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A: There are a number of ways to write it down. The key ingredient is the vector cross product map $\times:\mathbb{R}^n\times\mathbb{R}^n\to \mathbb{R}^n$ and the identity map $id:S^n\to S^n$. In your case it is $x\times y = y-x$. It turns out there is a formula for the two spaces in terms of the sphere. Namely: $\mathbb{R}^{2n+2}$ is the space of vectors $x=(x_1,x_2,\ldots,x_{2n+2})$, and $S^n$ is the space of vectors $x=(x_1,x_2,\ldots,x_{2n+1})$ such that $x_1^2+\ldots+x_{2n+1}^2=1$. The two spaces are related by the map $S^{2n+1}\to \mathbb{R}^{2n+2}$ given by $x\mapsto x\times x$. It is an exercise to show that this map is a trivial fiber bundle, and so gives a fibration $S^{2n+1}\to S^{2n+1}\times S^1 \to S^{2n+1}$ (note: $S^1$ has one point). EDIT: In fact, there What's New in the? Markup Export and Support: Make exporting to PDF and ECR format a seamless part of the drawing process. 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