Duality Theorems for the Homology of Manifolds

  • William S. Massey
Part of the Graduate Texts in Mathematics book series (GTM, volume 70)


An n-dimensional manifold is a Hausdorff space such that every point has an open neighborhood which is homeomorphic to Euclidean n-space, R n (see Massey, [6], Chapter I). One of the main goals of this chapter will be to prove one of the oldest results of algebraic topology, the famous Poincaré duality theorem for compact, orientable manifolds. It is easy to state the Poincaré duality theorem but the proof is lengthy.


Homology Group Direct Limit Duality Theorem Homology Class Cohomology Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography for Chapter IX

  1. [1]
    M. Barratt and J. Milnor, An example of anomalous singular homology, Proc. Amer. Math. Soc., 13 (1962), 293–297.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    H. Cartan, Seminaire Henri Cartan 1948/49: Topologie Algébrique, W. A. Ben-jamin, Inc., New York, 1967.Google Scholar
  3. [3]
    R. Connelly, A new proof of Brown’s collaring theorem, Proc. Amer. Math. Soc., 27(1971), 180–182.MathSciNetMATHGoogle Scholar
  4. [4]
    A. Dold, Lectures on Algebraic Topology, Springer-Verlag, New York, 1972.MATHGoogle Scholar
  5. [5]
    N. Jacobson, Basic Algebra I, W. H. Freeman and Co., San Francisco, 1974.MATHGoogle Scholar
  6. [6]
    W. S. Massey, Algebraic Topology: An Introduction, Springer-Verlag, New York, 1978.Google Scholar
  7. [7]
    W. S. Massey, Homology and Cohomology Theory : An Approach Based on Alexander—Spanier Cochains, Marcel Dekker, Inc., New York, 1978.MATHGoogle Scholar
  8. [8]
    J. Milnor, Lectures on Characteristic Classes, Princeton University Press, Princeton, 1974.Google Scholar
  9. [9]
    E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.MATHGoogle Scholar
  10. [10]
    E. Spanier, Tautness for Alexander—Spanier cohomology, Pac. J. Math., 75 (1978), 561–563.MathSciNetMATHGoogle Scholar
  11. [11]
    J. Vick, Homology Theory, Academic Press, New York, 1973.MATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1980

Authors and Affiliations

  • William S. Massey
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

Personalised recommendations