities to be modified or eliminated via homotopy, while taut maps are more rigid, often reflecting deeper global topological constraints. Our aim is to characterize the homotopy types of spaces of fold maps in codimension 1, and to identify when these spaces coincide with their formal analogues. Special attention is given to the elimination of cusp singularities in flexible maps and the persistence of topological invariants in taut maps.