On codimensions k immersions of m-manifolds for k = 1 and k = m − 2
Tóm tắt
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).
Tài liệu tham khảo
Adachi M.: A remark on submersions and immersions with codimension one or two. J. Math. Kyoto Univ. 9, 393–404 (1969)
Biasi C., Daccach J., Saeki O.: A primary obstruction to topological embeddings and its applications. Manus. Math. 104, 97–110 (2001)
Li B.H., Li G.S.: Immersions with non-zero normal vector fields. Math. Proc. Camb. Phil. Soc. 112, 281–285 (1992)
Li B.H., Peterson F.P.: On immersions of k-manifolds in (2k − 1)-manifolds. Proc. Am. Math. Soc. 83, 159–162 (1981)
Li B.H., Peterson F.P.: On immersions of n-manifolds in (2n − 2)-manifolds. Proc. Am. Math. Soc. 3(97), (1986)
Biasi C., Saeki O.: On the self-intersection set and the image of a generic map. Math. Scand. 80, 5–24 (1997)
Haeflinger A.: Plongements differentiables dans les domaines stable. Comment. Math. Helv. 37, 155–176 (1962)
Hirsch M.W.: Immersions of manifolds. Trans. Am. Math. Soc. 93, 242–276 (1959)
Koschorke, U.: 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)
Milnor J., Stasheff J.: Characteristic Classes. Ann. of Math. Studies, vol. 76. Princeton University Press, Princeton (1974)
Paechter G.F.: The groups π r (V n,m ). Quart. J. Math. Oxford (2) 7, 249–268 (1956)
Spanier E.H.: Algebraic Topology. TATA McGraw-Hill Publ. Company Ltd., Bombay (1966)