What Are Stiefel-Whitney Homology Classes?

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Discussion Overview

The discussion centers around Stiefel-Whitney homology classes, exploring their definitions, references for further reading, and connections to cohomology classes. Participants seek to clarify the distinction between Stiefel-Whitney classes and homology classes, as well as their applications in topology and geometry.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants inquire about good references for Stiefel-Whitney homology classes.
  • Others suggest standard sources such as Wikipedia, Amazon, and specific textbooks, including Bredon's "Topology and Geometry" and Milnor and Stasheff's "Characteristic Classes."
  • It is noted that most literature focuses on Stiefel-Whitney classes as cohomology classes, with a mention that Sullivan defined the homology classes.
  • A participant explains that one can obtain a homology class by taking the Poincaré dual of a cohomology class, referencing a paper by Halperin & Toledo that describes the relationship in terms of triangulation on manifolds.
  • There is a description of how the pth Stiefel-Whitney homology class of the tangent bundle can be represented using mod-2 cycles and barycentric subdivisions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definitions and implications of Stiefel-Whitney homology classes versus cohomology classes, indicating that multiple views and interpretations remain in the discussion.

Contextual Notes

Some limitations include the potential dependence on specific definitions of homology and cohomology classes, as well as the unresolved nature of the distinctions between these concepts in the literature.

wofsy
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Can anyone explain/give a good reference for Stiefel-Whitney homology classes?
 
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Have you tried the standard sources: Amazon.com ( in the book-review section),
Wikipedia, Planet Math.? They recommend books on some of their articles. If not,
you can always post it on sci.math.

HTH.
 
There's a small section about those things in Bredon's book Topology and Geometry (6 pages). In it, they give the as reference the book Characteristic Classes of Milnor and Stasheff for ppl interested in learning more.
 
Hatcher's Vector Bundles & K-Theory (http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html" ) has a nice chapter on characteristic classes.
 
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thanks - most books cover Stiefel-Whitney classes. These are cohomology classes. Sulliva n supposedly defined the homology classes.
 
One can of course take the Poincaré dual of a cohomology class and obtain a homology class, see for example the paper "Stiefel-Whitney homology classes" by Halperin & Toledo (Ann. of Math.). In fact, applied to the tangent bundle, these classes have a very simple description in terms of a triangulation K (simplicial structure) on the manifold: the pth Stiefel-Whitney homology class of TM is represented by the mod-2 cycle which is the sum of all p-simplices of the first barycentric subdivision of K.
 
yyat said:
One can of course take the Poincaré dual of a cohomology class and obtain a homology class, see for example the paper "Stiefel-Whitney homology classes" by Halperin & Toledo (Ann. of Math.). In fact, applied to the tangent bundle, these classes have a very simple description in terms of a triangulation K (simplicial structure) on the manifold: the pth Stiefel-Whitney homology class of TM is represented by the mod-2 cycle which is the sum of all p-simplices of the first barycentric subdivision of K.

thanks. that is really great.
 

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