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Transactions of the American Mathematical Society

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Global surjectivity of submersions via contractibility of the fibers


Author: Patrick J. Rabier
Journal: Trans. Amer. Math. Soc. 347 (1995), 3405-3422
MSC: Primary 58C15
DOI: https://doi.org/10.1090/S0002-9947-1995-1321587-3
MathSciNet review: 1321587
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Abstract: We give a sufficient condition for a $ {C^1}$ submersion $ F:X \to Y$, $ X$ and $ Y$ real Banach spaces, to be surjective with contractible fibers $ {F^{ - 1}}(y)$. Roughly speaking, this condition "interpolates" two well-known but unrelated hypotheses corresponding to the two extreme cases: Hadamard's criterion when $ Y \simeq X$ and $ F$ is a local diffeomorphism, and the Palais-Smale condition when $ Y = \mathbb{R}$. These results may be viewed as a global variant of the implicit function theorem, which unlike the local one does not require split kernels. They are derived from a deformation theorem tailored to fit functionals with a norm-like nondifferentiability.


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  • [1] R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, tensor analysis and applications, 2nd ed., Springer-Verlag, New York, 1988. MR 960687 (89f:58001)
  • [2] M. S. Berger and R. A. Plastock, On the singularities of nonlinear Fredholm operators of positive index, Proc. Amer. Math. Soc. 76 (1980), 217-221. MR 565342 (81d:58011)
  • [3] G. E. Bredon, Topology and geometry, Springer-Verlag, New York, 1993. MR 1224675 (94d:55001)
  • [4] D. Burghelea and N. M. Kuiper, Hilbert manifolds, Ann. of Math. 90 (1969), 379-417. MR 0253374 (40:6589)
  • [5] G. Cerami, Un criterio di esistenza per i punti critici su varietà illimitati, Rend. Acad. Sci. Lett. Istituto Lombardo 112 (1978), 332-336.
  • [6] W. J. Davis, D. W. Dean and I. Singer, Complemented subspaces and $ \Lambda $ systems in Banach spaces, Israel J. Math. 6 (1968), 303-309. MR 0234259 (38:2576)
  • [7] K. Deimling, Nonlinear functional analysis, Springer-Verlag, New York, 1985. MR 787404 (86j:47001)
  • [8] C. J. Earle and J. Eells, Foliations and fibrations, J. Differential Geom. 1 (1967), 33-41. MR 0215320 (35:6161)
  • [9] M. W. Hirsch, Differential topology, Springer-Verlag, New York, 1976. MR 0448362 (56:6669)
  • [10] J. Lindenstrauss and L. Tzafriri, On the complemented subspace problem, Israel J. Math. 9 (1971), 263-269. MR 0276734 (43:2474)
  • [11] N. Moulis, Approximation de fonctions differentiables sur certains espaces de Banach, Ann. Inst. Fourier 21 (1971), 293-345. MR 0375379 (51:11573)
  • [12] R. S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115-132. MR 0259955 (41:4584)
  • [13] P. J. Rabier, On global diffeomorphisms of Euclidian space, Nonlinear Analysis, TMA 21 (1993), 925-947. MR 1249211 (95b:57033)
  • [14] G. Restrepo, Differentiable norms in Banach spaces, Bull. Amer. Math. Soc. 70 (1964), 413-414. MR 0161129 (28:4338)
  • [15] E. Zeidler, Nonlinear functional analysis and its applications, Springer-Verlag, New York, 1985. MR 768749 (90b:49005)

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DOI: https://doi.org/10.1090/S0002-9947-1995-1321587-3
Article copyright: © Copyright 1995 American Mathematical Society

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