On the uniformly proper classification of open manifolds

J. Eichhorn

Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 6, pp. 3-63.

Résumé

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, $n\ge2$. Our main approach consists in splitting this set into generalized components and then try 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 $\mathrm{GH}$- and $\mathrm{L}$-cohomologies. The $\mathrm{GH}$-components are particularly appropriate to introduce geometric bordism theory for open manifolds, the $\mathrm{L}$-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.

Bibliographie

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

[2] Attie O., A surgery theory for manifolds of bounded geometry, 2008, arXiv: math/0312017v3

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

[4] De Baun D., “$L_2$-cohomology of noncompact surfaces”, Trans. Amer. Math. Soc., 284 (1984), 543–565 | MR

[5] Cheeger J., Gromov M., “On the characteristic numbers of complete manifolds of bounded geometry and finite volume”, Differential Geometry and Complex Analysis, Springer, Berlin, 1985, 115–154 | DOI | MR

[6] Cheeger J., Gromov M., “Chopping Riemannian manifolds”, Differential Geometry, Pitman Monogr. Surv. Pure Appl. Math., 52, Longman Sci. Tech., Harlow, 1991, 85–94 | MR

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

[8] Dodziuk J., “Finite-difference approach to the Hodge theory of harmonic forms”, Amer. J. Math., 98 (1976), 79–104 | DOI | MR | Zbl

[9] Dodziuk J., “Sobolev spaces of differential forms and the de Rham–Hodge isomorphism”, J. Diff. Geom., 16 (1981), 63–73 | DOI | MR | Zbl

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

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

[12] Eichhorn J., “Spaces of Riemannian metrics on open manifolds”, Results Math., 27 (1995), 256–283 | DOI | MR | Zbl

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

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

[15] Eichhorn J., “Characteristic numbers, bordism theory and the Novikov conjecture for open manifolds”, Banach Center Publ., 76, Polish Acad. Sci., Warsaw, 2007, 269–312 | DOI | MR | Zbl

[16] Eichhorn J., Global Analysis on Open Manifolds, Nova Sci. Publ., New York, 2007 | MR | Zbl

[17] Eichhorn J., On the Classification of Open Manifolds, In preparation | MR

[18] Eschenburg J.-H., Comparison Theorems in Riemannian Geometry, Lect. Notes Ser. Dip. Mat. Univ. Studi Trento, 3, Trento, 1994

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

[20] Farrell F. T., Wagoner J., “Algebraic torsion for infinite simple homotopy types”, Comm. Math. Helv., 47 (1972), 502–513 | DOI | MR | Zbl

[21] Farrell F. T., Wagoner J., “Infinite matrices in algebraic K-theory and topology”, Comm. Math. Helv., 47 (1972), 474–501 | DOI | MR | Zbl

[22] Golstein V. M., Kuzminov V. I., Shvedov I. A., “The de Rham isomorphism of $L_p$-cohomology of noncompact Riemannian manifolds”, Sib. Math. J., 29 (1988), 34–44 | MR

[23] Greene R., “Complete metrics of bounded curvature on noncompact manifolds”, Arch. Math. (Basel), 31 (1978/1979), 89–95 | DOI | MR | Zbl

[24] Grimaldi R., Pansu P., Bounded geometry, growth and topology, arXiv: math/1008.4887v1 | MR

[25] Gromov M., Lafontaine J., Pansu P., Stuctures metriques pour les varietes Riemanniennes, Math. Texts, 1, CEDIC, Paris, 1981 | MR

[26] Higson N., Roe J., “Mapping surgery to analysis I: analytic signatures”, K-Theory, 33 (2004), 277–299 | DOI | MR | Zbl

[27] Higson N., Roe J., “Mapping surgery to analysis II: geometric signatures”, K-Theory, 33 (2004), 301–324 | DOI | MR | Zbl

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

[29] Hilton P. J., Wylie S., Homology theory, New York, 1960 | MR

[30] Hughes B., Ranicki A., Ends of complexes, Cambridge Tracts in Mathematics, 123, Cambridge, 1996 | MR | Zbl

[31] Maumary S., Proper surgery groups for non-compact manifolds of finite dimension, Preprint, U. C., Berkeley, 1972 | MR

[32] Maumary S., “Proper surgery groups and Wall–Novikov groups”, Algebraic $K$-Theory, Lect. Notes Math., 343, Springer, Berlin, 1973, 526–539 | DOI | MR

[33] Megginson R. E., An Introduction to Banach Space Theory, Springer, Berlin, 1988 | MR

[34] Pedersen E., Ranicki A., “Projective surgery theory”, Topology, 19 (1980), 239–254 | DOI | MR | Zbl

[35] Ranicki A., Sullivan D., “A semilocal combinatorial formula for the signature of a $4k$-manifold”, J. Diff. Geom., 11 (1976), 23–29 | DOI | MR | Zbl

[36] Richards I., “On the classification of noncompact surfaces”, Trans. Amer. Math. Soc., 106 (1963), 259–269 | DOI | MR | Zbl

[37] Rinow W., Lehrbuch der Topologie, Deutscher Verlag d. Wiss., Berlin, 1975 | MR | Zbl

[38] Roe J., Coarse Cohomology and Index Theory on Complete Riemannian Manifolds, Mem. Amer. Math. Soc., 497, Amer. Math. Soc., Providence, 1993 | MR | Zbl

[39] Roe J., Index Theory, Coarse Geometry, and Topology of Manifolds, CBMC Reg. Conf. Ser. Math., 90, Amer. Math. Soc., Providence, 1996 | MR | Zbl

[40] Schick T., Analysis on $\partial$-manifolds of bounded geometry, Hodge–de Rham isomorphism and $L_2$-index theorem, Thesis, Mainz, 1996

[41] Schubert H., Topologie, Teubner, Stuttgart, 1964 | MR | Zbl

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

[43] Siebenmann L., The obstruction to finding a boundary for an open manifold of dimension greater than five, Thesis, Princeton Univ., 1965 | MR | Zbl

[44] Stöcker R., Zieschang H., Algebraische Topologie, Teubner, Stuttgart, 1994 | MR | Zbl

[45] Stong R., Notes on Cobordism Theory, Math. Notes, Princeton Univ. Press, Princeton, 1968 | MR | Zbl

[46] Taylor L. R., Surgery on paracompact manifolds, Thesis, U. C., Berkeley, 1971

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

[48] Wall C. T. C., Surgery on Compact Manifolds, Academic Press, New York, 1970 | MR | Zbl