Collapse map of degree one onto a sphere ($M^n⟶M/(M\setminus \mathrm{int}{D}^n)\cong S^n$)

Choose an oriented embedded closed disc in a closed oriented positive-dimensional manifold and collapse its complement to a point. A target point away from the collapsed value has one preimage with local degree $+1$, proving global degree one by the degree as a sum of local degrees.