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<p>For each based map <i>f</i>: <i>X</i> → ℝP<sup>2</sup> from a closed surface into the real projective plane, we compute its absolute and twisted degrees and describe the action of the fundamental group of ℝP<sup>2</sup> over the based homotopy class of <i>f</i>. We emphasize the finding that for any nonorientable closed surface, there exists only one based homotopy class of maps from it into ℝP<sup>2</sup> whose maps have twisted degree zero and absolute degree nonzero–which shows that, unlike the absolute degree, the twisted degree is not able to detect the strong surjectivity in this setting. In all the other scenarios, the absolute degree of each map is either equal to the twisted degree or its absolute value–and so the twisted degree detects strong surjectivity.</p>

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Twisted and Absolute Degrees of Maps into the Projective Plane

  • 期刊论文

  • 发表日期：

    2025年5月15日

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  • Marcio Colombo Fenille,
  • Daciberg Lima Gonçalves

摘要

For each based map f: X → ℝP2 from a closed surface into the real projective plane, we compute its absolute and twisted degrees and describe the action of the fundamental group of ℝP2 over the based homotopy class of f. We emphasize the finding that for any nonorientable closed surface, there exists only one based homotopy class of maps from it into ℝP2 whose maps have twisted degree zero and absolute degree nonzero–which shows that, unlike the absolute degree, the twisted degree is not able to detect the strong surjectivity in this setting. In all the other scenarios, the absolute degree of each map is either equal to the twisted degree or its absolute value–and so the twisted degree detects strong surjectivity.