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 $3$? Are there any references or results?


**Note:** When $m=3$, $S^3$ is a Lie group hence $TS^3$ is a trivial bundle. Hence $\tau (S^3)$ is a trivial bundle and
$$
H^\*(\tau(S^3);\mathbb{Z})=H^\*(S^2;\mathbb{Z})\otimes H^\*(S^3;\mathbb{Z}).
$$



cohomology module of unit tangent vector bundles over spheres
If $m$ is even, $d\_m$ is multiplication by $2$: 


For every unit sphere bundle of an $m$-dimendional vector bundle over a space $X$, $d\_m$ sends $[S^{m-1}] \otimes 1$ to the euler class $e(x) \in H^m(X)$. In your case, we know that $$\langle e, [S^m]\rangle = \chi(S^m) = 1+(-1)^m$$ which is $2$ if $m$ is even. You can read off all the cohomology of $\tau(S^m)$ from that.


