On codimensions k immersions of m-manifolds for k = 1 and k = m − 2

Let us consider M a closed smooth connected m -manifold, N a smooth (2 m − 2)-manifold and a continuous map, with . We prove that if is injective, then f is homotopic to an immersion. Also we give conditions to a map between manifolds of codimension one to be homotopic to an immersion. This work complements some results of Biasi et al. (Manu. Math. 104 , 97–110, 2001; Koschorke in The singularity method and immersions of m -manifolds into manifolds of dimensions 2 m − 2, 2 m − 3 and 2 m − 4. Lecture Notes in Mathematics, vol. 1350. Springer, Heidelberg, 1988; Li and Li in Math. Proc. Camb. Phil. Soc. 112 , 281–285, 1992).