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

Abstract

Let us consider M a closed smooth connected m-manifold, N a smooth (2m − 2)-manifold and \({ f : M \longrightarrow N}\) a continuous map, with \({ m \equiv 1(4)}\). We prove that if \({ {f}_* : {H}_{1}(M; \, Z_2) \longrightarrow \check{H}_{1}(f(M) ; \, Z_2)}\) 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 2m − 2, 2m − 3 and 2m − 4. Lecture Notes in Mathematics, vol. 1350. Springer, Heidelberg, 1988; Li and Li in Math. Proc. Camb. Phil. Soc. 112, 281–285, 1992).