Pull-Back of Metric Currents and Homological Boundedness of BLD-Elliptic Spaces

Pekka Pankka 1  and Elefterios Soultanis 2
  • 1 University of Helsinki, , Helsinki, Finland
  • 2 SISSA, , University of Freiburg, , Trieste, Italy


Using the duality of metric currents and polylipschitz forms, we show that a BLD-mapping f : XY between oriented cohomology manifolds X and Y induces a pull-back operator f* : Mk,loc(Y) → Mk,loc(X) between the spaces of metric k-currents of locally finite mass. For proper maps, the pull-back is a right-inverse (up to multiplicity) of the push-forward f* : Mk,loc(X) → Mk,loc(Y).

As an application we obtain a non-smooth version of the cohomological boundedness theorem of Bonk and Heinonen for locally Lipschitz contractible cohomology n-manifolds X admitting a BLD-mapping ℝnX.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] Martina Aaltonen and Pekka Pankka. Local monodromy of branched covers and dimension of the branch set. Ann. Acad. Sci. Fenn. Math., 42(1):487–496, 2017.

  • [2] Y. A. Abramovich and C. D. Aliprantis. An invitation to operator theory, volume 50 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.

  • [3] Frederick J. Almgren Jr. Almgren’s big regularity paper, volume 1 of World Scientific Monograph Series in Mathematics. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. Q-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2, With a preface by Jean E. Taylor and Vladimir Scheffer.

  • [4] Luigi Ambrosio and Bernd Kirchheim. Currents in metric spaces. Acta Math., 185(1):1–80, 2000.

  • [5] Mario Bonk and Juha Heinonen. Quasiregular mappings and cohomology. Acta Math., 186(2):219–238, 2001.

  • [6] Glen E. Bredon. Sheaf theory, volume 170 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1997.

  • [7] Camillo De Lellis and Emanuele Nunzio Spadaro. Q-valued functions revisited. Mem. Amer. Math. Soc., 211(991):vi+79, 2011.

  • [8] Th. De Pauw, R. M. Hardt, and W. F. Pfeffer. Homology of normal chains and cohomology of charges. Mem. Amer. Math. Soc., 247(1172):v+115, 2017.

  • [9] Herbert Federer and Wendell H. Fleming. Normal and integral currents. Ann. of Math. (2), 72:458–520, 1960.

  • [10] Misha Gromov. Metric structures for Riemannian and non-Riemannian spaces. Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston, MA, english edition, 2007. Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates.

  • [11] Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002.

  • [12] Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, and Jeremy Tyson. Sobolev spaces on metric measure spaces: an approach based on upper gradients. New Mathematical Monographs. Cambridge University Press, United Kingdom, first edition, 2015.

  • [13] Juha Heinonen and Seppo Rickman. Geometric branched covers between generalized manifolds. Duke Math. J., 113(3):465–529, 2002.

  • [14] Sze-tsen Hu. Theory of retracts. Wayne State University Press, Detroit, 1965.

  • [15] I. Kangasniemi. Sharp cohomological bound for uniformly quasiregularly elliptic manifolds. ArXiv e-prints, November 2017.

  • [16] Bernd Kirchheim. Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Amer. Math. Soc., 121(1):113–123, 1994.

  • [17] Urs Lang. Local currents in metric spaces. J. Geom. Anal., 21(3):683–742, 2011.

  • [18] Rami Luisto. A characterization of BLD-mappings between metric spaces. J. Geom. Anal., 27(3):2081–2097, 2017.

  • [19] O. Martio and J. Väisälä. Elliptic equations and maps of bounded length distortion. Math. Ann., 282(3):423–443, 1988.

  • [20] P. Mattila and S. Rickman. Averages of the counting function of a quasiregular mapping. Acta Math., 143(3-4):273–305, 1979.

  • [21] Ayato Mitsuishi. The coincidence of the current homology and the measure homology via a new topology on spaces of Lipschitz maps. arXiv:1403.5518 [math.AT].

  • [22] Jani Onninen and Kai Rajala. Quasiregular mappings to generalized manifolds. J. Anal. Math., 109:33–79, 2009.

  • [23] Pekka Pankka. Mappings of bounded mean distortion and cohomology. Geom. Funct. Anal., 20(1):229–242, 2010.

  • [24] Pekka Pankka and Elefterios Soultanis. Metric currents and Polylipschitz forms. arXiv:1902.06106 [math.MG].

  • [25] Eden Prywes. A Bound on the Cohomology of Quasiregularly Elliptic Manifolds. Preprint. arXiv:1806.05306 [math.DG].

  • [26] Seppo Rickman. Quasiregular mappings, volume 26 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1993.

  • [27] Jussi Väisälä. Discrete open mappings on manifolds. Ann. Acad. Sci. Fenn. Ser. A I No., 392:10, 1966.

  • [28] Frank W. Warner. Foundations of differentiable manifolds and Lie groups, volume 94 of Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition.

  • [29] Stefan Wenger. Isoperimetric inequalities of Euclidean type in metric spaces. GAFA, 15(2):534–554, 2005.

  • [30] Stefan Wenger. Flat convergence for integral currents in metric spaces. Calc. Var. Partial Differential Equations, 28(2):139–160, 2007.


Journal + Issues

Analysis and Geometry in Metric Spaces (AGMS) is a fully peer-reviewed, open access electronic journal that publishes cutting-edge original research on analytical and geometrical problems in metric spaces and their mathematical applications. It features articles making connections among relevant topics in this field.