# Definition of 1st degree obstruction class

Recently I go through obstruction class illustrated by Milnor. He defined $$\mathfrak{o}_i$$by an element in $$H^i(M; \pi_{i-1}(V_{n-i+1}(F))$$, which is cohomology with local coefficients.

But the 0th homotopy group has no group structure, and the definition for $$\mathfrak{o}_1$$ doesn’t work. So does 1st degree obstruction class exist and if exists, how do you define it?

When $$i=1$$, the Stiefel manifold is $$V_n({\mathbb R})$$ which is the bundle of $$n$$-frames in an $$n$$-dimensional vector space. As such it is homeomorphic to $$GL(n,{\mathbb R})$$ and hence $$\pi_0$$ has a group structure. It has two components and it is easy to see that $${\mathfrak o}_1$$ measures orientability in the way that you might expect.
• Then $\pi_0 (V_n(R^n)) = Z/2$, right? Apr 27, 2020 at 17:44