# REMARKS ON HIGHER TYPE ADJUNCTION INEQUALITIES OF 4-MANIFOLDS OF NON-SIMPLE TYPE

• Published : 2002.08.01

#### Abstract

Recently P. Ozsv$\'{a}$th Z. Szab$\'{o}$ proved higher type adjunction inequalities for embedded surfaces in 4-manifolds of non-simple type. The aim of this short paper is to give a simple and direct proof of such higher type adjunction inequalities for smoothly embedded surfaces with negative self-intersection number in smooth 4-manifolds of non-simple type. This will be achieved through a relation between the Seiberg-Witten invariants used to get adjunction inequalities of 4-manifolds of simple type and a blow-up formula.

#### References

1. Turkish J. Math. v.19 Immersed shperes in 4-manifold and the immersed Thom conjecture R. Fintushel;R. Stern
2. Math. Res. Lett. v.1 The Genus of embedded surfaces in the projective plane P. Kronheimer;T. Mrowka https://doi.org/10.4310/MRL.1994.v1.n6.a14
3. J. Diff. Geo. v.44 A product formula for Seiberg-Witten invariants and the generalized Thom conjecture J. Morgan;Z. $Szab\'{o}$ https://doi.org/10.4310/jdg/1214459408
4. Ann. Math. v.151 The symjplectic Thjom conjecture P. $Ozsv\'{a}th$;Z. $Szab\'{o}$ https://doi.org/10.2307/121113
5. J. Diff. Geo. v.55 Higher type adjunction inequalities in Seiberg-Witten theory P. $Ozsv\'{a}th$;Z. $Szab\'{o}$ https://doi.org/10.4310/jdg/1090341259
6. Math. Res. Lett. v.1 Monopoles and four-manifolds E. Witten https://doi.org/10.4310/MRL.1994.v1.n6.a13