Spin structures, frame bundles, and trivializations over the 2-skeleton

Question

While reading an introduction to Spin- and Spin$^{\operatorname{c}}$ structures (found here), I encountered the following definition:

Let $E\to X$ be an oriented $\mathbb{R}^n$-bundle over a CW complex $X$, where $\operatorname{Fr}(E)$ denotes the oriented frame bundle (and we fix an inner product/metric on $E\to X$ to reduce $\operatorname{Fr}(E)$ from a $\operatorname{GL}(n)$-bundle to a $\operatorname{SO}(n)$-bundle) . We define a spin structure on $E \to X$ to be an element of $H^1(\operatorname{Fr}(E); \mathbb{Z}_2)$ that restricts to the generator of $H^1(\operatorname{SO}(n);\mathbb{Z}_2)$ on each fiber of $\operatorname{Fr}(E)$.

My question is about the very next line:

Another way to say this is that there is a trivialization of $E$ restricted to the two-skeleton of $X$.

Why are these two definitions equivalent?

Answer 1

I believe this is an application of obstruction theory applied to fibrations. I haven't checked the details, but the material is in Davis & Kirk's Lecture Notes in Algebraic Topology, available online in revised form.

You should look at section 7.10, more precisely theorem 7.37, and the discussion following it.

Here's the actual theorem (though the discussion in the book will also likely be useful):

I am not familiar with your particular bundles so I can't say anything as to why the hypotheses are fulfilled.

Answer 2

An oriented $n$-bundle is always trivializable over the 1-skeleton. (this is in fact equivalent to being orientable)

Now you can ask yourself when does such a trivialization extend over the 2-skeleton? To consider this we look at the trivial bundle over a 2-cell $e$ and the transitions maps on $\partial e \cong S^1$ which are coming from the trivialization on the 1-skeleton. Since the bundle is orientable we can obtain a map $S^1 \to SO(n)$ for this cell, i.e. an element in the fundamental group $\pi_1(SO(n))$.

Now note that an element in $H^1(Fr(E),\mathbb Z_2) = Hom (\pi_1(Fr(E),\mathbb Z_2)$ is just a homomorphism on the fundamental group. Respecting the restriction to the fiber means that this homomorphism measures the extendability over the 2-skeleton I described above. (because if we can restrict compatibly it means that we get actually information about the vector bundle, since it respects the choice of basis, which is precisely what we want to extend).