We can define the signature of a manifold in $4k$ dimensions.
1) If I understand correctly, the signature $\sigma$ of the manifold of the product space of spheres would always be zero:
$$\sigma(S^n \times S^m \times S^p \times S^q \times \dots )=0$$
2) Am I correct that the signature of the manifold of the (multiple) fibrations over spheres will always be zero?
$$S^n \hookrightarrow X_1 \rightarrow S^m$$
$$X_1 \hookrightarrow X_2 \rightarrow S^p$$
$$X_2 \hookrightarrow X_3 \rightarrow S^o$$
$$\sigma(X_j)=0?$$ (for $j=1,2,3,...$, $m,n,o$ are positive integers.)
If I am wrong, is this statement true in 4-dimensions?
Or, can one give one or two simplest counterexamples in 4-dimensions? (or other dimensions?)
See also this post.