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Generic immersions and totally real embeddings

Department of Mathematics, Kyoto Sangyo University, Kamigamo-Motoyama, Kita-ku, Kyoto 603-8555, Japan

E-mail Address: nkasuya@cc.kyoto-su.ac.jp

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Masamichi Takase

Faculty of Science and Technology, Seikei University, 3-3-1 Kichijoji-kitamachi, Musashino, Tokyo 180-8633, Japan

E-mail Address: mtakase@st.seikei.ac.jp

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https://doi.org/10.1142/S0129167X18500738Cited by:2 (Source: Crossref)

Abstract

We show that, for a closed orientable $n$ $n$ -manifold, with $n$ $n$ not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex $(n - 1)$ $(n - 1)$ -space ensures the existence of a totally real embedding into complex $n$ $n$ -space. This implies that a closed orientable $(4 k + 1)$ $(4 k + 1)$ -manifold with non-vanishing Kervaire semi-characteristic possesses no CR-regular embedding into complex $4 k$ $4 k$ -space. We also pay special attention to the cases of CR-regular embeddings of spheres and of simply-connected 5-manifolds.

Dedicated to Professor Takashi Nishimura on the occasion of his $60$ $60$ th birthday

Keywords:

AMSC: 32V40, 53C40, 57R20, 57R40, 57R42

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