Maps with finitely many critical points into high dimensional manifolds


Assume that there exists a smooth map between two closed manifolds \(M^m\rightarrow N^k\), where \(2\le k\le m\le 2k-1\), with only finitely many singular points, all of which are cone-like. If \((m,k)\not \in \{(2,2), (4,3), (5,3), (8,5), (16,9)\}\), then \(M^m\) admits a locally trivial topological fibration over \(N^k\) and there exists a smooth map \(M^m\rightarrow N^k\) with at most one critical point.

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The author is grateful to Norbert A’Campo, Cornel Pintea, Nicolas Dutertre, Valentin Poenaru and Mihai Tibǎr for useful discussions and to the referees for corrections and helpful remarks. The author was partially supported by the GDRI Eco-Math.

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Correspondence to Louis Funar.

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Funar, L. Maps with finitely many critical points into high dimensional manifolds. Rev Mat Complut (2020).

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  • Isolated singularity
  • Open book decomposition
  • Poenaru–Mazur contractible manifold

Mathematics Subject Classification

  • 57R45
  • 57 R70
  • 58K05