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Embedded surfaces in four-manifolds, branched covers, and SO(3)-invariants

Published online by Cambridge University Press:  24 October 2008

D. Kotschick  and
G. Matić
D. Kotschick
Affiliation:
Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland
G. Matić
Affiliation:
Department of Mathematics, The University of Georgia, Athens, GA 30602, U.S.A.
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Extract

One of the outstanding problems in four-dimensional topology is to find the minimal genus of an oriented smoothly embedded surface representing a given homology class in a smooth four-manifold. For an arbitrary homology class in an arbitrary smooth manifold not even a conjectural lower bound is known. However, for the classes represented by smooth algebraic curves in (simply connected) algebraic surfaces, it is possible that the genus of the algebraic curve, given by the adjunction formula

is the minimal genus. This is usually called the (generalized) Thom conjecture. It is mentioned in Kirby's problem list [11] as Problem 4·36.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 2 , March 1995 , pp. 275 - 286
Copyright
Copyright © Cambridge Philosophical Society 1995

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