Manifold with infinitely many fibrations over the sphere

TL;DR

This work shows that the manifold $X= S^2\times S^3$ admits infinitely many fiber-bundle structures over the base $S^2$ with varying lens-space fibres $L(p,1)$. The authors construct a family of links $L_k$ as the links of affine cones over products built from Veronese and Segre embeddings, identifying each $L_k$ as a lens-space bundle over $S^2$ and applying Ehresmann's fibration theorem together with Giblin’s criterion to conclude the total space is diffeomorphic to $S^2\times S^3$. Consequently, for every $k\ge1$ there is a fibration $L(k,1)\to X\to S^2$, providing an infinite family of distinct fibrations. This connects algebraic-geometric constructions with 3-manifold topology and extends the phenomenon of infinite fibrations to even-dimensional base spheres.

Abstract

We show that the manifold $X=S^2\times S^3$ has infinitely many structures of a fiber bundle over the base $B=S^2.$ In fact for every lens space $L(p,1)$ there is a fibration $L(p,1)\to X\to B.$

Theorems & Definitions (10)

• Definition 2.1
• Definition 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• Remark 3.3
• Theorem 3.4
• proof
• Theorem 3.5
• proof