nLab
Borel-Moore homology

Contents

Contents

Idea

What is called Borel-Moore homology is a homology theory for suitably nice (locally compact) topological spaces, which was introduced in terms of sheaf cohomology in (Borel-Moore 60). Later it was observed that the homology groups obtained this way have the following equivalent description: They arise simply by generalizing in the ordinary definition of singular homology the finite formal linear combinations to locally finite ones. Accordingly, one speaks also of locally finite homology (Spanier 93). A detailed account is in (Hughes-Ranicki 96).

A key property of Borel-Moore/locally finite homology is that on nice enough topological spaces XX it is naturally isomorphic to the ordinary homology of the one-point compactification of XX. It is in this way that the theory is often used in practice, as a means for computing homology of one-point compactifications.

The proof of this statement is indicated in (Hughes-Ranicki 96), it passes through comparison with Steenrod homology via results of (Milnor 95).

References

The original article is

The formulation in terms of locally finite singular homology was highlighted in

  • Edwin Spanier, Singular homology and cohomology with local coefficients and duality for manifolds, Pacific J. Math. 1993

and the equivalence to ordinary homology of one-point compactifications is based on results in

  • John Milnor, On the Steenrod homology theory, Cambridge 1995

Textbook accounts include

  • Bruce Hughes, Andrew Ranicki, Ends of complexes, Cambridge University Press 1996 (pdf, pdf)

  • Glen Bredon, Borel-Moore Homology, chapter in Sheaf Theory, Volume 170 of the series Graduate Texts in Mathematics pp 279-416

See also

  • Wikipedia, Borel-Moore homology

  • Joel Lemay, Borel-Moore Homology, 2012 (pdf)

  • Martin Olsson, Borel-Moore homology, Riemann-Roch transformations, and local terms (pdf)

Last revised on July 21, 2017 at 12:29:01. See the history of this page for a list of all contributions to it.