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# Mathematical Research Letters

## Volume 21 (2014)

### Number 1

### Lantern substitution and new symplectic 4-manifolds with $b_2{}^{+} = 3$

Pages: 1 – 17

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n1.a1

#### Authors

#### 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 $K3 \# 2 \overline{\mathbb{CP}}^2$ 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 $M = 3 \overline{\mathbb{CP}}^2 \# (19 − k)\overline{\mathbb{CP}}^2$ for $1 \leq k \leq 4$.

#### Keywords

4-manifold, mapping class group, Lefschetz fibration, lantern relation, rational blowdown

#### 2010 Mathematics Subject Classification

Primary 57R55. Secondary 57R17.