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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by A. Duane Randall

Trans. Amer. Math. Soc. 156 (1971), 45-58

DOI: https://doi.org/10.1090/S0002-9947-1971-0286121-1

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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.

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