On the mapping class groups of $\mathbb{P}^1$-bundles over $\mathbb{P}^2$

TengLin Hu

geometric topology

On the mapping class groups of $P^{1}$ -bundles over $P^{2}$

arXiv:2512.18590

summary

The paper determines the mapping class group of the total space of a sphere bundle over the complex projective plane when the first Pontryagin number of the underlying 3‑dimensional real vector bundle equals 8n+5, covering examples such as the Milnor hypersurface and its odd‑k generalizations.

Abstract

In this article we compute the mapping class group of the total space $S (I^¾)$ of the sphere bundle of a 3-dimensional real vector bundle $I^¾$ over the complex projective plane $P^{2}$ with $⟨ p_{1} (I^¾), [P^{2}]⟩ = 8 n + 5$ . Examples of these manifolds include the Milnor hypersurface $M_{1}$ and its generalizations $M_{k} = {(x, y) \in P^{2} \times P^{2} ∣ \sum x_{i}^{k} y_{i} = 0}$ with $k$ odd.

Topics & keywords

#mapping class groups #sphere bundles #complex projective plane #Milnor hypersurface #characteristic classesmapping class groupsphere bundlereal vector bundlePontryagin classMilnor hypersurfaceCP2

On the mapping class groups of P1-bundles over P2

Abstract

Topics & keywords

On the mapping class groups of $P^{1}$ -bundles over $P^{2}$