Topology of boundary special generic maps into Euclidean spaces

We introduce boundary special generic maps, a class of submersions from manifolds with boundary to Euclidean spaces whose restriction to the boundary has only boundary definite fold points as its singular points. We derive the differential-topological restrictions imposed by the existence of such maps on the global structure of the source manifolds. Furthermore, we apply our results to the non-singular extension problem, which asks when a map on a closed manifold extends to a non-singular map on a manifold with boundary, and obtain new results on non-singular extensions of special generic maps.