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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On transversality with deficiency
and a conjecture of Sard


Authors: Carlos Biasi and Osamu Saeki
Journal: Trans. Amer. Math. Soc. 350 (1998), 5111-5122
MSC (1991): Primary 57N75; Secondary 57R45, 55M10
MathSciNet review: 1458297
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Abstract: Let $f : M \to N$ be a $C^{r}$ map between $C^{r}$ manifolds $(r \geq 1)$ 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 $g : K \to N$ 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 $f(R_{q}(f))$.


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Additional Information

Carlos Biasi
Affiliation: Departamento de Matemática, ICMSC-USP, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil
Email: biasi@icmsc.sc.usp.br

Osamu Saeki
Affiliation: Department of Mathematics, Faculty of Science, Hiroshima University, Higashi- Hiroshima 739, Japan
Email: saeki@top2.math.sci.hiroshima-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02088-1
PII: S 0002-9947(98)02088-1
Received by editor(s): November 14, 1996
Additional Notes: The second author has been partially supported by CNPq, Brazil, and by the Grant-in-Aid for Encouragement of Young Scientists (no. 08740057), Ministry of Education, Science and Culture, Japan.
Article copyright: © Copyright 1998 American Mathematical Society