Enumerating embeddings of \(n\)-manifolds into complex projective \(n\)-space
type: Item
instance of: scholarly article
title: Enumerating embeddings of \(n\)-manifolds into complex projective \(n\)-space
zbMATH Open document ID: 0944.57024
author: Tsutomu Yasui
published in: Hiroshima Mathematical Journal
publication date: 2000-09-17T00:00:00Z
review text: Let \(\text{Emb}(M,N)\) be the space of differentiable embeddings between closed connected differentiable manifolds \(M, N\). For \(f\in \text{Emb}(M,N)\), \([M\subset N]_f\) denotes the set \(\pi_1(N^M, \text{Emb}(M,N),f)\). It is known that there is a \(\pi_1(N^M,f)\)-action on \([M\subset N]_f\) such that \({[M\subset N]_f}/\pi_1(N^M,f)\) is equivalent to the set \([M\subset N]_{[f]}\) of isotopy classes of embeddings homotopic to \(f\). This paper mainly studies the set \([M^n\subset \mathbb CP^n]_f\), where \(\mathbb CP^n\) is complex projective \(n\)-space. For \(n>3\), under further suitable conditions, the author describes the set \([M^n\subset \mathbb CP^n]_f\) basically in terms of the cohomology of \(M^n\). As corollaries, also results on \([M^n\subset \mathbb CP^n]_{[f]}\) (for \(1\)-connected \(M^n\)), and results on \([\mathbb CP^n\subset \mathbb CP^{2n}]_{[f]}\) or on \([\mathbb RP^n\subset \mathbb CP^n]_{[f]}\) are presented. As remarked by the author, this paper and a paper by \textit{B. Li} and \textit{P. Zhang} [Syst. Sci. Math. Sci. 6, No. 1, 61-69 (1993; Zbl 0788.57017)] overlap to some extent. The proofs in the paper under review essentially use L. Larmore's method of computing the set \([M\subset N]_f\) and also results from some of the author's earlier papers.
reviewed by: Július Korbaš
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 55S45
Mathematics Subject Classification ID: 57R52
zbMATH DE Number: 1423817
zbMATH Keywords: embedding
zbMATH Keywords: manifold
zbMATH Keywords: isotopy
zbMATH Keywords: Postnikov tower
zbMATH Keywords: \(2\)-fold symmetric product
zbMATH Keywords: Stiefel-Whitney class
zbMATH Keywords: Steenrod square
zbMATH Keywords: Thom class
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206124855
OpenAlex ID: W1515293330
DOI: 10.32917/HMJ/1206124855
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description: scientific article; zbMATH DE number 1423817