Bounds for dimensions of odd order nonsingular immersions of 𝑅𝑃ⁿ

Abstract

Introduction. Operations of exterior power in KO-theory as well as StiefelWhitney classes are useful means to estimate the lowest dimension of an affine space in which real projective spaces and their product spaces are immersed or embedded (see, e.g., Milnor [1], Atiyah [2] and Suzuki [3]). The purposes of the present paper are to show, by similar means, that the operations in KOtheory as well as characteristic classes are applicable to higher order nonsingular immersions of real projective spaces, and to compare proofs and results of the former with those of the latter. In fact, proofs of the two methods are quite analogous. Theorems (1.1), (1.4) are higher order nonimmersion theorems for differentiable manifolds. They are applied to real projective spaces in Theorems (1.2), (1.6) and we find bounds of dimensions of affine spaces in which real projective spaces are immersed without odd order singularities. The results (Theorem (1.1), (1.2)) are due to characteristic class arguments. Theorem (1.2) includes Feldman's examples for real projective planes [5. I] and we shall show more examples of the theorem. However, Corollary (1.5), Theorem (1.6) are derived from KOtheory and they are useful for real projective spaces of some dimensions, where Theorem (1.2) does not work. In ?2, we explain symmetric tensor products of vector space bundle and introduce symmetric power operations in KO-theory. These may be more than what we need below. We also explain, according to Pohl [4] and Feldman [5], the bundle of pth order tangent vectors of the differentiable manifold and its relation to the immersion of the manifold in the affine space without singularities of order p (p ? 1). Lemma (2.3) is stated in a proposition of Feldman [5. I] without proof. We shall add a brief proof of it. In the last section, we prove all theorems.