Some immersion theorems for manifolds

Randall, A. Duane

doi:10.1090/S0002-9947-1971-0286121-1

Some immersion theorems for manifolds
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Trans. Amer. Math. Soc. 156 (1971), 45-58 Request permission

Abstract:

In this paper we obtain several results on immersing manifolds into Euclidean spaces. For example, a spin manifold ${M^n}$ immerses in ${R^{2n - 3}}$ for dimension $n \equiv 0\bmod 4$ and n not a power of 2. A spin manifold ${M^n}$ immerses in ${R^{2n - 4}}$ for $n \equiv 7\bmod 8$ and $n > 7$. Let ${M^n}$ be a 2-connected manifold for $n \equiv 6\bmod 8$ and $n > 6$ such that ${H_3}(M;Z)$ has no 2-torsion. Then M immerses in ${R^{2n - 5}}$ and embeds in ${R^{2n - 4}}$. The method of proof consists of expressing k-invariants in Postnikov resolutions for the stable normal bundle of a manifold by means of higher order cohomology operations. Properties of the normal bundle are used to evaluate the operations.

References

J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20–104. MR 141119, DOI 10.2307/1970147
José Adem, On secondary cohomology operations, Bol. Soc. Mat. Mexicana (2) 7 (1962), 95–110 (Spanish). MR 167978
José Adem and Samuel Gitler, Secondary characteristic classes and the immersion problem, Bol. Soc. Mat. Mexicana (2) 8 (1963), 53–78. MR 167990
J. Adem, S. Gitler, and M. Mahowald, Embedding and immersion of projective spaces, Bol. Soc. Mat. Mexicana (2) 10 (1965), 84–88. MR 220303
M. F. Atiyah, Thom complexes, Proc. London Math. Soc. (3) 11 (1961), 291–310. MR 131880, DOI 10.1112/plms/s3-11.1.291
Albrecht Dold, Erzeugende der Thomschen Algebra ${\mathfrak {N}}$, Math. Z. 65 (1956), 25–35 (German). MR 79269, DOI 10.1007/BF01473868
S. Feder, Non-immersion theorems for complex and quaternionic projective spaces, Bol. Soc. Mat. Mexicana (2) 11 (1966), 62–67. MR 232396
S. Gitler and M. Mahowald, The geometric dimension of real stable vector bundles, Bol. Soc. Mat. Mexicana (2) 11 (1966), 85–107. MR 231367
S. Gitler, M. Mahowald, and R. James Milgram, Secondary cohomology operations and complex vector bundles, Proc. Amer. Math. Soc. 22 (1969), 223–229. MR 243515, DOI 10.1090/S0002-9939-1969-0243515-4
André Haefliger and Morris W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. MR 149494, DOI 10.1016/0040-9383(63)90028-4
Morris W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR 119214, DOI 10.1090/S0002-9947-1959-0119214-4
C. S. Hoo and M. E. Mahowald, Some homotopy groups of Stiefel manifolds, Bull. Amer. Math. Soc. 71 (1965), 661–667. MR 177412, DOI 10.1090/S0002-9904-1965-11387-8
Mark Mahowald, On obstruction theory in orientable fiber bundles, Trans. Amer. Math. Soc. 110 (1964), 315–349. MR 157386, DOI 10.1090/S0002-9947-1964-0157386-8
Mark Mahowald, Some Whitehead products in $S^{n}$, Topology 4 (1965), 17–26. MR 178467, DOI 10.1016/0040-9383(65)90046-7
Mark E. Mahowald and Franklin P. Peterson, Secondary cohomology operations on the Thom class, Topology 2 (1963), 367–377. MR 157378, DOI 10.1016/0040-9383(63)90016-8
M. E. Mahowald and R. F. Williams, The stable homotopy of $K(Z,\,n)$, Bol. Soc. Mat. Mexicana (2) 11 (1966), 22–28. MR 235563
W. S. Massey and F. P. Peterson, On the dual Stiefel-Whitney classes of a manifold, Bol. Soc. Mat. Mexicana (2) 8 (1963), 1–13. MR 163325
W. S. Massey and F. P. Peterson, The cohomology structure of certain fibre spaces. I, Topology 4 (1965), 47–65. MR 189032, DOI 10.1016/0040-9383(65)90048-0
G. F. Paechter, The groups $\pi _{r}(V_{n,\,m})$. I, Quart. J. Math. Oxford Ser. (2) 7 (1956), 249–268. MR 131878, DOI 10.1093/qmath/7.1.249
A. Duane Randall, Some immersion theorems for projective spaces, Trans. Amer. Math. Soc. 147 (1970), 135–151. MR 253357, DOI 10.1090/S0002-9947-1970-0253357-4
B. J. Sanderson and R. L. E. Schwarzenberger, Non-immersion theorems for differentiable manifolds, Proc. Cambridge Philos. Soc. 59 (1963), 319–322. MR 148080, DOI 10.1017/s0305004100036938
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
Emery Thomas, Seminar on fiber spaces, Lecture Notes in Mathematics, vol. 13, Springer-Verlag, Berlin-New York, 1966. Lectures delivered in 1964 in Berkeley and 1965 in Zürich; Berkeley notes by J. F. McClendon. MR 0203733, DOI 10.1007/BFb0097864
Emery Thomas, Postnikov invariants and higher order cohomology operations, Ann. of Math. (2) 85 (1967), 184–217. MR 210135, DOI 10.2307/1970439
Emery Thomas, Real and complex vector fields on manifolds, J. Math. Mech. 16 (1967), 1183–1205. MR 0210136
Emery Thomas, The index of a tangent $2$-field, Comment. Math. Helv. 42 (1967), 86–110. MR 215317, DOI 10.1007/BF02564413
Emery Thomas, The span of a manifold, Quart. J. Math. Oxford Ser. (2) 19 (1968), 225–244. MR 234487, DOI 10.1093/qmath/19.1.225
Emery Thomas, On the existence of immersions and submersions, Trans. Amer. Math. Soc. 132 (1968), 387–394. MR 225332, DOI 10.1090/S0002-9947-1968-0225332-8
Emery Thomas, An exact sequence for principal fibrations, Bol. Soc. Mat. Mexicana (2) 12 (1967), 35–45. MR 243526
Emery Thomas, Vector fields on low dimensional manifolds, Math. Z. 103 (1968), 85–93. MR 224109, DOI 10.1007/BF01110620
J. J. Ucci, Immersions and embeddings of Dold manifolds, Topology 4 (1965), 283–293. MR 187250, DOI 10.1016/0040-9383(65)90012-1