Skip to main content
SpringerLink
Log in

Approximately fibering a manifold over an aspherical one

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

The paper is devoted to the problem when a map from some closed connected manifold to an aspherical closed manifold approximately fibers, i.e., is homotopic to manifold approximate fibration. We define obstructions in algebraic K-theory. Their vanishing is necessary and under certain conditions sufficient. Basic ingredients are Quinn’s thin h-cobordism theorem and end theorem, and knowledge about the Farrell–Jones conjectures in algebraic K- and L-theory and the MAF-rigidity conjecture by Hughes–Taylor–Williams.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In the special case where B is a (real) hyperbolic manifold, \(\dim M>\dim B+4\), and \({{\mathrm{Wh}}}(\pi _1(M)\times {\mathbb {Z}}^n)=0\) for all \(n\ge 0\), the conclusion of Theorem 1.4 was proven by Farrell-Jones in [29, Theorem 10.7].

References

  1. Armentrout, S.: Cellular decompositions of 3-manifolds that yield 3-manifolds. Memoirs of the American Mathematical Society, No. 107. American Mathematical Society, Providence (1971)

  2. Bak, A.: The computation of surgery groups of odd torsion groups. Bull. Am. Math. Soc. 80(6), 1113–1116 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bartels, A., Echterhoff, S., Lück, W.: Inheritance of isomorphism conjectures under colimits. In: Cortinaz, G. Cuntz, J., Karoubi, M., Nest, R. and Weibel, C.W. (eds.), K-Theory and Noncommutative Geometry, EMS-Series of Congress Reports, pp. 41–70. European Mathematical Society (2008)

  4. Bartels, A., Farrell, F.T., Lück, W.: The Farrell–Jones conjecture for cocompact lattices in virtually connected Lie groups. J. Am. Math. Soc. 27(2), 339–388 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bartels, A., Lück, W.: On twisted group rings with twisted involutions, their module categories and \({L}\)-theory. In: Cohomology of Groups and Algebraic \(K\)-Theory, vol. 12 of Advanced Lectures in Mathematics, pp. 1–55, Somerville. International Press (2009)

  6. Bartels, A., Lück, W.: The Borel conjecture for hyperbolic and CAT(0)-groups. Ann. Math. 2(175), 631–689 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bartels, A., Lück, W., Reich, H.: The \(K\)-theoretic Farrell–Jones conjecture for hyperbolic groups. Invent. Math. 172(1), 29–70 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bartels, A., Lück, W., Reich, H., Rüping, H.: K- and L-theory of group rings over GL\(_n(\bf {Z})\). Publ. Math., Inst. Hautes Étud. Sci. 119:97–125 (2014)

  9. Bartels, A., Reich, H.: Coefficients for the Farrell–Jones Conjecture. Adv. Math. 209(1), 337–362 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bass, H., Murthy, M.P.: Grothendieck groups and Picard groups of abelian group rings. Ann. Math. 2(86), 16–73 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  11. Borevich, A. I., Shafarevich, I. R.: Number Theory. Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, vol. 20. Academic Press, New York (1966)

  12. Bredon, G.E.: Topology and Geometry. Springer, New York (1997, corrected third printing of the 1993 original)

  13. Chapman, T.A.: Topological invariance of Whitehead torsion. Am. J. Math. 96, 488–497 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chapman, T.A.: Homotopy conditions which detect simple homotopy equivalences. Pac. J. Math. 80(1), 13–46 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chapman, T.A.: Approximation results in Hilbert cube manifolds. Trans. Am. Math. Soc. 262(2), 303–334 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chapman, T.A.: Approximation results in topological manifolds. Mem. Am. Math. Soc. 34(251):iii+64 (1981)

  17. Chapman, T.A., Ferry, S.C.: Constructing approximate fibrations. Trans. Am. Math. Soc. 276(2), 757–774 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  18. Coram, D.S., Duvall Jr., P.F.: Approximate fibrations. Rocky Mt. J. Math. 7(2), 275–288 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  19. Davis, J.F., Lück, W.: Spaces over a category and assembly maps in isomorphism conjectures in \(K\)- and \(L\)-theory. K-Theory 15(3), 201–252 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Davis, M.W., Januszkiewicz, T.: Hyperbolization of polyhedra. J. Differ. Geom. 34(2), 347–388 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Edwards, R.D.: Topological regular neighborhoods. Unpublished manuscript. arXiv:0904.4665v1

  22. Edwards, D.A., Geoghegan, R.: Shapes of complexes, ends of manifolds, homotopy limits and the Wall obstruction. Ann. Math. 2(101), 521–535 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  23. Farrell, F.T.: The obstruction to fibering a manifold over a circle. Bull. Am. Math. Soc. 73, 737–740 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  24. Farrell, F.T.: The obstruction to fibering a manifold over a circle. Actes du Congrès International des Mathématiciens (Nice. 1970), Tome 2, pp. 69–72. Gauthier-Villars, Paris (1971)

  25. Farrell, F.T.: The obstruction to fibering a manifold over a circle. Indiana Univ. Math. J. 21:315–346 (1971/1972)

  26. Farrell, F.T., Hsiang, W.-C.: A geometric interpretation of the Künneth formula for algebraic \(K\)-theory. Bull. Am. Math. Soc. 74, 548–553 (1968)

    Article  MATH  Google Scholar 

  27. Farrell, F. T., Hsiang, W.-C.: A formula for \({K}_{1}{R}_{\alpha }\,[{T}]\). In: Applications of Categorical Algebra (Proceedings of Symposium on Pure Mathematics, Vol. XVII, New York, 1968), pp. 192–218. American Mathematical Society, Providence (1970)

  28. Farrell, F.T., Hsiang, W.C.: Manifolds with \(\pi _{1}=G\times _{\alpha } T\). Am. J. Math. 95, 813–848 (1973)

    Article  Google Scholar 

  29. Farrell, F.T., Jones, L.E.: A topological analogue of Mostow’s rigidity theorem. J. Am. Math. Soc. 2(2), 257–370 (1989)

    MathSciNet  MATH  Google Scholar 

  30. Farrell, F.T., Jones, L.E.: Isomorphism conjectures in algebraic \({K}\)-theory. J. Am. Math. Soc. 6(2), 249–297 (1993)

    MathSciNet  MATH  Google Scholar 

  31. Farrell, F.T., Lück, W., Steimle, W.: Obstructions to fibering a manifold. Geom. Dedic. 148, 35–69 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ferry, S.C.: Approximate fibrations with nonfinite fibers. Proc. Am. Math. Soc. 64(2), 335–345 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  33. Freedman, M.H., Quinn, F.: Topology of 4-Manifolds. Princeton University Press, Princeton (1990)

    MATH  Google Scholar 

  34. Hempel, J.: 3-Manifolds. Ann. Math. Stud. 86 (1976)

  35. Hirsch, M.W.: On combinatorial submanifolds of differentiable manifolds. Comment. Math. Helv. 36, 103–111 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  36. Hu, S.-T.: Theory of Retracts. Wayne State University Press, Detroit (1965)

    MATH  Google Scholar 

  37. Hughes, B.: Controlled topological equivalence of maps in the theory of stratified spaces and approximate fibrations. Sci. Bull. Josai Univ. (Special issue 2), 31–50 (1997) (Surgery and geometric topology (Sakado, 1996))

  38. Hughes, B., Khan, Q.: Dihedral manifold approximate fibrations over the circle. Geom. Dedic. 148, 205–243 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Hughes, C.B.: Approximate fibrations on topological manifolds. Michigan Math. J. 32(2), 167–183 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  40. Hughes, C.B.: Spaces of approximate fibrations on Hilbert cube manifolds. Compos. Math. 56(2), 131–151 (1985)

    MathSciNet  MATH  Google Scholar 

  41. Hughes, C.B.: Topology with control and approximate fibrations. In: Proceedings of the Guilford College Sesquicentennial Topology Conference, pp. 1–6. Guilford College, Greensboro, NC (1988)

  42. Hughes, C.B., Taylor, L.R., Williams, E.B.: Bundle theories for topological manifolds. Trans. Am. Math. Soc. 319(1), 1–65 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  43. Hughes, C.B., Taylor, L.R., Williams, E.B.: Manifold approximate fibrations are approximately bundles. Forum Math. 3(4), 309–325 (1991)

    MathSciNet  MATH  Google Scholar 

  44. Hughes, C.B., Taylor, L.R., Williams, E.B.: Rigidity of fibrations over nonpositively curved manifolds. Topology 34(3), 565–574 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  45. Husch, L.S.: Approximating approximate fibrations by fibrations. Can. J. Math. 29(5), 897–913 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  46. Kammeyer, H., Lück, W., Rüping, H.: The Farrell–Jones conjecture for arbitrary lattices in virtually connected Lie groups. Geom. Topol. 20(3), 1275–1287 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kirby, R.C., Siebenmann, L.C., Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. Princeton University Press, Princeton (1977). (With notes by J. Milnor and M. F, Atiyah, Annals of Mathematics Studies, p. 88 (1977))

  48. Kleiner, B., Lott, J.: Notes on Perelman’s papers. Geom. Topol. 12(5), 2587–2855 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  49. Kwun, K.W., Szczarba, R.H.: Product and sum theorems for Whitehead torsion. Ann. Math. 2(82), 183–190 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  50. Lacher, R.C.: Cell-like mappings and their generalizations. Bull. Am. Math. Soc. 83(4), 495–552 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  51. Lück, W.: Transformation Groups and Algebraic \({K}\)-Theory. Lecture Notes in Mathematics, vol. 1408. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  52. Lück, W.: \(K\)- and \(L\)-theory of the semi-direct product of the discrete 3-dimensional Heisenberg group by \({\mathbb{Z}}/4\). Geom. Topol. 9, 1639–1676 (2005). (Electronic)

    Article  MathSciNet  MATH  Google Scholar 

  53. Lück, W.: Survey on aspherical manifolds. In: Ran, A., te Riele, H., Wiegerinck, J. (eds.) Proceedings of the 5-th European Congress of Mathematics Amsterdam 14–18 July 2008, pp. 53–82. EMS (2010)

  54. Lück, W., Reich, H.: The Baum–Connes and the Farrell–Jones conjectures in \(K\)- and \(L\)-theory. In: Handbook of \(K\)-Theory. vol. 1, 2, pp. 703–842. Springer, Berlin (2005)

  55. Lück, W., Steimle, W.: Delooping \(K\)-theory for additive categories. In: Ausoni, C., Hess, K., Johnson, K., Lück, W., Scherer, J. (eds.) An Alpine Expedition through Algebraic Topology, Volume 617 of Contemporary Mathematics, pp. 205–236. AMS (2014)

  56. Lück, W., Steimle, W.: Splitting the relative assembly map, nil-terms and involutions. Ann. K-Theory 1(4), 339–377 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  57. Milnor, J.: Introduction to algebraic \({K}\)-theory. Princeton University Press, Princeton. Ann. Math. Stud. No. 72 (1971)

  58. Morgan, J., Tian, G.: Ricci flow and the Poincaré Conjecture. Clay Mathematics Monographs, vol. 3. American Mathematical Society, Providence (2007)

    MATH  Google Scholar 

  59. Morgan, J., Tian, G.: The geometrization conjecture, volume 5 of Clay Mathematics Monographs. American Mathematical Society, Providence, Clay Mathematics Institute, Cambridge (2014)

  60. Mostow, G.D.: Strong rigidity of locally symmetric spaces. Princeton University Press, Princeton. Ann. Math. Stud. 78 (1973)

  61. Pedersen, E.K., Weibel, C.A.: A non-connective delooping of algebraic \({K}\)-theory. In: Algebraic and Geometric Topology; Proceedings of Conference Rutgers University, New Brunswick 1983, volume 1126 of Lecture Notes in Mathematics, pp. 166–181. Springer (1985)

  62. Quinn, F.: A geometric formulation of surgery. PhD thesis, Princeton, (1969)

  63. Quinn, F.: A geometric formulation of surgery. In: Topology of Manifolds (Proceedings of Inst., University of Georgia, Athens, 1969), pp. 500–511. Markham, Chicago (1970)

  64. Quinn, F.: Ends of maps. I. Ann. Math 110(2), 275–331 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  65. Quinn, F.: Ends of maps. II. Invent. Math. 68(3), 353–424 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  66. Quinn, F.: Applications of topology with control. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Berkeley, 1986), pp. 598–606, Providence, RI, 1987. American Mathematical Society (1987)

  67. Quinn, F.: Homotopically stratified sets. J. Am. Math. Soc. 1(2), 441–499 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  68. Raghunathan, M.S.: Discrete subgroups of Lie groups. Springer, New York (1972). (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68 (1972))

  69. Ranicki, A.A.: Algebraic \(L\)-theory. III. Twisted Laurent extensions. In: Algebraic K-Theory, III: Hermitian K-Theory and Geometric Application (Proceedings of Conference Seattle Research Center, Battelle Memorial Institute, 1972), pp. 412–463. Lecture Notes in Mathematics, vol. 343. Springer, Berlin (1973)

  70. Rüping, H.: The Farrell–Jones conjecture for \(S\)-arithmetic groups. J. Topol. 9(1), 51–90 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  71. Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15(5), 401–487 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  72. Siebenmann, L.: The obstruction to finding a boundary for an open manifold of dimension greater than five. Ph.D. thesis, Princeton (1965). http://www.maths.ed.ac.uk/~aar/surgery/index.htm

  73. Siebenmann, L.C.: A total Whitehead torsion obstruction to fibering over the circle. Comment. Math. Helv. 45, 1–48 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  74. Siebenmann, L.C.: Approximating cellular maps by homeomorphisms. Topology 11, 271–294 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  75. Steimle, W.: Homotopy coherent diagrams and approximate fibrations. Preprint (2011). arXiv:1107.5213

  76. Vogell, W.: The canonical involution on the algebraic \(K\)-theory of spaces. In: Algebraic Topology (Aarhus, 1982), Volume 1051 of Lecture Notes in Mathematics, pp. 156–172. Springer, Berlin (1984)

  77. Vogell, W.: The involution in the algebraic \(K\)-theory of spaces. In: Algebraic and geometric topology (New Brunswick, N.J., 1983), Volume 1126 of Lecture Notes in Math. pp. 277–317. Springer, Berlin (1985)

  78. Vogell, W.: Algebraic \(K\)-theory of spaces, with bounded control. Acta Math. 165(3–4), 161–187 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  79. Vogell, W.: Boundedly controlled algebraic \(K\)-theory of spaces and its linear counterparts. J. Pure Appl. Algebra 76(2), 193–224 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  80. Wall, C.: On the classification of hermitian forms. VI: Group rings. Ann. Math. 103, 1–80 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  81. Wall, C.T.C.: Poincaré complexes. I. Ann. Math. 2(86), 213–245 (1967)

    Article  MATH  Google Scholar 

  82. Wall, C.T.C.: Surgery on compact manifolds, Volume 69 of Mathematical Surveys and Monographs, 2nd ed. American Mathematical Society, Providence, RI (1999). (Edited and with a foreword by A. A. Ranicki)

  83. Wegner, C.: The \(K\)-theoretic Farrell–Jones conjecture for CAT(0)-groups. Proc. Am. Math. Soc. 140(3), 779–793 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  84. Wegner, C.: The Farrell–Jones conjecture for virtually solvable groups. J. Topol. 8(4), 975–1016 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  85. Weinberger, S.: The Topological Classification of Stratified Spaces. University of Chicago Press, Chicago (1994)

    MATH  Google Scholar 

  86. Weiss, M., Williams, B.: Duality in Waldhausen categories. Forum Math. 10(5), 533–603 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  87. West, J.E.: Mapping Hilbert cube manifolds to ANR’s: a solution of a conjecture of Borsuk. Ann. Math. 106(1), 1–18 (1977)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This paper is financially supported by the Leibniz-Preis of the second author, granted by the Deutsche Forschungsgemeinschaft DFG. Moreover the first named author was partially supported by NSF Grant DMS-1206622. The third named author was partly supported by the ERC Advanced Grant 288082 and by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92). We thank the referee for his detailed report and very valuable comments.

Author information

Authors and Affiliations

  1. Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

    Tom Farrell

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, China

    Tom Farrell

  3. Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Wolfgang Lück

  4. Institut für Mathematik, Universität Augsburg, 86135, Augsburg, Germany

    Wolfgang Steimle

Authors
  1. Tom Farrell
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wolfgang Lück
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wolfgang Steimle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wolfgang Steimle.

Additional information

Communicated by Thomas Schick.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Farrell, T., Lück, W. & Steimle, W. Approximately fibering a manifold over an aspherical one. Math. Ann. 370, 669–726 (2018). https://doi.org/10.1007/s00208-017-1535-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-017-1535-1

Mathematics Subject Classification

Search

Navigation