##
**Gauge theory for embedded surfaces. I.**
*(English)*
Zbl 0799.57007

Every 2-dimensional homology class of a closed oriented 4-manifold is represented by an embedded surface. In this paper, the authors give lower bounds on the genus of such a surface in terms of its self-intersection number. They work in a smooth simply-connected 4-manifold which has non- trivial polynomial invariants with \(b_ 2^ +\) odd and not less than 3.

As an application, the “local Thom conjecture” is solved. It states that if \(C\) is a smooth algebraic curve in complex 2-space and \(B^ 4\) is an embedded 4-ball whose boundary meets \(C\) transversely in a knot \(K\) then the surface \(C\cap B^ 4\) realizes the slice genus of \(K\). This result implies that the unknotting number of a knot which arises from singularities in algebraic curves equals the genus of the Milnor fibre.

As an application, the “local Thom conjecture” is solved. It states that if \(C\) is a smooth algebraic curve in complex 2-space and \(B^ 4\) is an embedded 4-ball whose boundary meets \(C\) transversely in a knot \(K\) then the surface \(C\cap B^ 4\) realizes the slice genus of \(K\). This result implies that the unknotting number of a knot which arises from singularities in algebraic curves equals the genus of the Milnor fibre.

Reviewer: P.Teichner (Mainz)

### MSC:

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57R57 | Applications of global analysis to structures on manifolds |

57R95 | Realizing cycles by submanifolds |