Return to Answer

Questionable statement removed, just to be sure.

Source Link

edited Dec 24, 2009 at 15:53

Chris Schommer-Pries

27.5k
3
91
171

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). Also every closed manifold will have an E fundamental class, even unorientable manifolds.

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.

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). Also every closed manifold will have an E fundamental class, even unorientable manifolds.

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.

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.

Source Link

created Dec 24, 2009 at 15:29

Chris Schommer-Pries

27.5k
3
91
171

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). Also every closed manifold will have an E fundamental class, even unorientable manifolds.

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.