Proof of a conjecture of Whitney

Massey, William S.

doi:10.2140/pjm.1969.31.143

Cited by 78 publications

(63 citation statements)

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“…Theorem 1.5 (1) holds also for maps with triple points. Note also that Theorem 1.5 (3) is closely related to the well-known Whitney conjecture which has been proved by Massey [14] (see also [9]). See Remark 6.1 in §6.…”

Section: Introductionmentioning

confidence: 74%

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Immersed 𝑛-manifolds in 𝐑²ⁿ and the double points of their generic projections into 𝐑²ⁿ⁻¹

Saeki

Sakuma

1996

Trans. Amer. Math. Soc.

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Abstract. We give two congruence formulas concerning the number of nontrivial double point circles and arcs of a smooth map with generic singularities -the Whitney umbrellas -of an n-manifold into R 2n−1 , which generalize the formulas by Szücs for an immersion with normal crossings. Then they are applied to give a new geometric proof of the congruence formula due to Mahowald and Lannes concerning the normal Euler number of an immersed n-manifold in R 2n . We also study generic projections of an embedded nmanifold in R 2n into R 2n−1 and prove an elimination theorem of Whitney umbrella points of opposite signs, which is a direct generalization of a recent result of Carter and Saito concerning embedded surfaces in R 4 . The problem of lifting a map into R 2n−1 to an embedding into R 2n is also studied.

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Section: Introductionmentioning

confidence: 74%

“…Note that the above inequality has been proved for general embeddings by Massey [14] and also by Kamada [9]. The proofs by Massey and Kamada use the double branched covering of S 4 = R 4 ∪ {∞} branched along f (M 2 ) and they are completely different from our approach.…”

Section: Does Not Contain a Whitney Umbrella Point And There Does Notmentioning

confidence: 87%

Immersed 𝑛-manifolds in 𝐑²ⁿ and the double points of their generic projections into 𝐑²ⁿ⁻¹

Saeki

Sakuma

1996

Trans. Amer. Math. Soc.

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“…Let Σ be a compact nonorientable surface with Euler characteristics χ(Σ) = χ. By a result of Massey [14], Σ can be embedded into C 2 with χ(N Σ) ∈ {2χ − 4, 2χ, . .…”

Section: Proofs Of Theoremsmentioning

confidence: 99%

Real Surfaces in Elliptic Surfaces

Slapar

2005

Int. J. Math.

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“…where 2t denotes the twisted integer coefficient system on M. Both groups are infinite cyclic and, in each case, the generator can be taken to be the fundamental class [9]. Since n* maps fundamental class to fundamental class it follows that n* is an isomorphism.…”

Section: Immersions With Flat Normal Bundlementioning

confidence: 99%