Isotoping mappings to open mappings
HTML articles powered by AMS MathViewer
- by John J. Walsh PDF
- Trans. Amer. Math. Soc. 250 (1979), 121-145 Request permission
Abstract:
Let f be a quasi-monotone mapping from a compact, connected manifold ${M^m} (m \geqslant 3)$ onto a space Y; then there is an open mapping g from M onto Y such that, for each $y \in Y, {g^{ - 1}}(y)$ is not a point and ${g^{ - 1}}(y)$ and ${f^{ - 1}}(y)$ are equivalently embedded in M (in particular, ${g^{ - 1}}(y)$ and ${f^{ - 1}}(y)$ have the same shape). Applying the result with f equal to the identity mapping on M yields a continuous decomposition of M into cellular sets each of which is not a point.References
- R. D. Anderson, Open mappings of compact continua, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 347β349. MR 78682, DOI 10.1073/pnas.42.6.347
- R. D. Anderson, Monotone interior dimension-raising mappings, Duke Math. J. 19 (1952), 359β366. MR 48798, DOI 10.1215/S0012-7094-52-01936-4
- R. D. Anderson, Continuous collections of continuous curves, Duke Math. J. 21 (1954), 363β367. MR 62429, DOI 10.1215/S0012-7094-54-02136-5
- R. D. Anderson, On monotone interior mappings in the plane, Trans. Amer. Math. Soc. 73 (1952), 211β222. MR 50269, DOI 10.1090/S0002-9947-1952-0050269-5
- R. H. Bing, Complementary domains of continuous curves, Fund. Math. 36 (1949), 303β318. MR 38063, DOI 10.4064/fm-36-1-303-318
- Ludmila Keldych, Transformation of a monotone irreducible mapping into a monotone-interior mapping and a monotone-interior mapping of the cube onto the cube of higher dimension, Dokl. Akad. Nauk SSSR (N.S.) 114 (1957), 472β475 (Russian). MR 0091455
- K. Kuratowski and R. C. Lacher, A theorem on the space of monotone mappings, Bull. Acad. Polon. Sci. SΓ©r. Sci. Math. Astronom. Phys. 17 (1969), 797β800 (English, with Russian summary). MR 275386
- Louis F. McAuley, Open mappings and open problems, Topology Conference (Arizona State Univ., Tempe, Ariz., 1967) Arizona State Univ., Tempe, Ariz., 1968, pp.Β 184β202. MR 0240789 . A. B. Sosinskii, Monotonically-open mappings of a sphere,, Amer. Math. Soc. Transl. (2) 78 (1968), 67-101.
- John J. Walsh, Monotone and open mappings on manifolds. I, Trans. Amer. Math. Soc. 209 (1975), 419β432. MR 375326, DOI 10.1090/S0002-9947-1975-0375326-0
- John J. Walsh, Light open and open mappings on manifolds. II, Trans. Amer. Math. Soc. 217 (1976), 271β284. MR 394674, DOI 10.1090/S0002-9947-1976-0394674-2
- John J. Walsh, Monotone and open mappings onto $ANRβs$, Proc. Amer. Math. Soc. 60 (1976), 286β288 (1977). MR 425888, DOI 10.1090/S0002-9939-1976-0425888-6
- David Wilson, Open mappings on manifolds and a counterexample to the Whyburn conjecture, Duke Math. J. 40 (1973), 705β716. MR 320989
- David C. Wilson, Open mappings of the universal curve onto continuous curves, Trans. Amer. Math. Soc. 168 (1972), 497β515. MR 298630, DOI 10.1090/S0002-9947-1972-0298630-0
- Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR 0007095, DOI 10.1090/coll/028 . β, Topological analysis, Princeton Univ. Press, Princeton, N. J., 1958.
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 250 (1979), 121-145
- MSC: Primary 57N37; Secondary 54C10, 57N25, 57N60
- DOI: https://doi.org/10.1090/S0002-9947-1979-0530046-8
- MathSciNet review: 530046