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

TL;DR

This paper computes the mapping class group of certain 6-manifolds formed as sphere bundles over the complex projective plane, including Milnor hypersurfaces and their generalizations, revealing their topological symmetries.

Contribution

It provides the first explicit computation of the mapping class groups for these specific sphere bundle manifolds over , extending understanding of their topological automorphisms.

Findings

Determined the structure of the mapping class group for these manifolds.

Identified the role of the first Pontryagin class in the classification.

Connected the results to known examples like Milnor hypersurfaces.

Abstract

In this article we compute the mapping class group of the total space $S(\xi)$ of the sphere bundle of a 3-dimensional real vector bundle $\xi$ over the complex projective plane $\mathbb{P}^2$ with $\langle p_1(\xi), [\mathbb{P}^2] \rangle =8n+5$. Examples of these manifolds include the Milnor hypersurface $M_1$ and its generalizations $M_k=\{(x,y)\in \mathbb{P}^2\times\mathbb{P}^2 \ | \ \sum x^k_iy_i=0\}$ with $k$ odd.