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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.

##### MSC:
 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 57P99 Generalized manifolds 57M12 Low-dimensional topology of special (e.g., branched) coverings
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##### References:
 [1] J. W. Alexander, Note on Riemannian spaces , Bull. Amer. Math. Soc. 26 (1920), 370–372. · JFM 47.0529.02 [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 [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 [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 [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 [6] C. J. Bishop and P. W. Jones, Hausdorff dimension and Kleinian groups , Acta Math. 179 (1997), 1–39. · Zbl 0921.30032 [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 [10] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces , Geom. Funct. Anal. 9 (1999), 428–517. · Zbl 0942.58018 [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 [14] R. J. Daverman, Decompositions of Manifolds , Pure Appl. Math. 124 , Academic Press, Orlando, Fla., 1986. · Zbl 0608.57002 [15] G. David, OpÃ©rateurs d’intÃ©grale singuliÃ\"re sur les surfaces rÃ©guliÃ\"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-delÃ des graphes lipschitziens , AstÃ©risque 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 [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 [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 PoincarÃ©, 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 [24] â EUR”, Definitions of quasiconformality , Invent. Math. 120 (1995), 61–79. · Zbl 0832.30013 [25] â EUR”, Quasiconformal maps in metric spaces with controlled geometry , Acta Math. 181 (1998), 1–61. · Zbl 0915.30018 [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 [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 [28] J. Heinonen and D. Sullivan, On the locally branched Euclidean metric gauge , to appear in Duke Math. J. · Zbl 1019.58002 [29] U. Hirsch, On branched coverings of the $$3$$-sphere , Math. Z. 157 (1977), 225–236. · Zbl 0347.57002 [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 [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 [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 [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 [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 [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. PoincarÃ©Anal. Non LinÃ©aire 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 [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 [48] â EUR”, Finding curves on general spaces through quantitative topology, with applications for Sobolev and PoincarÃ©inequalities , Selecta Math. (N.S.) 2 (1996), 155–295. · Zbl 0870.54031 [49] â EUR”, Good metric spaces without good parameterizations , Rev. Mat. Iberoamericana 12 (1996), 187–275. · Zbl 0854.57018 [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 [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 [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 [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 [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 [62] E. Villamor and J. J. Manfredi, An extension of Reshetnyak’s theorem , Indiana Univ. Math. J. 47 (1998), 1131–1145. · Zbl 0931.30014
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