Hello,


'ordinary' Stiefel-Whitney classes are elements of the singular cohomology ring and are constructed using the Thom isomorphism and Steenrod squares. So I think they should exist for any (generalized) multiplicative cohomology theory for which the Thom isomorphism and cohomology operations like the Steenrod squares exist. If I am not wrong, I would be really happy about some references on this.


Thanks in advance (and merry christmas)


Jonas



Characteristic classes in generalized cohomology theories?
Stiefel-Whitney classes exist for any real-oriented cohomology theory. This is a (multiplicative) cohomology theory E equipped with an isomorphism 


$E^\* ( \mathbb{R} P^{\infty} ) \cong E^\*(pt) [[x]]$


The two most well known examples are ordinary cohomology (i.e. singular cohomology) **with $\mathbb{Z}/2$-coefficients** and also unoriented bordism theory MO (this is the Thom spectrum MO, not "MathOverflow"). Note that such an isomorphism might not exist. For example it doesn't exist for ordinary Z cohomology, nor does it exist for K-theory.


The choice of this generator x is the first Stiefel-Whitney class and the other classes can be constructed using the splitting principle. For such a theory you will have Thom isomorphisms for all real bundles (not necessarily oriented).


I believe this is all explained in Switzer's book "Algebraic Topology", but I'm not sure. The course I learned it from didn't have a text book. This is often standard material for a second semester of graduate level algebraic topology, at least it was at UC Berkeley.


