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<p>In 2023, Zhang and Zhao presented the first examples of aspherical manifolds lacking the bounded index property (BIP) for fixed points. This answered a question posed by Jiang in 1998 in the negative. In this paper, we first extend the notion BIP to that of iterates of selfmaps (BIP<sub><i>k</i></sub>), and then demonstrate BIP<sub><i>k</i></sub> for certain products <i>M</i> × <i>N</i> with <i>N</i> a nilmanifold. Finally, we give characterizations for the Lefschetz number, the Nielsen number, and the minimal number of fixed points of self-homotopy equivalences of <i>M</i> × <i>N</i>.</p>

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On Jiang’s bounded index property for products with nilmanifolds

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • Peng Wang,
  • Qiang Zhang,
  • Xuezhi Zhao

摘要

In 2023, Zhang and Zhao presented the first examples of aspherical manifolds lacking the bounded index property (BIP) for fixed points. This answered a question posed by Jiang in 1998 in the negative. In this paper, we first extend the notion BIP to that of iterates of selfmaps (BIPk), and then demonstrate BIPk for certain products M × N with N a nilmanifold. Finally, we give characterizations for the Lefschetz number, the Nielsen number, and the minimal number of fixed points of self-homotopy equivalences of M × N.