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On transversality with deficiency and a conjecture of Sard

Authors:	Carlos Biasi Osamu Saeki

Institution:	Departamento de Matemática, ICMSC-USP, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil ; Department of Mathematics, Faculty of Science, Hiroshima University, Higashi- Hiroshima 739, Japan

Abstract:	Let $f : M \to N$ be a $C^{r}$ map between $C^{r}$ manifolds $(r \geq 1)$ and a $C^{r}$ manifold. In this paper, by using the Sard theorem, we study the topological properties of the space of $C^{r}$ maps $g : K \to N$ which satisfy a certain transversality condition with respect to in a weak sense. As an application, by considering the case where 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 where the rank of the differential of is less than or equal to . 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 $f(R_{q}(f))$ .

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