Let us fix positive integers $k, m$. Let $A^k_{4m} \subset \Omega^{\text{SO}}_{4m} \otimes \mathbb Q$ be the subgroup generated by oriented cobordism classes of manifolds fibered over $S^k$.
The signature of any such manifold vanishes, i.e., we have
$$A^k_{4m} \subset \text{ker} \left( \Omega^{\text{SO}}_{4m} \otimes \mathbb Q \xrightarrow{\mathcal \sigma = L_m} \mathbb Q\right).$$
My question is whether, for each $k$, this inclusion is an isomorphism if $m$ is big enough.
In other words, I wish to understand if, for all $k$, the signature is the only rational cobordism obstruction for a manifold to fiber over a $k$-sphere in sufficiently high dimensions.
If $k \leq 4$, the answer is yes: this follows from a sequence of papers in the 60s/70s/80s by Burdick, Neumann, Alexander–Kahn, and Kahn, addressing the corresponding integral question for fixed values of $k$.
