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On the Uniformly Proper Classification of Open Manifolds

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We give a brief account on the uniformly proper classification of open manifolds, i.e., the classification under bounded, uniformly proper maps. The small category of diffeomorphism classes of open n-manifolds has uncountably many homotopy types, n ≥ 2. Our main approach consists in splitting this set into generalized components and then trying to classify these components and thereafter the elements inside a component. To define these components, we introduce Gromov–Hausdorff and Lipschitz metrizable uniform structures and corresponding GH- and L-cohomologies. The GH-components are particularly appropriate to introduce geometric bordism theory for open manifolds, and the L-components are appropriate to establish surgery. We present independent generators for the bordism groups. The fundamental contributions of Farrell, Wagoner, Siebenmann, Maumary, and Taylor play a decisive role. In our approach, we suppose the manifolds to be endowed with a metric of bounded geometry and restrict ourselves to bounded uniformly proper morphisms. Finally, we discuss the question under which conditions bounded geometry and uniform properness are preserved by surgery, and sketch some proper surgery groups.

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References

  1. O. Attie, “Quasi-isometry classification of some manifolds of bounded geometry,” Math. Z., 216, 501–527 (1994).

    Article  MathSciNet  Google Scholar 

  2. O. Attie, A Surgery Theory for Manifolds of Bounded Geometry, Preprint, arXiv:math/0312017v3 (2008).

    Google Scholar 

  3. T. Aubin, “Espaces de Sobolev sur les varietes Riemanniennes,” Bull. Soc. Math. (France), 100, 149–173 (1976).

    MathSciNet  MATH  Google Scholar 

  4. D. de Baun, “L2-cohomology of noncompact surfaces,” Trans. Am. Math. Soc., 284, 543–565 (1984).

    Google Scholar 

  5. J. Cheeger and M. Gromov, “On the characteristic numbers of complete manifolds of bounded geometry and finite volume,” Differential Geometry and Complex Analysis, Springer, Berlin (1985), pp. 115–154.

    MATH  Google Scholar 

  6. J. Cheeger and M. Gromov, “Chopping Riemannian manifolds,” Differential Geometry, Pitman Monogr. Surv. Pure Appl. Math., Vol. 52, Longman Sci. Tech., Harlow (1991), pp. 85–94.

  7. J. Cheeger, M. Gromov, and M. Taylor, “Finite propagation speed, kernel estimates for functions of the Laplacian and geometry of complete Riemannian manifolds,” J. Differ. Geom., 17, 15–53 (1983).

    Article  Google Scholar 

  8. J. Dodziuk, “Finite-difference approach to the Hodge theory of harmonic forms,” Am. J. Math., 98, 79–104 (1976).

    Article  MathSciNet  Google Scholar 

  9. J. Dodziuk, “Sobolev spaces of differential forms and the de Rham–Hodge isomorphism,” J. Differ. Geom., 16, 63–73 (1981).

    Article  MathSciNet  Google Scholar 

  10. J. Dodziuk and V. K. Patodi, “Riemannian structures and triangulation of manifolds,” J. Indian Math. Soc., 40, 1–52 (1976).

    MathSciNet  MATH  Google Scholar 

  11. J. Eichhorn, “The manifold structure of maps between open manifolds,” Ann. Global Anal. Geom., 11, 253–300 (1993).

    MathSciNet  MATH  Google Scholar 

  12. J. Eichhorn, “Spaces of Riemannian metrics on open manifolds,” Results Math., 27, 256–283 (1995).

    Article  MathSciNet  Google Scholar 

  13. J. Eichhorn, “Uniform structures of metric spaces and open manifolds,” Results Math., 40, 144–191 (2001).

    Article  MathSciNet  Google Scholar 

  14. J. Eichhorn, “Invariants for proper metric spaces and open Riemannian manifolds,” Math. Nachr., 253, 8–34 (2003).

    Article  MathSciNet  Google Scholar 

  15. J. Eichhorn, “Characteristic numbers, bordism theory and the Novikov conjecture for open manifolds,” Banach Center Publ., 76, 269–312 (2007).

    Article  MathSciNet  Google Scholar 

  16. J. Eichhorn, Global Analysis on Open Manifolds, Nova Sci. Publ., New York (2007).

    MATH  Google Scholar 

  17. J. Eichhorn, On the Classification of Open Manifolds, in preparation.

  18. J.-H. Eschenburg, Comparison Theorems in Riemannian Geometry, Lect. Notes Ser. Dip. Mat. Univ. Studi Trento, No. 3, Trento (1994).

  19. F. T. Farrell, L. Taylor, and J. Wagoner, “The Whitehead theorem in the proper category,” Compositio Math., 27, 1–23 (1973).

    Article  MathSciNet  Google Scholar 

  20. F. T. Farrell and J. Wagoner, “Algebraic torsion for infinite simple homotopy types,” Commun. Math. Helv., 47, 502–513 (1972).

    Article  MathSciNet  Google Scholar 

  21. F. T. Farrell and J.Wagoner, “Infinite matrices in algebraic K-theory and topology,” Commun. Math. Helv., 47, 474–501 (1972).

    Article  MathSciNet  Google Scholar 

  22. V. M. Golstein, V. I. Kuzminov, and I. A. Shvedov, “The de Rham isomorphism of Lp–cohomology of noncompact Riemannian manifolds,” Sib. Math. J., 29, 34–44 (1988).

    Google Scholar 

  23. R. Greene, “Complete metrics of bounded curvature on noncompact manifolds,” Arch. Math. (Basel), 31, 89–95 (1978/79).

    Article  MathSciNet  Google Scholar 

  24. R. Grimaldi and P. Pansu, Bounded Geometry, Growth and Topology, arXiv:math/1008.4887v1.

  25. M. Gromov, J. Lafontaine, and P. Pansu, Stuctures metriques pour les varietes Riemanniennes, Math. Texts 1, CEDIC, Paris (1981).

  26. N. Higson and J. Roe, “Mapping surgery to analysis I: analytic signatures,” K-Theory, 33, 277–299 (2004).

    Article  MathSciNet  Google Scholar 

  27. N. Higson and J. Roe, “Mapping surgery to analysis II: geometric signatures,” K-Theory, 33, 301–324 (2004).

    Article  MathSciNet  Google Scholar 

  28. N. Higson, J. Roe, “Mapping surgery to analysis III: exact sequences,” K-Theory, 33, 325–346 (2004).

    Article  MathSciNet  Google Scholar 

  29. P. J. Hilton and S. Wylie, Homology Theory, New York (1960).

  30. B. Hughes and A. Ranicki, Ends of Complexes, Cambridge Tracts Math., Vol. 123, Cambridge (1996).

  31. S. Maumary, Proper surgery groups for non-compact manifolds of finite dimension, Preprint U.C., Berkeley (1972).

    Google Scholar 

  32. S. Maumary, “Proper surgery groups andWall–Novikov groups,” in: Algebraic K-Theory, Lect. Notes Math., Vol. 343, Springer, Berlin (1973), pp. 526–539.

  33. R. E. Megginson, An Introduction to Banach Space Theory, Springer, Berlin (1988).

    MATH  Google Scholar 

  34. E. Pedersen and A. Ranicki, “Projective surgery theory,” Topology, 19, 239–254 (1980).

    Article  MathSciNet  Google Scholar 

  35. A. Ranicki and D. Sullivan, “A semilocal combinatorial formula for the signature of a 4k–manifold,” J. Differ. Geom., 11, 23–29 (1976).

    Article  Google Scholar 

  36. I. Richards, “On the classification of noncompact surfaces,” Trans. Am. Math. Soc., 106, 259–269 (1963).

    Article  MathSciNet  Google Scholar 

  37. W. Rinow, Lehrbuch der Topologie, Deutscher Verlag d. Wiss., Berlin (1975).

    MATH  Google Scholar 

  38. J. Roe, Coarse Cohomology and Index Theory on Complete Riemannian Manifolds, Mem. Amer. Math. Soc., Vol. 497, Amer. Math. Soc., Providence (1993).

  39. J. Roe, Index Theory, Coarse Geometry, and Topology of Manifolds, CBMC Reg. Conf. Ser. Math., Vol. 90, Amer. Math. Soc., Providence (1996).

  40. T. Schick, Analysis on ∂–manifolds of bounded geometry, Hodge–de Rham isomorphism and L2–index theorem, Thesis, Mainz (1996).

    MATH  Google Scholar 

  41. H. Schubert, Topologie, Teubner, Stuttgart (1964).

    MATH  Google Scholar 

  42. L. Siebenmann, “Infinite simple homotopy types,” Indag. Math., 32, 479–495 (1970).

    Article  MathSciNet  Google Scholar 

  43. L. Siebenmann, The Obstruction to Finding a Boundary for an Open Manifold of Dimension Greater than Five, Thesis, Princeton U. (1965).

    Google Scholar 

  44. R. Stöcker and H. Zieschang, Algebraische Topologie, Teubner, Stuttgart (1994).

    Book  Google Scholar 

  45. R. Stong, Notes on Cobordism Theory, Math. Notes, Princeton Univ. Press, Princeton (1968).

    MATH  Google Scholar 

  46. L. R. Taylor, Surgery on Paracompact Manifolds, Thesis, U.C. Berkeley (1971).

    Google Scholar 

  47. L. Vanhecke, “Geometry in normal and tubular neighborhoods,” Rend. Sem. Fac. Sci. Univ. Cagliari, 58, Suppl., 73–176 (1988).

  48. C. T. C. Wall, Surgery on Compact Manifolds, Academic Press, New York (1970).

    MATH  Google Scholar 

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  1. Institut für Mathematik und Informatik, Walter-Rathenau-Strasse 47, D-17487, Greifswald, Germany

    J. Eichhorn

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  1. J. Eichhorn
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Correspondence to J. Eichhorn.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 6, pp. 3–63, 2016.

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Eichhorn, J. On the Uniformly Proper Classification of Open Manifolds. J Math Sci 248, 668–708 (2020). https://doi.org/10.1007/s10958-020-04905-y

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