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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On transversality with deficiency and a conjecture of Sard
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by Carlos Biasi and Osamu Saeki PDF
Trans. Amer. Math. Soc. 350 (1998), 5111-5122 Request permission

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
  • 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.
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 5111-5122
  • MSC (1991): Primary 57N75; Secondary 57R45, 55M10
  • DOI: https://doi.org/10.1090/S0002-9947-98-02088-1
  • MathSciNet review: 1458297