Geometric branched covers between generalized manifolds. (English) Zbl 1017.30023
The authors develop a basic theory of geometrically controlled branched covers between generalized metric manifolds. A generalized \(n\)-manifold is a finite-dimensional locally compact Hausdorff space whose local cohomology groups over the integers in dimensions \(n-1\) and higher are similar to those of an \(n\)-manifold. The authors study the relationship between regular maps and maps of bounded length distortion, or BLD-maps, on the generalized manifolds. It turns out that under quite general circumstances, the sense-preserving regular maps are precisely the BLD-maps of finite maximal multiplicity. Examples that illustrate the relationship between various classes of maps are given. A particular kind of generalized metric manifolds, called spaces of “type A”, is introduced. It is shown that BLD-maps between such spaces behave in many respects like BLD-maps between Riemannian manifolds. In particular, for maps from spaces of type A into \(\mathbb R^n\), an analytic description akin to the one used by Martio and Väisälä in \(\mathbb R^n\) is available. Some further analytic and metric properties of BLD-maps between spaces of type A, including a result on value distribution are pointed out. A proof of the generalized Berstein-Edmonds theorem together with some examples and applications are given. In particular, the authors carefully study the geometric decomposition spaces of Semmes, and show how they admit BLD-branched coordinates in \(\mathbb R^3\). It is also pointed out how one obtains bounded quasiregular maps in the open 3-ball in \(\mathbb R^3\) which fail to have radial limits in a large set on the boundary. Moreover, Lipschitz maps \(S^3\to S^3\) with nonzero degree having as fibers some strange continua, for example, the Whitehead continuum, but with dilatation function in a local Lebesgue space \(L^p\) for \(p\) arbitrarily close to 2 are constructed. These examples are quite interesting in view of the fact that each Lipschitz map \(S^3\to S^3\) is a branched cover if the integrability exponent for the dilatation exceeds 2. Finally, preserving the Poincaré inequalities by BLD-maps is discussed.
Reviewer: Vasily A.Chernecky (Odessa)
MSC:
| 30C65 | Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations |
| 57P99 | Generalized manifolds |
| 57M12 | Low-dimensional topology of special (e.g., branched) coverings |
Keywords:
regular maps; maps of bounded length distortion; sense-preserving regular maps; generalized manifolds; orientation; local degree; sense-preserving branched covers; geometrically controlled branched covers; change-of-variables formula; approximate differentiability; separating complex sheets; geometric decomposition spaces; metric orientation; radial limits; Lipschitz map with strange fibers; Poincaré inequalitiesReferences:
| [1] | J. W. Alexander, Note on Riemannian spaces , Bull. Amer. Math. Soc. 26 (1920), 370–372. · JFM 47.0529.02 · doi:10.1090/S0002-9904-1920-03319-7 |
| [2] | L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces , Math. Ann. 318 (2000), 527–555. \CMP1 800 768 · Zbl 0966.28002 · doi:10.1007/s002080000122 |
| [3] | J. M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter , Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 315–328. · Zbl 0478.46032 · doi:10.1017/S030821050002014X |
| [4] | S. Bates, W. B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman, Affine approximation of Lipschitz functions and nonlinear quotients , Geom. Funct. Anal. 9 (1999), 1092–1127. · Zbl 0954.46014 · doi:10.1007/s000390050108 |
| [5] | I. Berstein and A. L. Edmonds, On the construction of branched coverings of low-dimensional manifolds , Trans. Amer. Math. Soc. 247 (1979), 87–124. JSTOR: · Zbl 0359.55001 · doi:10.2307/1998776 |
| [6] | C. J. Bishop and P. W. Jones, Hausdorff dimension and Kleinian groups , Acta Math. 179 (1997), 1–39. · Zbl 0921.30032 · doi:10.1007/BF02392718 |
| [7] | A. Borel, Seminar on Transformation Groups , Ann. of Math. Stud. 46 , Princeton Univ. Press, Princeton, 1960. · Zbl 0091.37202 |
| [8] | G. E. Bredon, Sheaf Theory , 2d ed., Grad. Texts in Math. 170 , Springer, New York, 1997. · Zbl 0874.55001 |
| [9] | R. D. Canary, Ends of hyperbolic \(3\)-manifolds , J. Amer. Math. Soc. 6 (1993), 1–35. JSTOR: · Zbl 0810.57006 · doi:10.2307/2152793 |
| [10] | J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces , Geom. Funct. Anal. 9 (1999), 428–517. · Zbl 0942.58018 · doi:10.1007/s000390050094 |
| [11] | A. V. ÄŒernavskiÄ-, Finite-to-one open mappings of manifolds (in Russian), Mat. Sb. (N.S.) 65 ( 107 ) (1964), 357–369. · Zbl 0129.15003 |
| [12] | â EUR”, Addendum to the paper “Finite-to-one open mappings of manifolds” (in Russian), Mat. Sb. (N.S.) 66 ( 108 ) (1965), 471–472. |
| [13] | P. T. Church, Discrete maps on manifolds , Michigan Math. J. 25 (1978), 351–357. · Zbl 0371.54024 · doi:10.1307/mmj/1029002116 |
| [14] | R. J. Daverman, Decompositions of Manifolds , Pure Appl. Math. 124 , Academic Press, Orlando, Fla., 1986. · Zbl 0608.57002 |
| [15] | G. David, Opérateurs d’intégrale singuliÃ\"re sur les surfaces réguliÃ\"res , Ann. Sci. Ã\?cole Norm. Sup. (4) 21 (1988), 225–258. |
| [16] | G. David and S. Semmes, Singular Integrals and Rectifiable Sets in \(R^n\): Au-delà des graphes lipschitziens , Astérisque 193 , Soc. Math. France, Montrouge, 1991. · Zbl 0743.49018 |
| [17] | â EUR”, Quantitative rectifiability and Lipschitz mappings , Trans. Amer. Math. Soc. 337 (1993), 855–889. · Zbl 0792.49029 · doi:10.2307/2154247 |
| [18] | â EUR”, Fractured Fractals and Broken Dreams: Self-Similar Geometry through Metric and Measure , Oxford Lecture Ser. Math. Appl. 7 , Oxford Univ. Press, New York, 1997. · Zbl 0887.54001 |
| [19] | â EUR”, Regular mappings between dimensions , Publ. Mat. 44 (2000), 369–417. · Zbl 1041.42010 · doi:10.5565/PUBLMAT_44200_02 |
| [20] | A. Dold, Lectures on Algebraic Topology , Grundlehren Math. Wiss. 200 , Springer, New York, 1972. · Zbl 0234.55001 |
| [21] | H. Federer, Geometric Measure Theory , Grundlehren Math. Wiss. 153 , Springer, New York, 1969. · Zbl 0176.00801 |
| [22] | P. HajÅ\?asz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688. · Zbl 0954.46022 |
| [23] | J. Heinonen and P. Koskela, Sobolev mappings with integrable dilatations , Arch. Rational Mech. Anal. 125 (1993), 81–97. · Zbl 0792.30016 · doi:10.1007/BF00411478 |
| [24] | â EUR”, Definitions of quasiconformality , Invent. Math. 120 (1995), 61–79. · Zbl 0832.30013 · doi:10.1007/BF01241122 |
| [25] | â EUR”, Quasiconformal maps in metric spaces with controlled geometry , Acta Math. 181 (1998), 1–61. · Zbl 0915.30018 · doi:10.1007/BF02392747 |
| [26] | J. Heinonen and S. Rickman, Quasiregular maps \(\mathbfS^3 \to \mathbfS^3\) with wild branch sets , Topology 37 (1998), 1–24. · Zbl 0895.30016 · doi:10.1016/S0040-9383(97)00015-3 |
| [27] | J. Heinonen and S. Semmes, Thirty-three yes or no questions about mappings, measures, and metrics , Conform. Geom. Dyn. 1 (1997), 1–12., · Zbl 0885.00006 · doi:10.1090/S1088-4173-97-00012-X |
| [28] | J. Heinonen and D. Sullivan, On the locally branched Euclidean metric gauge , to appear in Duke Math. J. · Zbl 1019.58002 · doi:10.1215/S0012-7094-02-11412-4 |
| [29] | U. Hirsch, On branched coverings of the \(3\)-sphere , Math. Z. 157 (1977), 225–236. · Zbl 0347.57002 · doi:10.1007/BF01214354 |
| [30] | J. F. P. Hudson, Piecewise Linear Topology , Univ. Chicago Math. Lecture Notes, Benjamin, New York, 1969. · Zbl 0189.54507 |
| [31] | W. Hurewicz and H. Wallman, Dimension Theory , Princeton Math. Ser. 4 , Princeton Univ. Press, Princeton, 1941. · Zbl 0060.39808 |
| [32] | T. Iwaniec and V. Å verÃ\?k, On mappings with integrable dilatation , Proc. Amer. Math. Soc. 118 (1993), 181–188. JSTOR: · Zbl 0784.30015 · doi:10.2307/2160025 |
| [33] | B. Kirchheim, Rectifiable metric spaces: Local structure and regularity of the Hausdorff measure , Proc. Amer. Math. Soc. 121 (1994), 113–123. JSTOR: · Zbl 0806.28004 · doi:10.2307/2160371 |
| [34] | T. J. Laakso, Plane with \(A_\infty\)-weighted metric not bilipschitz embeddable to \(R^n\) , to appear in Bull. London Math. Soc. · Zbl 1029.30014 · doi:10.1112/S0024609302001200 |
| [35] | J. J. Manfredi and E. Villamor, Mappings with integrable dilatation in higher dimensions , Bull. Amer. Math. Soc. (N.S.) 32 (1995), 235–240. · Zbl 0857.30020 · doi:10.1090/S0273-0979-1995-00583-5 |
| [36] | O. Martio and U. Srebro, Locally injective automorphic mappings in \(\mathbfR^n\) , Math. Scand. 85 (1999), 49–70. · Zbl 0953.30010 |
| [37] | O. Martio and J. VÃ\?isÃ\?lÃ\?, Elliptic equations and maps of bounded length distortion , Math. Ann. 282 (1988), 423–443. · Zbl 0632.35021 · doi:10.1007/BF01460043 |
| [38] | S. MÃ\frac{1}{4} ller, T. Qi [Q. Tang], and B. S. Yan, On a new class of elastic deformations not allowing for cavitation , Ann. Inst. H. PoincaréAnal. Non Linéaire 11 (1994), 217–243. · Zbl 0863.49002 |
| [39] | J. R. Munkres, Elementary Differential Topology , Ann. of Math. Stud. 54 , Princeton Univ. Press, Princeton, 1966. · Zbl 0161.20201 |
| [40] | E. A. PoleckiÄ- [PoletskiÄ-], The method of moduli for nonhomeomorphic quasiconformal mappings (in Russian), Mat. Sb. (N.S.) 83 ( 125 ) (1970), 261–272. |
| [41] | Ju. G. ReÅ\?etnjak [Yu. G. Reshetnyak], Spatial mappings with bounded distortion (in Russian), Sibirsk. Mat. Å\frac{1}{2} . 8 (1967), 629–658. |
| [42] | â EUR”, Space Mappings with Bounded Distortion , Transl. Math. Monogr. 73 , Amer. Math. Soc., Providence, 1989. · Zbl 0667.30018 |
| [43] | S. Rickman, The analogue of Picard’s theorem for quasiregular mappings in dimension three , Acta Math. 154 (1985), 195–242. · Zbl 0617.30024 · doi:10.1007/BF02392472 |
| [44] | â EUR”, Quasiregular Mappings , Ergeb. Math. Grenzgeb. (3) 26 , Springer, Berlin, 1993. |
| [45] | â EUR”, “Construction of quasiregular mappings” in Quasiconformal Mappings and Analysis (Ann Arbor, Mich., 1995) , Springer, New York, 1998, 337–345. · Zbl 0888.30018 |
| [46] | H. Seifert and W. Threlfall, Seifert and Threlfall: A Textbook of Topology , Pure Appl. Math. 89 , Academic Press, New York, 1980. · Zbl 0469.55001 |
| [47] | S. Semmes, Chord-arc surfaces with small constant, II: Good parameterizations , Adv. Math. 88 (1991), 170–199. · Zbl 0733.42016 · doi:10.1016/0001-8708(91)90007-T |
| [48] | â EUR”, Finding curves on general spaces through quantitative topology, with applications for Sobolev and Poincaréinequalities , Selecta Math. (N.S.) 2 (1996), 155–295. · Zbl 0870.54031 · doi:10.1007/BF01587936 |
| [49] | â EUR”, Good metric spaces without good parameterizations , Rev. Mat. Iberoamericana 12 (1996), 187–275. · Zbl 0854.57018 · doi:10.4171/RMI/198 |
| [50] | â EUR”, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about \(A_\infty\)-weights , Rev. Mat. Iberoamericana 12 (1996), 337–410. · Zbl 0858.46017 · doi:10.4171/RMI/201 |
| [51] | â EUR”, Some Novel Types of Fractal Geometry , Oxford Math. Monogr., Clarendon Press, Oxford, 2001. \CMP1 815 356 · Zbl 0970.28001 |
| [52] | N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces , Rev. Mat. Iberoamericana 16 (2000), 243–279. · Zbl 0974.46038 · doi:10.4171/RMI/275 |
| [53] | L. Siebenmann and D. Sullivan, “On complexes that are Lipschitz manifolds” in Geometric Topology (Athens, Ga., 1977) , Academic Press, New York, 1979, 503–525. · Zbl 0478.57008 |
| [54] | D. Sullivan, “Hyperbolic geometry and homeomorphisms” in Geometric Topology (Athens, Ga., 1977) , Academic Press, New York, 1979, 543–555. · Zbl 0478.57007 |
| [55] | C. J. Titus and G. S. Young, The extension of interiority, with some applications , Trans. Amer. Math. Soc. 103 (1962), 329–340. JSTOR: · Zbl 0113.38001 · doi:10.2307/1993663 |
| [56] | T. Toro, Surfaces with generalized second fundamental form in \(L^2\) are Lipschitz manifolds , J. Differential Geom. 39 (1994), 65–101. · Zbl 0806.53020 |
| [57] | â EUR”, Geometric conditions and existence of bi-Lipschitz parameterizations , Duke Math. J. 77 (1995), 193–227. · Zbl 0847.42011 · doi:10.1215/S0012-7094-95-07708-4 |
| [58] | J. VÃ\?isÃ\?lÃ\?, Minimal mappings in euclidean spaces , Ann. Acad. Sci. Fenn. Ser. A I Math. 1965 , no. 366. · Zbl 0144.22103 |
| [59] | â EUR”, Discrete open mappings on manifolds , Ann. Acad. Sci. Fenn. Ser. A I Math. 1966 , no. 392. · Zbl 0144.22202 |
| [60] | â EUR”, Lectures on \(n\)-dimensional Quasiconformal Mappings , Lecture Notes in Math. 229 , Springer, Berlin, 1971. · Zbl 0221.30031 |
| [61] | â EUR”, Local topological properties of countable mappings , Duke Math. J. 41 (1974), 541–546. · Zbl 0288.54012 · doi:10.1215/S0012-7094-74-04159-3 |
| [62] | E. Villamor and J. J. Manfredi, An extension of Reshetnyak’s theorem , Indiana Univ. Math. J. 47 (1998), 1131–1145. · Zbl 0931.30014 · doi:10.1512/iumj.1998.47.1323 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
