Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}\_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. How to compute the total Stiefel-Whitney class of the tangent bundle of $F(S^m,2)/\mathbb{Z}\_2$
$$
w(TF(S^m,2)/\mathbb{Z}\_2)?
$$


Is $F(S^m,2)/\mathbb{Z}\_2$ homeomorphic to $\mathbb{R}P^m\times \mathbb{R}^m$ or not?


Moreover, if we change $S^m$ to other manifolds, for example, projective spaces, Grassmannians, then how to compute the corresponding Stiefel-Whitney class? The question is on [characteristic classes of tangent bundle of 2-nd unordered configuration space](https://mathoverflow.net/questions/217408/characteristic-classes-of-tangent-bundle-of-2-nd-unordered-configuration-space)



Stiefel-Whitney class of unordered configuration space
The configuration space $F(S^m, 2)/\mathbb Z\_2$ is diffeomorphic to the total space of the $m$-dimensional vector bundle $E=\gamma^\perp$ over $\mathbb RP^m$, where $\gamma\subset \mathbb R^{m+1}$ is the tautological 1-dimensional line bundle and $\gamma^\perp$ is its orthogonal complement.


To define a diffeomorphism $\phi\colon Tot(E)\to F(S^m, 2)/\mathbb Z\_2$ let us pick a point $([v], w)\in Tot(E)$ (where $[v]\in\mathbb R P^m$ and $w\in v^\perp$),
such that a vector $v\in \mathbb R^{m+1}$ has the unit norm $||v||\_2=1$. 


Let $\phi \colon ([v],w)\mapsto\bigl(\frac{w+v}{||w+v||\_2}, \frac{w-v}{||w-v||\_2}\bigr)\in F(S^m, 2)/\mathbb Z\_2$. You can easily verify, that $\phi$ is diffeomorphism.


Computation of the total Stiefel-Whitney class is now straighforward. Let $\pi\colon Tot(E)\to \mathbb R P^m$ be the natural projection, then one has the long exact sequence:
$$
0\to \pi^\* E\to TTot(E)\to\pi^\*T\mathbb RP^m\to 0,
$$
hence 
$$
w(TF(S^m, 2)/\mathbb Z\_2)=w(\pi^\*E)w(\pi^\*T\mathbb RP^m)=\pi^\*(1+a)^{-1}\cdot\pi^\*(1+a)^{m+1}=(1+\pi^\*a)^m,$$ where $a\in H^1(\mathbb RP^m,\mathbb Z\_2)$ is the generator.


