Structure of Torus Fibration Under the First Betti Number Restriction

We study torus bundles with affine structure groups. First, we establish a rigidity result under constraints on the first Betti numbers: If ${\text{b}}_1(M)-{\text{b}}_1(N)=\dim M-\dim N$ holds for a torus bundle $M$ with an affine structure group over a closed manifold $N$, then $M$ can be classified. Second, we obtain some necessary and sufficient conditions for the topological splitting of principal torus bundles. These results improve the understanding of the geometry of collapsing sequences under the first Betti number constraints, thereby extending the prior work by Huang-Wang.