Structure of Torus Fibration Under the First Betti Number Restriction

TL;DR

This paper investigates the structure and classification of torus bundles with affine structures under first Betti number constraints, providing new rigidity results and conditions for topological splitting.

Contribution

It establishes a rigidity theorem for torus bundles with affine structures based on Betti number constraints and offers criteria for topological splitting, extending previous research.

Findings

Rigidity result for torus bundles with specific Betti number relations

Necessary and sufficient conditions for topological splitting of principal torus bundles

Enhanced understanding of collapsing sequences in geometric analysis

Abstract

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.