The minimal genus problem in \(\mathbb{CP}^2\# \mathbb{CP}^2\) (Q2441255)

For a smooth, closed, oriented, simply connected 4-manifold \(X\), the minimal genus problem is to find the minimal genus of a surface \(\Sigma\) representing a given homology class in \(H_2(X; \mathbb{Z})\), where \(\Sigma\) ranges over closed, connected, oriented surfaces smoothly embedded in \(X\). This problem was solved for \(\mathbb{CP}^2\) by \textit{P. B. Kronheimer} and \textit{T. S. Mrowka} [Math. Res. Lett. 1, No. 6, 797--808 (1994; Zbl 0851.57023)], and for \(S^2 \times S^2\) and \(\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}\) by \textit{D. Ruberman} [Turk. J. Math. 20, No. 1, 129--133 (1996; Zbl 0870.57025)]. For other results on this problem, see the expository paper by \textit{T. Lawson} [Expo. Math. 15, No. 5, 385--431 (1997; Zbl 0894.57013)]. In this reference, Lawson gave the following conjecture: The minimal genus of \((m,n) \in H_2(\mathbb{CP}^2 \# \mathbb{CP}^2)=H_2(\mathbb{CP}^2) \oplus H_2(\mathbb{CP}^2)\) is given by \({m-1\choose 2} + {n-1\choose 2}\), and it is realized by the connected sum of the complex projective curves in each factor. In this paper, the author exhibits counterexamples and positive examples to this conjecture. The main results are as follows. There are two infinite families of counterexamples: Lawson's conjecture fails for the following two cases: (1) \((2p, d) \in H_2(\mathbb{CP}^2 \#\mathbb{CP}^2)\) for any \(p \geq 2\) and a possible degree \(d\) of the \((p, 4p-1)\)-torus knot in \(\mathbb{CP}^2\), and (2) \((m,0) \in H_2(\mathbb{CP}^2 \#\mathbb{CP}^2)\) for any \(m \geq 3\). There are two nontrivial positive examples: The minimal genera of \((3,3)\) and \((6,6) \in H_2(\mathbb{CP}^2 \# \mathbb{CP}^2)\) are respectively \(2\) and \(20\). The examples are based on twisting operations of knots, and the author treats the \(\mathbb{CP}^2\)-genera in the proof.