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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Some immersion theorems for manifolds


Author: A. Duane Randall
Journal: Trans. Amer. Math. Soc. 156 (1971), 45-58
MSC: Primary 57.20
DOI: https://doi.org/10.1090/S0002-9947-1971-0286121-1
MathSciNet review: 0286121
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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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DOI: https://doi.org/10.1090/S0002-9947-1971-0286121-1
Keywords: Immersion, embedding, spin manifold, normal bundle, Postnikov resolution, k-invariant, twisted cohomology operation, higher order cohomology operation, Thom complex, Poincaré duality, Dold manifold
Article copyright: © Copyright 1971 American Mathematical Society