Are there simple examples of $n+1$D TQFT that assign 1-dimensional Hilbert spaces to both $n$-torus and $n$-sphere but higher dimensional Hilbert spaces to some other $n$-manifolds? Here I am assuming that the manifolds under consideration are all orientable.

For $n=2$, the requirement of 1-dimensional Hilbert spaces on both 2-torus and 2-sphere implies that the corresponding modular tensor category associated to the TQFT has only 1 simple objects. Such TQFT have 1-dimensional Hilbert spaces on all oriented 2-manifold. Therefore, examples I am looking for should not exist with $n=2$. For $n>2$, I cannot think of any reason why such example cannot exist.

Are there simple examples of $n+1$D TQFT that assign 1-dimensional Hilbert spaces to both $n$-torus and $n$-sphere but higher dimensional Hilbert spaces to some other $n$-manifolds? Here I am assuming that the manifolds under consideration are all orientable.

For $n=2$, the requirement of 1-dimensional Hilbert spaces on both 2-torus and 2-sphere implies that the corresponding modular tensor category associated to the TQFT has only 1 simple object. Such TQFT have 1-dimensional Hilbert spaces on all oriented 2-manifold. Therefore, examples I am looking for should not exist with $n=2$. For $n>2$, I cannot think of any reason why such example cannot exist.