## 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 $f(R_{q}(f))$.## References

- P. T. Church,
*Differentiable open maps on manifolds*, Trans. Amer. Math. Soc.**109**(1963), 87–100. MR**154296**, DOI 10.1090/S0002-9947-1963-0154296-6 - P. T. Church,
*On points of Jacobian rank $k$*, Trans. Amer. Math. Soc.**110**(1964), 413–423. MR**157390**, DOI 10.1090/S0002-9947-1964-0157390-X - P. T. Church,
*On points of Jacobian rank $k$. II*, Proc. Amer. Math. Soc.**16**(1965), 1035–1038. MR**182017**, DOI 10.1090/S0002-9939-1965-0182017-7 - S. K. Donaldson and P. B. Kronheimer,
*The geometry of four-manifolds*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR**1079726** - M. Golubitsky and V. Guillemin,
*Stable mappings and their singularities*, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR**0341518** - Morris W. Hirsch,
*Differential topology*, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR**1336822** - Sergio Sispanov,
*Generalización del teorema de Laguerre*, Bol. Mat.**12**(1939), 113–117 (Spanish). MR**3** - James Munkres,
*Obstructions to the smoothing of piecewise-differentiable homeomorphisms*, Bull. Amer. Math. Soc.**65**(1959), 332–334. MR**112150**, DOI 10.1090/S0002-9904-1959-10345-1 - J. J. Corliss,
*Upper limits to the real roots of a real algebraic equation*, Amer. Math. Monthly**46**(1939), 334–338. MR**4** - Arthur Sard,
*Images of critical sets*, Ann. of Math. (2)**68**(1958), 247–259. MR**100063**, DOI 10.2307/1970246 - Arthur Sard,
*Hausdorff measure of critical images on Banach manifolds*, Amer. J. Math.**87**(1965), 158–174. MR**173748**, DOI 10.2307/2373229 - Edwin H. Spanier,
*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112** - H. Whitney,
*Differentiable manifolds*, Ann. of Math.**37**(1936), 645–680.

## 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