This Week's Finds in Mathematical Physics (Week 149)
Volker Braun
> [...] In fact, complex cobordism theory is the "universal" complex
> oriented cohomology theory - it's the most informative of the whole lot.
> All the rest are like watered-down versions of this one. Ordinary
> cohomology is the most watered-down of all. Complex K-theory is a bit
> less watered-down. And elliptic cohomology is still less watered-down!
I cannot follow you at all here - what is the mathematical
definition of "watered down"? You seem to say that complex
K-theory gives more information than ordinary cohomology,
but this does not make any sense to me.
Take e.g. S^1 and S^3. Ordinary cohomology perfectly
distinguishes between these two manifolds (compute e.g.
H^3), while K-groups (rings) are the same:
K^i(S^1) = Z = K^i(S^3) for all i
One even can (in principle) calculate K-groups from ordinary
cohomology, so the statement should be the other way around.
Volker
John Baez
>In sci.physics.research John Baez <ba...@math.ucr.edu> wrote:
>> [...] In fact, complex cobordism theory is the "universal" complex
>> oriented cohomology theory - it's the most informative of the whole lot.
>> All the rest are like watered-down versions of this one. Ordinary
>> cohomology is the most watered-down of all. Complex K-theory is a bit
>> less watered-down. And elliptic cohomology is still less watered-down!
>I cannot follow you at all here - what is the mathematical
>definition of "watered down"? You seem to say that complex
>K-theory gives more information than ordinary cohomology,
>but this does not make any sense to me.
It wasn't a very precise or clear statement, but one thing I
meant was that there's a map from complex cobordism theory to
elliptic cohomology, and a map from elliptic cohomology to
complex K-theory, and finally a map from complex K-theory to
ordinary cohomology (the Chern character). All these maps
preserve the complex orientation, so corresponding to these we
get maps between their formal group laws. Complex cobordism
theory is special because it's it's the "initial" complex
oriented cohomology theory - there's a unique homomorphism from
it to any other one (where we require that the homomorphism
preserve the complex orientation). Its formal group law is
the initial formal group law, too.
People talking about elliptic cohomology often summarize this
with a chart where complex cobordism theory is on the top, then
a big gap, then elliptic cohomology, then complex K-theory, and
then ordinary cohomology at the bottom.
Often initial objects are "maximally informative". But I
guess in this case that's sort of misleading.
>Take e.g. S^1 and S^3. Ordinary cohomology perfectly
>distinguishes between these two manifolds (compute e.g.
>H^3), while K-groups (rings) are the same:
> K^i(S^1) = Z = K^i(S^3) for all i
That's true, and it's a good point! In some sense here
the problem is just that K-theory is "rolled up" - it's
a Z/2-graded theory instead of a Z-graded one. But you're
right, that makes it unable to distinguish certain spaces
that ordinary cohomology can.
>One even can (in principle) calculate K-groups from ordinary
>cohomology, so the statement should be the other way around.
Well, over the rationals the Chern character is an isomorphism
between complex K-theory and the "rolled-up" or Z/2-graded version
of ordinary cohomology. But if we don't tensor with the rationals,
the Chern character can fail to be an isomorphism.... and I thought
this lead to the possibility that the K-theory groups could distinguish
between certain spaces that the ordinary cohomology groups can't.
But I guess if it's not surjective, it can work the other way around.
I was going to look up a paper I noticed a while ago in order to
demonstrate that physicists are getting interested in this stuff, but
then I noticed that you wrote that paper! :-)
hep-th/0005103
K-theory Torsion
Author: Volker Braun
The Chern isomorphism determines the free part of the K-groups from
ordinary cohomology. Thus to really understand the implications of
K-theory for physics one must look at manifolds with K-torsion.
Unfortunately there are not many explicit examples, and usually
for very symmetric spaces. Cartesian products of RP^n are examples
where the order of the torsion part differs between K-theory and
ordinary cohomology. The dimension of corresponding branes is also
discussed. An example of a Calabi-Yau manifold with K-torsion is given.
So maybe you can educate me on the relation between the torsion parts
of K-theory and ordinary cohomology!