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

Anar Akhmedov (School of Mathematics, University of Minnesota, Minneapolis, Minn., U.S.A.)

Jun-Yong Park (School of Mathematics, University of Minnesota, Minneapolis, Minn., U.S.A.)

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.

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