Topologically flat embedded 2-spheres in specific simply connected 4-manifolds (Q2058916)

For a simply connected closed 4-manifold \(X\), any element in \(H_2(X; \mathbb{Z})\) can be represented by an embedded closed oriented surface. In the paper under review, the authors give specific examples, and consider whether specific elements can be represented by embedded spheres. Let \(M=8 \mathbb{CP}^2 \# \overline{\mathbb{CP}^2}\) and \(M'=8 \mathbb{CP}^2 \# \overline{\star \mathbb{CP}^2}\), where \(\star \mathbb{CP}^2\) is a fake \(\mathbb{CP}^2\), that is, a manifold that is homotopy equivalent to \(\mathbb{CP}^2\) but with non-trivial Kirby-Siebenmann invariant. The groups \(H_2(M; \mathbb{Z})\), \(H_2(M'; \mathbb{Z})\) and \(H_2(\mathbb{CP}^2 \# M; \mathbb{Z})\) are equipped with the evident bases, where the basis of \(H_2(M; \mathbb{Z})\) consists of 9 spheres \(e_1, \ldots, e_9\) such that \(e_1, \ldots, e_8\) have self-intersection \(1\), and \(e_9\) has self-intersection \(-1\). Let \(x=(1, \ldots, 1, 3) \in H_2(M; \mathbb{Z})\), \(x'= (1, \ldots, 1, 3) \in H_2(M'; \mathbb{Z})\) and \((0,x) \in H_2(\mathbb{CP}^2 \# M; \mathbb{Z})\). The authors give a remark that \(x\) cannot be represented by a smoothly embedded sphere. The main results are as follows. The element \(x\) cannot be represented by a topologically flat embedding \(S^2 \to M\). However, the element \(x'\) can be represented by a topologically flat embedding \(S^2 \to M'\). The element \((0,x)\) can be represented by a topologically flat embedding \(S^2 \to \mathbb{CP}^2 \# M\), but not by a smooth embedding. The authors prove the results twice, first using the Kirby-Siebenmann invariant and Freedman's classification of simply connected manifolds, and for the second time using the Kervaire-Milnor invariant. For the entire collection see [Zbl 1459.37002].