\'{E}tale Homotopy Obstructions of Arithmetic Spheres

Let $K$ be a field of characteristic $\ne 2$ and let $X$ be the affine variety over $K$ defined by the equation $$X:\ a_0x_0^2 + \cdots + a_nx_n^2 = 1$$ where $n\ge 0$ and $a_i\in K$. In this paper we compute the lowest mod 2 \'{e}tale homological obstruction class to the existence of a $K$-rational point on $X$, and show that it is the cup product of the form $$o_{n+1} = [a_0]\cup\cdots\cup[a_n].$$ Our computation is an \'{e}tale-homotopy analogue of the topological fact that Stiefel-Whitney classes are the homological obstructions to find a section to the unit sphere bundle of a real vector bundle.