Let $S^m$ be the $m$-sphere and $\tau (S^m)$ the sphere bundle consisting of unit tangent vectors in the tangent bundle $TS^m$. Then we have a fibration
$$
S^{m-1}\longrightarrow \tau(S^m)\longrightarrow S^m.
$$
The Serre spectral sequence has $E_2$-page
$$
E_2^{p,q}=H^p(S^{m};\mathbb{Z})\otimes H^q(S^{m-1};\mathbb{Z})=\mathbb{Z}[x_{m}]/(x_m^2)\otimes \mathbb{Z}[x_{m-1}]/(x_{m-1}^2)
$$
where the dimensions of the generators are $|x_{m-1}|=m-1$, $|x_m|=m$, and converges to 
$$
H^*(\tau(S^m);\mathbb{Z}).
$$
My unknown part is the differential $$d_{m}: \mathbb{Z}x_{m-1}\longrightarrow \mathbb{Z}x_m.
$$

**Question:** Suppose $m$ is even. Is it possible that 
$$
H^{m-1}(\tau(S^m);\mathbb{Z})
$$
and 
$$
H^{m}(\tau(S^m);\mathbb{Z})
$$
are both torsion (do not have $\mathbb{Z}$-part) and both of their torsions are prime to $3$?