Skip to main content
SpringerLink
Log in

Equivariant finiteness obstruction and its geometric applications - A survey

  • Survey Articles
  • Conference paper
  • First Online:
Algebraic Topology Poznań 1989

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1474))

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D.R. Anderson: Torsion invariants and actions of finite groups, Michigan Math. J. 29 (1982), 27–42.

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Andrzejewski: The equivariant Wall finiteness obstruction and Whitehead torsion, Transformation Groups, Poznaʼn 1985, pp. 11–25, Lecture Notes in Math. 1217, Springer Vlg 1986.

    Google Scholar 

  3. P. Andrzejewski: An application of equivariant finiteness obstruction — equivariant version of Siebenmann's theorem, (preprint, to appear).

    Google Scholar 

  4. P. Andrzejewski: A complement to the theory of equivariant finiteness obstruction (preprint 1989).

    Google Scholar 

  5. A.H. Assadi: Extensions of group actions from submanifolds of disks and spheres (preprint).

    Google Scholar 

  6. J.A. Baglivo: An equivariant Wall obstruction theory, Trans. Amer. Math. Soc. 256 (1979), 305–324.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Bass, A. Heller, R. Swan: The Whitehead group of a polynomial extension, Publ. Math. IHES 22 (1964), 67–79.

    MathSciNet  MATH  Google Scholar 

  8. G.E. Bredon: Introduction to compact transformation groups, Academic Press, N.Y. 1972.

    MATH  Google Scholar 

  9. S.E. Cappell, S. Weinberger: Homology propagation of group actions, Comm. Pure and Appl. Math. 40 (1987), 723–744.

    Article  MathSciNet  MATH  Google Scholar 

  10. T.A. Chapman: Controlled simple homotopy theory and applications, Lecture Notes in Math. 1009, Springer Vlg 1983.

    Google Scholar 

  11. M.M.Cohen: A course in simple-homotopy theory, Graduate Texts in Math., Springer Vlg 1973.

    Google Scholar 

  12. T. tom Dieck: Über projektive Moduln und Endlichkeitschindernisse bei Transformationsgruppen, Manuscripta Math. 34 (1981), 135–155.

    Article  MathSciNet  MATH  Google Scholar 

  13. T. tom Dieck: Transformation groups, de Gruyter, Berlin 1987.

    Book  MATH  Google Scholar 

  14. T. tom Dieck, T. Petrie: Homotopy representations of finite groups, Publ. Math. IHES 56 (1982), 129–170.

    Article  MathSciNet  MATH  Google Scholar 

  15. W. Dorabiała: On the equivariant homotopy type of G-fibrations, (preprint, to appear).

    Google Scholar 

  16. K.H. Dovermann: personal communication.

    Google Scholar 

  17. K.H. Dovermann, M. Rothenberg: An equivariant surgery sequence and equivariant diffeomorphism and homeomorphism classification, Topology Symposium (Siegen 1979), pp. 257–280, Lecture Notes in Math. 788, Springer Vlg 1980.

    Google Scholar 

  18. K.H. Dovermann, M. Rothenberg: Equivariant surgery and classification of finite group actions on manifolds, Memoirs Amer. Math. Soc. 379 (1988).

    Google Scholar 

  19. S. Ferry: A simple-homotopy approach to the finiteness obstruction, Shape Theory and Geometric Topology, pp. 73–81, Lecture Notes in Math. 870, Springer Vlg 1981.

    Google Scholar 

  20. S.M. Gersten: A product formula for Wall's obstruction, Amer. J. Math. 88 (1966), 337–346.

    Article  MathSciNet  MATH  Google Scholar 

  21. K. Iizuka: Finiteness conditions for G-CW-complexes, Japan. J. Math. 10 (1984), 55–69.

    MathSciNet  MATH  Google Scholar 

  22. S. Illman: Smooth equivariant triangulations of G-manifolds for G a finite group, Math. Ann. 233 (1978), 199–220.

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Illman: The equivariant triangulation theorem for actions of compact Lie groups, Math. Ann. 262 (1983), 487–501.

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Illman: A product formula for equivariant Whitehead torsion and geometric applications, Transformation Groups, Poznaʼn 1985, pp. 123–142, Lecture Notes in Math. 1217, Springer Vlg 1986.

    Google Scholar 

  25. S. Illman: Actions of compact Lie groups and the equivariant Whitehead group, Osaka J. Math. 23 (1986), 881–927.

    MathSciNet  MATH  Google Scholar 

  26. S. Illman: On some recent questions in equivariant simple homotopy theory, Transformation Groups and Whitehead Torsion, Proc. RIMS 633, Kyoto Univ. 1987, pp.19–33.

    Google Scholar 

  27. S. Illman: The restriction homomorphism Res H : Wh G (X) → Wh H (X) for G a compact Lie group, (preprint, 1989).

    Google Scholar 

  28. R.C. Kirby, L.C. Siebenmann: Foundational essays on topological manifolds, smoothings and triangulations, Ann. Math. Studies 88 (1977), Princeton Univ. Press.

    Google Scholar 

  29. H.T. Ku, M.C. Ku: Obstruction theory for finite group actions, Osaka J. Math. 18 (1981), 509–523.

    MathSciNet  MATH  Google Scholar 

  30. S. Kwasik: On the equivariant homotopy type of G-ANR's, Proc. Amer. Math. Soc. 267 (1981), 193–194.

    MathSciNet  MATH  Google Scholar 

  31. S. Kwasik: On equivariant finiteness, compositio Math. 48 (1983), 363–372.

    MathSciNet  MATH  Google Scholar 

  32. S. Kwasik: Locally smooth G-manifolds, Amer. J. Math. 108 (1986), 27–37.

    Article  MathSciNet  MATH  Google Scholar 

  33. W. Lück: The geometric finiteness obstruction, Proc. London Math. Soc. 54 (1987), 367–384

    Article  MathSciNet  MATH  Google Scholar 

  34. W. Lück: Transformation groups and algebraic K-theory, Lecture Notes in Math. 1408, Springer Vlg 1989.

    Google Scholar 

  35. I. Madsen, C.B. Thomas, C.T.C. Wall: The topological spherical space form problem-II: existence of free actions, Topology 15 (1976), 375–382.

    Article  MathSciNet  MATH  Google Scholar 

  36. I. Madsen, M. Rothenberg: On the classification of G-spheres III: TOP automorphism groups, preprint 14 (1985), Aarhus Univ.

    Google Scholar 

  37. M. Murayama: On G-ANR's and their G-homotopy types, Osaka J. Math. 20 (1983), 479–512.

    MathSciNet  MATH  Google Scholar 

  38. R. Oliver: Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177.

    Article  MathSciNet  MATH  Google Scholar 

  39. R. Oliver: Smooth compact Lie group actions on disks, Math. Zeit. 149 (1976), 79–96.

    Article  MathSciNet  MATH  Google Scholar 

  40. R. Oliver: G-actions on disks and permutation representations II, Math. Zeit. 157 (1977), 237–263.

    Article  MathSciNet  MATH  Google Scholar 

  41. R. Oliver: G-actions on disks and permutation representations, J. Algebra 50 (1978), 44–62.

    Article  MathSciNet  MATH  Google Scholar 

  42. R. Oliver, T. Petrie: G-CW-surgery and K o (ZG), Math. Zeit. 179 (1982), 11–42.

    Article  MathSciNet  MATH  Google Scholar 

  43. T. Petrie: G-maps and the projective class group, Comment. Math. Helv. 51 (1976), 611–626.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. A.A. Ranicki: The algebraic theory of finiteness obstruction, Math. Scand. 57 (1985), 105–126.

    MathSciNet  MATH  Google Scholar 

  46. L.C. Siebenmann: The obstruction to finding a boundary for an open manifold in dimension greater than five, Ph. D. thesis, Princeton 1965.

    Google Scholar 

  47. M. Steinberger: The equivariant topological s-cobordism theorem, Invent. Math. 91 (1988), 61–104.

    Article  MathSciNet  MATH  Google Scholar 

  48. M. Steinberger, J.E. West: Equivariant h-cobordisms and finiteness obstructions, Bull. Amer. Math. Soc. 12 (1985), 217–220.

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Steinberger, J.E. West: On the geometric topology of locally linear actions of finite groups, Geometric and Algebraic Topology, pp. 181–204, Banach Center Publ. 18, Warsaw 1986.

    Google Scholar 

  50. M. Steinberger, J.E. West: Equivariant controlled simple homotopy theory (in preparation)

    Google Scholar 

  51. M. Steinberger, J.E. West: Controlled finiteness is the obstruction to equivariant handle decomposition, (preprint).

    Google Scholar 

  52. M. Steinberger, J.E. West: Equivariant handles in finite group (preprint). actions, (preprint).

    Google Scholar 

  53. R.G. Swan: Periodic resolutions for finite groups, Ann. Math. 72 (1960), 267–291.

    Article  MathSciNet  MATH  Google Scholar 

  54. C.B. Thomas, C.T.C. Wall: The topological spherical space form problem-I, Compositio Math. 23 (1971), 101–114.

    MathSciNet  MATH  Google Scholar 

  55. C.T.C. Wall: Finiteness conditions for CW-complexes, Ann. Math. 81 (1965), 55–69.

    Article  MathSciNet  MATH  Google Scholar 

  56. C.T.C. Wall: Finiteness conditions for CW-complexes, II, Proc. Royal Soc. London, Ser. A, 295 (1966), 129–139.

    Article  MathSciNet  Google Scholar 

  57. A.G. Wasserman: Equivariant differential topology, Topology 8 (1969), 127–150.

    Article  MathSciNet  MATH  Google Scholar 

  58. D. Webb: Equivariantly finite manifolds with no handle structure, (preprint).

    Google Scholar 

  59. S. Weinberger: Constructions of group actions: a survey of some recent developments, Group actions on manifolds, pp. 269–298, Contemporary Math. 36 (1985).

    Google Scholar 

  60. S. Weinberger: an example in: Problems submitted to the AMS Summer Research Conference on Group Actions, Group actions on manifolds, ed. R.E. Schultz, pp. 513–568, Contemporary Math. 36 (1985).

    Google Scholar 

  61. S. Weinberger: Class numbers, the Novikov conjecture and transformation groups, Topology 27 (1988), 353–365.

    Article  MathSciNet  MATH  Google Scholar 

  62. J.E. West: Mapping Hilbert cube manifolds to ANRs, Ann. Math. 106 (1977), 1–18.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instytut Matematyki, Uniwersytet Szczeciʼnski ul., Wielkopolska 15, 70-451, Szczecin 3, Poland

    Pawel Andrzejewski

Authors
  1. Pawel Andrzejewski
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Stefan Jackowski Bob Oliver Krzystof Pawałowski

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag

About this paper

Cite this paper

Andrzejewski, P. (1991). Equivariant finiteness obstruction and its geometric applications - A survey. In: Jackowski, S., Oliver, B., Pawałowski, K. (eds) Algebraic Topology Poznań 1989. Lecture Notes in Mathematics, vol 1474. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084735

Download citation

  • DOI: https://doi.org/10.1007/BFb0084735

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54098-4

  • Online ISBN: 978-3-540-47403-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation