Some immersion theorems for projective spaces

Randall, A. Duane

doi:10.1090/s0002-9947-1970-0253357-4

Cited by 7 publications

(2 citation statements)

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“…We conclude the proof by listing the relations in the MPT of BSO(4ℓ −4) → BSO, the last of which yields the crucial fact (3.3). These are computed by the method initiated in [8] and utilized in such papers as [6], [12], and [15]. It is a matter of building a minimal resolution using Massey-Peterson algebras.…”

Section: Let H Cmentioning

confidence: 99%

Some New Immersion Results for Complex Projective Space

Davis

2008

Proceedings of the Edinburgh Mathematical Society

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Abstract. We prove the following two new optimal immersion results for complex projective space. First, if n ≡ 3 mod 8 but n ≡ 3 mod 64, and α(n) = 7, then CP n can be immersed in R 4n−14 . Second, if n is even and α(n) = 3, then CP n can be immersed in R 4n−4 . Here α(n) denotes the number of 1's in the binary expansion of n. The first contradicts a result of Crabb, which said that such an immersion does not exist, apparently due to an arithmetic mistake. We combine Crabb's method with that developed by the author and Mahowald. Main theoremsWe prove the following two new optimal immersion results for 2n-dimensional complex projective space CP n . Theorem 1.1. If n ≡ 3 mod 8 and n ≡ 3 mod 64, and α(n) = 7, then CP n can be immersed in R 4n−14 . Theorem 1.2. If n is even and α(n) = 3, then CP n can be immersed in R 4n−4 .Here and throughout, α(n) denotes the number of 1's in the binary expansion of n. Theorem 1.1 contradicts a result of Crabb ([2]). In Section 2, we prove Theorem 1.1 by an adaptation of Crabb's argument, and point out what we believe to be his mistake, apparently arithmetic. We prove Theorem 1.2 in Section 3.We now summarize what we feel to be the status of the immersion question for CP n . In addition to incorporating the two new immersion results above, we list as unresolved one immersion result claimed by Crabb. We will discuss our reason Date: February 21, 2006.

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Section: Let H Cmentioning

confidence: 99%

Some New Immersion Results for Complex Projective Space

Davis

2008

Proceedings of the Edinburgh Mathematical Society

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“…Since the early papers of Whitney ([14]), Hirsch ([8]) and Smale ([12]), the theory of embeddings focused on the question whether a smooth manifold can be immersed / embedded into a Euclidean space. In particular, there are a number of results concerning the best immersion or embedding of a projective space of a given dimension into a Euclidean space, and, on the other hand, topological obstructions to such immersability in some dimensions have been found (see, for instance, [1], [2], [4], [10], [11]).…”

Section: Introductionmentioning

confidence: 99%