I'nm reading Orthogonal Calculus by Michael Weiss, and trying to understand example 2.7, concerning the derivatives of the functor $BO$, which sends a (finite dimensional) inner product space to the classifying space of its orthogonal group.
My question concerns the homotopy equivalence between the homotopy fiber of the map $$ BO(V) \to BO(V\oplus \Bbb{R}) $$ and $O(\Bbb{R} \oplus V)/ O(V)$, and the subsequent homotopy equivalence of this, and the one-point compactificiation of $V$.
By guess for the first one is to use the fact that $BO(V) = Gr(\dim V, \Bbb{R}^\infty)$, where $Gr$ denotes the Grassmannian manifold, and that there is fibre sequence $$ O(V) \to Vr(\dim V, \Bbb{R}^\infty) \to Gr(\dim V+1, \Bbb{R}^\infty) $$ where $Vr$ denotes the Steifel manifold (which is contractible).
Although, I'm not overly sure how this helps.
I'm clueless on the second on currently.
Any references or suggestions would be greatly appreciated.
I haven't looked at the paper in a while, but I think you want
$$F^{(1)}(V)\simeq hofib\left(BO(V)\rightarrow BO(V\oplus\mathbb{R})\right)$$
without the loop functor, rather than with it, since we want $m=0$ in Proposition 2.2. (Here I'm using Weiss's notation of $F(V)=BO(V)$ from the Example 2.7.)
Then the map in the definition of $F^{(1)}$ is just the map induced by the incusion $O(V)\hookrightarrow O(V\oplus\mathbb{R})$ so clearly
$$F^{(1)}(V)\simeq O(V\oplus\mathbb{R})/O(V).$$
Now by definition $O(V\oplus\mathbb{R})$ acts transitively on the unit sphere $S(V\oplus\mathbb{R})$ of length one vectors in $V\oplus\mathbb{R}$, and the stabiliser of the line $\mathbb{R}$ is exactly $O(V)$. Hence $O(V\oplus\mathbb{R})/O(V)\cong S(V\oplus\mathbb{R})$. Finally applying stereographic projection we get $S(V\oplus\mathbb{R})\cong S^V$ (see, for instance, the nlab page https://ncatlab.org/nlab/show/representation+sphere). Therefore
$$F^{(1)}(V)\simeq S^V$$
as claimed.