Obstruction to reduction of structure group

Question

In the wiki article it's stated that the obstruction to reduction of structure group along a morphsim $H \to G$ can be stated in terms of classifying spaces via the cofibre $BG/BH$ as follows. A classifying map $X \to BG$ lifts (up to homotopy) to a classifying map $X \to BH$ if and only if the map $X \to BG \to BG/BH$ is nullhomotopic.

Why is this true? It seems like this is saying $BH \to BG \to BG/BH$ is a fibre sequence in addition to a cofibre sequence.

Answer 1

I am fairly confident that the Wikipedia article is wrong; as you say, there's no reason to expect a cofiber sequence to also be a fiber sequence in general. If others agree then we should remove that section of the Wikipedia article.

In general, you can describe a complete obstruction to reduction of the structure group along these lines whenever you can exhibit $BH$ as the homotopy fiber of a map from $BG$ to some auxiliary space. For example:

• $B \text{SO}(n) \to B \text{O}(n)$ is the homotopy fiber of the first Stiefel-Whitney class $w_1 : B \text{O}(n) \to B \mathbb{Z}_2$, so $w_1$ is a complete obstruction to a vector bundle having an orientation.
• $B \text{Spin}(n) \to B \text{SO}(n)$ is the homotopy fiber of the second Stiefel-Whitney class $w_2 : B \text{SO}(n) \to B^2 \mathbb{Z}_2$, so $w_2$ is a complete obstruction to an oriented vector bundle having a spin structure.

(I think that in neither of these cases is the auxiliary space also the homotopy cofiber. This is checkable at least in the first case using the long exact sequence in cohomology with coefficients in $\mathbb{Z}_2$.)

An obstruction to being able to do this is that the homotopy fiber of $BH \to BG$ itself must be a loop space (the loop space of the auxiliary space), and this just won't be the case in general.

If you don't get a nice homotopy fiber description then life is harder. First, a reduction of the structure group becomes equivalent to a section of an associated bundle (the homotopy pullback of the diagram $X \to BG \leftarrow BH$) with fiber the homotopy fiber of the map $BG \to BH$. Then you can run obstruction theory as usual on this bundle. (The nice case where $BG \to BH$ is itself a homotopy fiber corresponds to the case where this bundle is a principal bundle.)