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