On the normal bundle of a manifold
Mark E. Mahowald · Pacific Journal of Mathematics · 1964
This result is a slight sharpening of the theorem of Massey [4]; the proof is given in § 4 after some preliminary results in § § 2 and 3.Let χ be the Euler characteristic of M 2 .In Whitney's theorem the role of σ in Theorem 1 is played by χ.It is not hard to verify that for 2-dimension manifolds σ = χ mod 2. In addition, for 2-dimensional manifolds we can prove (section 6) THEOREM 2. For each k and each value of σ there is a manifold M 2 and an embedding of M 2 in R* with twisted Euler class 2σ + 4k.We have not been able to show that a single manifold has an embedding for each k.Whitney exhibited two embeddings of the Klein bottle, one with a trivial Euler class and one with a non-trivial one.We also have this weaker result (section 7) for other values of n.THEOREM 3.For every even n there exists a manifold M n and an embedding of M n in R 2n with no normal field.It is known that if n Φ 2> and n > 3, then every ^-manifold embeds