2022/11/24 by Yichen Tong, Tong, Yichen
Mathematics · #55P10 #55P62 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2211.13417
openalex publication_date 2022/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a closed connected oriented manifold M of dimension 2n, it was proved by Møller and Raussen that the components of the mapping space from M to S2n have exactly two different rational homotopy types. However, since this result was proved by the algebraic models for the components, it is unclear whether other homotopy invariants distinguish their rational homotopy types or not. The self-closeness number of a connected CW complex is the least integer k such that any of its self-map inducing an isomorphism in π_* for *≤ k is a homotopy equivalence, and there is no result on the components of mapping spaces so far. For a rational Poincaré complex X of dimension 2n with finite π1, we completely determine the self-closeness numbers of the rationalized components of the mapping space from X to S2n by using their Brown-Szczarba models. As a corollary, we show that the self-closeness number does distinguish the rational homotopy types of the components. Since a closed connected oriented manifold is a rational Poincaré complex, our result partially generalizes that of Møller and Raussen.