Symplectomorphisms and spherical objects in the conifold smoothing

Let $X$ denote the ‘conifold smoothing’, the symplectic Weinstein manifold which is the complement of a smooth conic in $T^*S^3$ or, equivalently, the plumbing of two copies of $T^*S^3$ along a Hopf link. Let $Y$ denote the ‘conifold resolution’, by which we mean the complement of a smooth divisor in $\mathcal {O}(-1) \oplus \mathcal {O}(-1) \to \mathbb {P}^1$ . We prove that the compactly supported symplectic mapping class group of $X$ splits off a copy of an infinite-rank free group, in particular is infinitely generated; and we classify spherical objects in the bounded derived category $D(Y)$ (the three-dimensional ‘affine $A_1$ -case’). Our results build on work of Chan, Pomerleano and Ueda and Toda, and both theorems make essential use of working on the ‘other side’ of the mirror.