### 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 |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Commentarii Mathematici Helvetici*,

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

**On the Betti number of the image of a generic map.** / Biasi, Carlos; Saeki, Osamu.

Research output: Contribution to journal › Article

*Commentarii Mathematici Helvetici*, vol. 72, no. 1, pp. 72-83. https://doi.org/10.1007/PL00000368

}

TY - JOUR

T1 - On the Betti number of the image of a generic map

AU - Biasi, Carlos

AU - Saeki, Osamu

PY - 1997/1/1

Y1 - 1997/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0031473939&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031473939&partnerID=8YFLogxK

U2 - 10.1007/PL00000368

DO - 10.1007/PL00000368

M3 - Article

AN - SCOPUS:0031473939

VL - 72

SP - 72

EP - 83

JO - Commentarii Mathematici Helvetici

JF - Commentarii Mathematici Helvetici

SN - 0010-2571

IS - 1

ER -