Lantern substitution and new symplectic 4-manifolds with ${b_{2}}^{+} = 3$

TL;DR

This paper constructs new symplectic 4-manifolds with specific Betti number properties by applying lantern substitutions to Lefschetz fibrations, demonstrating minimality and producing infinite families of non-diffeomorphic examples.

Contribution

It introduces a novel method of using lantern substitutions in Lefschetz fibrations to generate new symplectic 4-manifolds with controlled topological invariants.

Findings

Constructed symplectic 4-manifolds with ${b_{2}}^{+} = 3$

Computed Seiberg-Witten invariants showing minimality

Produced infinite families of non-diffeomorphic manifolds

Abstract

Motivated by the construction of H. Endo and Y. Gurtas, changing a positive relator in Dehn twist generators of the mapping class group by using lantern substitutions, we show that 4-manifold $K3#2\CPb$ equipped with the genus two Lefschetz fibration can be rationally blown down along six disjoint copies of the configuration $C_2$. We compute the Seiberg-Witten invariants of the resulting symplectic 4-manifold, and show that it is symplectically minimal. Using our example, we also construct an infinite family of pairwise non-diffeomorphic irreducible symplectic and non-symplectic 4-manifolds homeomorphic to $M = 3\CP# (19-k)\CPb$ for $1 \leq k \leq 4$.