What Are Stiefel-Whitney Homology Classes?

[wofsy]

Can anyone explain/give a good reference for Stiefel-Whitney homology classes?

 

[WWGD]

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.

 

[quasar987]

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.

 

[yyat]

Hatcher's Vector Bundles & K-Theory (http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html" ) has a nice chapter on characteristic classes.

 

[wofsy]

thanks - most books cover Stiefel-Whitney classes. These are cohomology classes. Sulliva n supposedly defined the homology classes.

 

[yyat]

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 p^th 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.

 

[wofsy]

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 p^th 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.