Enumerating embeddings of \(n\)-manifolds into complex projective \(n\)-space
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.
scientific article; zbMATH DE number 1423817