Something I read somewhere about Spin manifolds, I don't remember where?

[Jim Kata]

What I'm about to say is really sketchy since I don't even remember where I read this, and don't really understand the topic, but I thought it was cool. Basically, it said on a spin manifold, a manifold where can do a lift from, for example an SO(3) twisted bundle to Spin(3), the weights of SO(3) take their in values in Z/2. That is nothing new, but it said on a regular manifold, that is, I think, one where SO(3) is not the tangent bundle, SO(3) weights would have values in Z/4.

Can anyone elaborate?

 

[Jim Kata]

I found what I read. It is not as interesting sounding as I remembered, but it says

On the other hand, for
SO(3), the instanton number takes values in Z/4, or in Z/2
if M is a spin manifold.

Can anyone explain this?