## 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.
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

Zbl 0799.57008
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