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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?

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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.

Probably this question is more suited to MSE than Mathoverflow.

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  • $\begingroup$ Then $\pi_0 (V_n(R^n)) = Z/2$, right? $\endgroup$
    – XT Chen
    Commented Apr 27, 2020 at 17:44
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    $\begingroup$ That's right; there are two components, distinguished by the sign of the determinant. $\endgroup$
    – Danny Ruberman
    Commented Apr 28, 2020 at 18:33

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