# Stiefel-Whiney Homology Classes

• wofsy
In summary, Stiefel-Whitney homology classes are cohomology classes defined by Sullivan, which can also be obtained by taking the Poincaré dual of a cohomology class. They have a simple description in terms of a triangulation on the manifold and can be represented by the sum of simplices in the first barycentric subdivision of the triangulation. These classes are covered in many books, such as Bredon's Topology and Geometry, Milnor and Stasheff's Characteristic Classes, and Hatcher's Vector Bundles & K-Theory. Additional resources can be found on Amazon.com, Wikipedia, Planet Math, and sci.math.
wofsy
Can anyone explain/give a good reference for Stiefel-Whitney homology classes?

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.

Last edited by a moderator:
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.

## What are Stiefel-Whitney homology classes?

Stiefel-Whitney homology classes are a set of cohomology classes that can be used to classify vector bundles over a topological space. They were introduced by mathematicians Eduard Stiefel and Hassler Whitney in the 1940s.

## What is the significance of Stiefel-Whitney homology classes?

Stiefel-Whitney homology classes have many applications in topology and geometry. They can be used to study characteristic classes, cobordism theory, and differential geometry. They also have connections to physics, specifically in the study of gauge theories and the classification of fermionic fields.

## How are Stiefel-Whitney homology classes calculated?

Stiefel-Whitney homology classes are calculated using the cohomology ring of the topological space and the universal coefficient theorem. The classes are defined by a set of axioms, which can be used to compute their values for specific vector bundles.

## What is the relationship between Stiefel-Whitney homology classes and characteristic classes?

Stiefel-Whitney homology classes are a special case of characteristic classes, which are topological invariants associated with vector bundles. The Stiefel-Whitney classes can be used to calculate other characteristic classes, such as Chern classes and Pontryagin classes.

## What are some open questions about Stiefel-Whitney homology classes?

There are still many open questions about Stiefel-Whitney homology classes, including their relationship to other cohomology theories and their applications in physics. Additionally, there is ongoing research on generalizing Stiefel-Whitney classes to other types of bundles, such as complex and symplectic bundles.

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