Proper cohomologies and the proper classification problem

  • L. J. Hernández
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1298)


We attack the classification problem for proper maps using proper cohomologies with coefficients either in a tower of abelian groups or in a morphism of a tower of abelian groups into an abelian group. To compute these proper cohomology groups we have considered a universal coefficient formula which works in a sligtly different way that the clasic one. In order to apply this formula we have done an analysis of the projective objects in the category of tower of abelian groups tow-Ab and in the category (Ab, tow-Ab) whose objects are morphisms from a tower of abelian groups to an abelian group.

As a consequence of this study we prove that the group of proper homotopy classes of a finite dimension σ-compact simplicial complex K into the euclidean plane R2 is isomorphic to colim Hom (H1(Pi), Z) where {clousure of K-Pi} is cofinal in the set of compact subsets of K. In particular, we compute the group of proper homotopy classes of an open connected surface into the euclidean plane R2.

A.M.S. classification number

55N35 55S37 Key words proper map proper homotopy proper cohomology 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A-M]
    M. Artin, B. Mazur "Etale Homotopy" Lecture Notes in Math. 100, Springer-Verlag, 1969.Google Scholar
  2. [Br]
    E.M. Brown "On the proper homotpy type of simplicial complexes" Lecture Notes in Math., 375, Springer-Verlag, 1974.Google Scholar
  3. [B-K]
    A.K. Bousfield, D.M. Kan "Homotopy Limits, Complections and Localizations", Lecture Notes in Math., 304, Springer, 1972.Google Scholar
  4. [E-H]
    D.A. Edwards, H.M. Hastings "Čech and Steerod Homotopy Theories with Applications to Geometric Topology", Lecture Notes in Math., 542, Springer-Verlag, 1976.Google Scholar
  5. [Hem]
    J. Hempel "3-manifolds" Princeton University Press, Study 86, 1976.Google Scholar
  6. [Her1]
    L.J. Hernández "About the extension problem for proper maps" Top. and its Appl., 25, 51–64, (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  7. [Her2]
    " "Clases de homotopía propia de una superficie no compacta en el plano euclideo", Actas del II Seminario de Topología (U. Zaragoza), 1986.Google Scholar
  8. [H-P]
    ", T. Porter "Proper pointed maps from Rn+1 to a σ-compact space", preprint, 1986.Google Scholar
  9. [H-S]
    P.J. Hilton, U. Stammbach "A course in Homological Algebra", G.T.M. 4, Springer-Verlag, 1971.Google Scholar
  10. [O]
    B.L. Osofky "The subscript of Xn, projective dimension and the vanishing on lim(n)", Bull. A.M.S., 80, 9–26, 1974.Google Scholar
  11. [P]
    T. Porter "Homotopy groups for strong shape and proper homotopy", Convegno di Topologia, Serie II, no 4, 101–113, 1984.Google Scholar
  12. [R]
    I. Richards "On the classification on noncompact surfaces" Trans. Amer. Math. Soc., 106, 259–269, 1963.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • L. J. Hernández
    • 1
  1. 1.Department of Topology and GeometryUniversity of ZaragozaZaragozaSpain

Personalised recommendations