### Abstract

Let f : M → N be a differentiable map of a closed m-dimensional manifold into an (m + k)-dimerisional manifold with k > 0. VVe show, assuming that f is generic in a certain sense, that f is an embedding if and only if the (m - k + 1)-th Betti numbers with respect to the Čech homology of M and f(M) coincide, under a certain condition on the stable normal bundle of f. This generalizes the authors' previous result for immersions with normal crossings [BS1]. As a corollary, we obtain the converse of the Jordan-Brouwer theorem for codimension-1 generic maps, which is a generalization of the results of [BR, BMS1, BMS2, Sae1] for immersions with normal crossings.

Original language | English |
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Pages (from-to) | 72-83 |

Number of pages | 12 |

Journal | Commentarii Mathematici Helvetici |

Volume | 72 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 1997 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

Biasi, C., & Saeki, O. (1997). On the Betti number of the image of a generic map.

*Commentarii Mathematici Helvetici*,*72*(1), 72-83. https://doi.org/10.1007/PL00000368