### Abstract

Let f : M → N be a C^{r} map between C^{r} manifolds (r ≥ 1) and K a C^{r} manifold. In this paper, by using the Sard theorem, we study the topological properties of the space of C^{r} maps g : K → N which satisfy a certain transversality condition with respect to f in a weak sense. As an application, by considering the case where K is a point, we obtain some new results about the topological properties of f(R_{q}(f)), where R_{q}(f) is the set of points of M where the rank of the differential of f is less than or equal to q. In particular, we show a result about the topological dimension of f(R_{q}(f)), which is closely related to a conjecture of Sard concerning the Hausdorff measure of f(R_{q}(f)).

Original language | English |
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Pages (from-to) | 5111-5122 |

Number of pages | 12 |

Journal | Transactions of the American Mathematical Society |

Volume | 350 |

Issue number | 12 |

Publication status | Published - Dec 1 1998 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*350*(12), 5111-5122.