A fundamental problem in four-dimensional topology is to find the minimal genus of smoothly embedded surfaces in a four-manifold, in a given homology class. For example, the Thom conjecture implies the minimal genus of surfaces in the 2-dimensional complex projective plane are realized as smooth algebraic curves.

There is a relative version of the minimal genus problem by allowing boundaries of a 4-manifold and surfaces. Such a minimal genus is called the relative genus. When we consider $D^4$ as a 4-manifold with boundary, the minimal relative genus problems for torus knots were called Milnor conjecture. Both of the Thom conjecture and Milnor conjecture were solved by using gauge theory due to Kronheimer-Mrowka.

We provide a new lower bound of the relative genus obtained as a “real” variant of 10/8-inequality. The real 10/8-inequality will be proven by observing “the real part” of Manolescu’s Seiberg-Witten Floer homotopy type for branched covers. This work is joint work with Hokuto Konno and Jin Miyazawa.